Recent zbMATH articles in MSC 33https://zbmath.org/atom/cc/332024-08-14T19:23:59.529552ZWerkzeugDerivation of computational formulas for certain class of finite sums: approach to generating functions arising from \(p\)-adic integrals and special functionshttps://zbmath.org/1538.050102024-08-14T19:23:59.529552Z"Simsek, Yilmaz"https://zbmath.org/authors/?q=ai:simsek.yilmazSummary: The aim of this paper is to construct generating functions for certain families of special finite sums by using the Newton-Mercator series, hypergeometric functions, and \(p\)-adic integral. By using these generating functions with their functional and partial derivative equations, many novel computational formulas involving the special finite sums of (inverse) binomial coefficients, the Bernoulli type polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the (alternating) harmonic numbers, the Leibnitz polynomials, and others are derived. We also develop a computation algorithm for these finite sums and provide some of their special values. By using these finite sums and combinatorial numbers, we find some formulas involving multiple alternating zeta functions, the Bernoulli polynomials of higher order and the Euler polynomials of higher order. We then obtain a decomposition from these formulas, which are related to the multiple Hurwitz zeta functions.
{{\copyright} 2022 John Wiley \& Sons, Ltd.}Trigonometric identities and quadratic residueshttps://zbmath.org/1538.050212024-08-14T19:23:59.529552Z"Sun, Zhi-Wei"https://zbmath.org/authors/?q=ai:sun.zhi-weiThe paper proves the following trigonometric identities: for any \(n\) odd positive integer, and complex numbers \(x\), \(y\), \[\sum_{r=0}^{n-1} \frac{1}{1+\sin(2\pi\frac{x+r}{n})+\cos(2\pi\frac{x+r}{n})}=\frac{(-1)^{\frac{n-1}{2}}}{1+(-1)^{\frac{n-1}{2}}\sin 2\pi x +\cos 2\pi x},\] \[\sum_{j,k=0}^{n-1} \frac{1}{1+\sin(2\pi\frac{x+j}{n})+\cos(2\pi\frac{y+k}{n})}=\frac{(-1)^{\frac{n-1}{2}}n^2}{\sin 2\pi x +\cos 2\pi y}\] hold with a countably many exceptions. Furthermore, the values of \[\prod_{k=1}^{\frac{p-1}{2}}(1+\tan \pi\frac{k^2}{p}) \text{ and } \prod_{k=1}^{\frac{p-1}{2}}(1+\cot \pi\frac{k^2}{p})\] are determined for any odd prime \(p\). These results are closely connected to certain sums of quadratic residues. Several conjectures are formulated for the value of \(\prod_{k=1}^{\frac{p-1}{2}}(x-e^{2\pi i k^2/p})\) for odd prime \(p\) and a root of unity \(x\).
Reviewer: László A. Székely (Columbia)New \(q\)-analogues of a congruence of Sun and Taurasohttps://zbmath.org/1538.110042024-08-14T19:23:59.529552Z"Guo, Victor J. W."https://zbmath.org/authors/?q=ai:guo.victor-j-wIn this note the author proved two \(q\)-analogues of a congruence of \textit{Z.-W. Sun} and \textit{R. Tauraso} [Adv. Appl. Math. 45, No. 1, 125--148 (2010; Zbl 1231.11021)]. Further, their congruence has been generalized and another two \(q\)-analogues have been proposed as open problems.
Reviewer: C. L. Parihar (Indore)Three families of \(q\)-supercongruences modulo the square and cube of a cyclotomic polynomialhttps://zbmath.org/1538.110052024-08-14T19:23:59.529552Z"Guo, Victor J. W."https://zbmath.org/authors/?q=ai:guo.victor-j-w"Schlosser, Michael J."https://zbmath.org/authors/?q=ai:schlosser.michael-jSummary: In this paper, three parametric \(q\)-supercongruences for truncated very-well-poised basic hypergeometric series are proved, one of them modulo the square, the other two modulo the cube of a cyclotomic polynomial. The main ingredients of proof include a basic hypergeometric summation by George Gasper, the method of creative microscoping (a method recently introduced by the first author in collaboration with Wadim Zudilin), and the Chinese remainder theorem for coprime polynomials.A \(q\)-congruence for a truncated \(_{4}\varphi_{3}\) series.https://zbmath.org/1538.110062024-08-14T19:23:59.529552Z"Guo, Victor J. W."https://zbmath.org/authors/?q=ai:guo.victor-j-w"Wei, Chuanan"https://zbmath.org/authors/?q=ai:wei.chuananSummary: Let \(\Phi_n(q)\) denote the \(n\)th cyclotomic polynomial in \(q\). Recently, \textit{V. J. W. Guo}, \textit{M. J. Schlosser}, and \textit{W. Zudilin} [``New quadratic identities for basic hypergeometric series and \(q\)-congruences'', Preprint, \url{http://math.ecnu.edu.cn/~jwguo/maths/quad.pdf}] proved that for any integer \(n>1\) with \(n\equiv 1\pmod{4}\), \[\sum_{k=0}^{n-1}\frac{(q^{-1};q^2)_k^2(q^{-2};q^4)_k}{(q^2;q^2)_k^2(q^4;q^4)_k}q^{6k}\equiv 0\pmod{\Phi_n(q)^2},\] where \((a;q)_m=(1-a)(1-aq)\cdots(1-aq^{m-1})\). In this note, we give a generalization of the above \(q\)-congruence to the modulus \(\Phi_n(q)^3\) case. Meanwhile, we give a corresponding \(q\)-congruence modulo \(\Phi_n(q)^2\) for \(n\equiv 3\pmod{4}\). Our proof is based on the `creative microscoping' method, recently developed by \textit{V. J. W. Guo} and \textit{W. Zudilin} [Adv. Math. 346, 329--358 (2019; Zbl 1464.11028)], and a \(_4\varphi_3\) summation formula.Proof of some supercongruences concerning truncated hypergeometric serieshttps://zbmath.org/1538.110092024-08-14T19:23:59.529552Z"Wang, Chen"https://zbmath.org/authors/?q=ai:wang.chen.4|wang.chen|wang.chen.1"Hu, Dian-Wang"https://zbmath.org/authors/?q=ai:hu.dianwangSummary: In this paper, we prove some supercongruences concerning truncated hypergeometric series. For example, we show that for any prime \(p > 3\) and positive integer \(r\),
\[
\sum_{k=0}^{p^r-1}(3k+1)\frac{(\frac{1}{2})_k^3}{(1)_k^3}4^k\equiv p^r+\frac{7}{6}p^{r+3}B_{p-3}\pmod{p^{r+4}}
\]
and
\[
\sum_{k=0}^{(p^r-1)/2}(4k+1)\frac{(\frac{1}{2})_k^4}{(1)_k^4}\equiv p^r+\frac{7}{6}p^{r+3}B_{p-3}\pmod{p^{r+4}},
\]
where \((x)_k = x(x+1)\cdots(x+k-1)\) is the Pochhammer symbol and \(B_0, B_1, B_2, \dots\) are Bernoulli numbers. These two congruences confirm conjectures of \textit{Z. Sun} [Sci. China, Math. 54, No. 12, 2509--2535 (2011; Zbl 1256.11011)] and \textit{V. J. W. Guo} [Adv. Appl. Math. 120, Article ID 102078, 16 p. (2020; Zbl 1456.11024)], respectively.Some generalizations of a congruence by Sun and Taurasohttps://zbmath.org/1538.110102024-08-14T19:23:59.529552Z"Wang, Xiaoxia"https://zbmath.org/authors/?q=ai:wang.xiaoxia"Yu, Menglin Yu"https://zbmath.org/authors/?q=ai:yu.menglin-yuLet \((a;q)_n\) be the \(q\)-shifted factorial and \(\Phi_n(q)\) be the \(n\)-th cyclotomic polynomial. In this paper, the authors generalize the result of \textit{C.-Y. Gu} and \textit{V. J. W. Guo} [Period. Math. Hung. 82, No. 1, 82--86 (2021; Zbl 1499.11006)] and as well as the result of \textit{V. J. W. Guo} and \textit{J. Zeng} [Adv. Appl. Math. 45, No. 3, 303--316 (2010; Zbl 1231.11020)] on the combinatorial \(q\)-congruence. In particular, it has been proven \[\sum\limits_{k=0}^{n-1}\frac{(q^{2d+1};q^2)_k}{(q;q)_k}\,q^k\equiv(-1)^{\frac{n-1}2+d}q^{\frac{n^2-(2d+1)^2}4}\pmod{Phi_n(q)}\] where \(d\) is an integer and \(n\) is a positive odd integer with \(n>2\vert d\vert -1\). The above mentioned results of Gu and Guo (loc. cit.), Guo and Zeng (loc. cit.) are just the special cases by \(d=0,-1,1\).
Reviewer: Konstantin Malyutin (Kursk)\(d\)-Gaussian Jacobsthal, \(d\)-Gaussian Jacobsthal-Lucas polynomials and their matrix representationshttps://zbmath.org/1538.110492024-08-14T19:23:59.529552Z"Özkan, E."https://zbmath.org/authors/?q=ai:ozkan.erdogan-mehmet|ozkan.emre|ozkan.erhun|ozkan.engin"Uysal, M."https://zbmath.org/authors/?q=ai:uysal.mutlu|uysal.mesude-elif|uysal.murat|uysal.mitat|uysal.mehmet|uysal.mineSummary: In this paper, we define \(d\)-Gaussian Jacobsthal polynomials and \(d\)-Gaussian Jacobsthal-Lucas polynomials. We present the sum, generating functions and Binet formulas of these polynomials. We give the matrix representations of them. We present these matrices as binary representation according to the Riordan group matrix representation. By using Riordan method, we give factorizations of Pascal matrix involving \(d\)-Gaussian Jacobsthal polynomials and \(d\)-Gaussian Jacobsthal-Lucas polynomials. We give the inverse of matrices of these polynomials.Truncations of Gauss' square exponent theorem.https://zbmath.org/1538.110592024-08-14T19:23:59.529552Z"Liu, Ji-Cai"https://zbmath.org/authors/?q=ai:liu.jicai|liu.jicai.1"Zhao, Shan-Shan"https://zbmath.org/authors/?q=ai:zhao.shanshanSummary: We establish two truncations of Gauss' square exponent theorem and a finite extension of Euler's identity. For instance, we prove that for any positive integer \(n\), \[\sum_{k=0}^n(-1)^k\left [\begin{matrix}2n-k\\ k\end{matrix}\right ](q;q^2)_{n-k}q^{\binom{k+1}{2}}=\sum_{k=-n}^n(-1)^kq^{k^2},\] where \[\left [\begin{matrix} n\\ m\end{matrix}\right ]=\prod_{k=1}^m\frac{1-q^{n-k+1}}{1-q^k}\quad\text{and}\quad(a;q)_n=\prod_{k=0}^{n-1}(1-aq^k).\]Novel results for generalized Apostol type polynomials associated with Hermite polynomialshttps://zbmath.org/1538.110642024-08-14T19:23:59.529552Z"Khan, Waseem Ahmad"https://zbmath.org/authors/?q=ai:khan.waseem-ahmad"Nisar, Kottakkaran Sooppy"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaran"Duran, Ugur"https://zbmath.org/authors/?q=ai:duran.ugur"Acikgoz, Mehmet"https://zbmath.org/authors/?q=ai:acikgoz.mehmetSummary: In this paper, the authors introduce a new class of Hermite-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials. The authors then derive some basic properties and several implicit summation formulae by utilizing the series manipulation methods. The authors also investigate several symmetric identities, which are extensions of many earlier well-known results. Moreover, the authors consider a novel class of Hermite-based generalized Apostol-Bernoulli, Apostol-Euler, and Apustol-Genocchi polynomials including geometric and Bell polynomials, and give some basic properties.Some new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomialshttps://zbmath.org/1538.110672024-08-14T19:23:59.529552Z"Ramírez, W."https://zbmath.org/authors/?q=ai:ramirez.william"Cesarano, C."https://zbmath.org/authors/?q=ai:cesarano.clementeSummary: The aim of this paper is to study new classes of degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order \(\alpha\) and level \(m\) in the variable \(x\). Here the degenerate polynomials are a natural extension of the classic polynomials. In more detail, we derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. Most of the results are proved by using generating function methods.Generalized mixed type Bernoulli-Gegenbauer polynomialshttps://zbmath.org/1538.110792024-08-14T19:23:59.529552Z"Quintana, Yamilet"https://zbmath.org/authors/?q=ai:quintana.yamiletSummary: The generalized mixed type Bernoulli-Gegenbauer polynomials of order \(\alpha>-\frac{1}{2}\) are special polynomials obtained by use of the generating function method. These polynomials represent an interesting mixture between two classes of special functions, namely generalized Bernoulli polynomials and Gegenbauer polynomials. The main purpose of this paper is to discuss some of their algebraic and analytic properties.Bell polynomials and 2nd kind hypergeometric Bernoulli numbershttps://zbmath.org/1538.110802024-08-14T19:23:59.529552Z"Ricci, P. E."https://zbmath.org/authors/?q=ai:ricci.paolo-emilio"Natalini, P."https://zbmath.org/authors/?q=ai:natalini.pierpaoloSummary: After showing a recursive computation of the 2nd kind hypergeometric Bernoulli numbers, we exploit the Blissard problem to derive a connection of these Bernoulli-type numbers with the Bell polynomials.\(P\)-\(Q\) ``mixed'' modular equations of degree 15https://zbmath.org/1538.111282024-08-14T19:23:59.529552Z"Chandankumar, S."https://zbmath.org/authors/?q=ai:chandankumar.sathyanarayana"Hemanthkumar, B."https://zbmath.org/authors/?q=ai:hemanthkumar.bSummary: Ramanujan in his second notebook recorded total of seven \(P\)-\(Q\) modular equations involving theta-function \(f(-q)\) with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving his theta-functions \(\varphi(q)\) and \(\psi(-q)\) with moduli of orders 1, 3, 5 and 15. As a consequence, several values of quotients of theta-function and a continued fraction of order 12 are explicitly evaluated.Certain summation formulae and relations due to double series associated with the general hypergeometric type Hurwitz-Lerch zeta functionshttps://zbmath.org/1538.111512024-08-14T19:23:59.529552Z"Chandel, R. C. Singh"https://zbmath.org/authors/?q=ai:chandel.r-c-singh"Kumar, Hemant"https://zbmath.org/authors/?q=ai:kumar.hemantSummary: In this paper, we exhibit certain double series associated with general hypergeometric type Hurwitz-Lerch Zeta functions and then derive their summation formulae and relations due to their series and
integral identities. We also obtain various known and unknown results in terms of Hurwitz-Lerch Zeta functions and their generating relations.Identities of a general multiple Hurwitz-Lerch zeta function and applicationshttps://zbmath.org/1538.111522024-08-14T19:23:59.529552Z"Chandel, R. C. Singh"https://zbmath.org/authors/?q=ai:chandel.r-c-singh"Pathan, M. A."https://zbmath.org/authors/?q=ai:pathan.mahmood-ahmad"Kumar, Hemant"https://zbmath.org/authors/?q=ai:kumar.hemantSummary: In this article we introduce a general multiple Hurwitz-Lerch zeta function. Then its convergence conditions and identities are obtained under certain conditions. We also derive some of connections to the multiple Hurwitz-Lerch zeta function based upon Srivastava-Daoust hypergeometric series in several variables and other related functions of one and more variables found in the literature. Further, we study its integral representations and find their applications for deriving generating relations and solving the non-homogeneous fractional differential equation.Certain results of generalized Barnes type double series related to the Hurwitz-Lerch zeta functions of two variableshttps://zbmath.org/1538.111532024-08-14T19:23:59.529552Z"Kumar, Hemant"https://zbmath.org/authors/?q=ai:kumar.hemantSummary: In these researches we introduce a generalized Barnes type double series and then, discuss its convergent conditions. We obtain some of its results related to the known and new Hurwitz-Lerch zeta function of two variables and also derive Eulerian and Mellin-Barnes type integral representations of these functions and analyze various properties these functions.Relations and identities due to double series associated with general Hurwitz-Lerch type zeta functionshttps://zbmath.org/1538.111542024-08-14T19:23:59.529552Z"Kumar, Hemant"https://zbmath.org/authors/?q=ai:kumar.hemant"Chandel, R. C. Singh"https://zbmath.org/authors/?q=ai:chandel.r-c-singhSummary: In this paper, we introduce certain families of double series associated with general Hurwitz-Lerch type Zeta functions and then derive their summation formulae, series and integral identities. Again then using these identities, we obtain various known and unknown results and hypergeometric generating
relations.Some new identities of Rogers-Ramanujan typehttps://zbmath.org/1538.111772024-08-14T19:23:59.529552Z"Gu, Jing"https://zbmath.org/authors/?q=ai:gu.jing"Zhang, Zhizheng"https://zbmath.org/authors/?q=ai:zhang.zhizhengSummary: In this paper, we establish two transformation formulas for nonterminating basic hypergeometric series by using Carlitz's inversions formulas and Jackson's transformation formula. In terms of application, by specializing certain parameters in the two transformations, four Rogers-Ramanujan type identities associated with moduli 20 are obtained.Hypergeometric representations of Gelfond's constant and its generalisationshttps://zbmath.org/1538.111922024-08-14T19:23:59.529552Z"Rathie, Arjun K."https://zbmath.org/authors/?q=ai:rathie.arjun-kumar"Milovanović, Gradimir V."https://zbmath.org/authors/?q=ai:milovanovic.gradimir-v"Paris, Richard B."https://zbmath.org/authors/?q=ai:paris.richard-bruceSummary: The aim of this note is to provide a natural extension and generalisation of the well-known Gelfond constant \(e^\pi\) using a hypergeometric function approach. An extension is also found for the square root of this constant. Several known mathematical constants are also deduced in hypergeometric form from our newly introduced constant.On the generalized trapezoid and midpoint type inequalities involving Euler's beta functionhttps://zbmath.org/1538.260042024-08-14T19:23:59.529552Z"Sarikaya, Mehmet Zeki"https://zbmath.org/authors/?q=ai:sarikaya.mehmet-zeki"Kozan, Gizem"https://zbmath.org/authors/?q=ai:kozan.gizemSummary: The main object of this paper is to present some generalizations of fractional integral inequalities involving Euler's beta function of Hermite-Hadamard type which cover the previously published result such as Riemann integral, Riemann-Liouville fractional integral, \(k\)-Riemann-Liouville fractional integral.Fractional integral operators in relationship with \(R\)-functionhttps://zbmath.org/1538.260162024-08-14T19:23:59.529552Z"Malik, Naseer Ahmad"https://zbmath.org/authors/?q=ai:malik.naseer-ahmad"Ahmad, Farooq"https://zbmath.org/authors/?q=ai:ahmad.farooq"Jain, D. K."https://zbmath.org/authors/?q=ai:jain.deepak-kumarSummary: The aim of the present paper is to establish certain new results of the \(R\)-Function in relationship with the Marichev-Saigo Maeda fractional integral operator. The results are presented in terms of the Wright type function. Some special cases of the main results are also pointed out in the last section of the paper.Fractional integral operators in relationship with \(R\)-functionhttps://zbmath.org/1538.260172024-08-14T19:23:59.529552Z"Malik, Naseer Ahmad"https://zbmath.org/authors/?q=ai:malik.naseer-ahmad"Ahmad, Farooq"https://zbmath.org/authors/?q=ai:ahmad.farooq"Jain, D. K."https://zbmath.org/authors/?q=ai:jain.deepak-kumarSummary: The aim of the present paper is to establish certain new results of the \(R\)-function in relationship with the Marichev-Saigo Maeda fractional integral operator. The results are presented in terms of the Wright type function. Some special cases of the main results are also pointed out in the last section of the paper.Fractional calculus of product of \(M\)-series and \(I\)-function of two variableshttps://zbmath.org/1538.260192024-08-14T19:23:59.529552Z"Sachan, Dheerandra Shanker"https://zbmath.org/authors/?q=ai:sachan.dheerandra-shanker"Jalori, Harsha"https://zbmath.org/authors/?q=ai:jalori.harsha"Jaloree, Shailesh"https://zbmath.org/authors/?q=ai:jaloree.shaileshSummary: The object of this paper is to develop the generalized fractional calculus formulas for the product of generalized \(M\)-series and \(I\)-function of two variables which is based on generalized fractional integration and differentiation operators of arbitrary complex order involving Appell hypergeometric function \(F_3\) as a kernel due to Saigo and Maeda. On account of general nature of the Saigo-Maeda operators, a large number of results involving Saigo and Riemann-Liouville operetors are found as corollaries. Again due to general nature of \(I\)-function of two variables and \(M\)-series, some special cases also have been considered.A particular family of absolutely monotone functions and relations to branching processeshttps://zbmath.org/1538.260252024-08-14T19:23:59.529552Z"Möhle, M."https://zbmath.org/authors/?q=ai:mohle.martinA function \(f: [0,1)\to\mathbb{R}\) is \textit{absolutely monotone} if \(f^{(n)}(z)\ge 0\) for all \(n\ge 0\) and \(z\in [0,1)\). Clearly, if \(f\) is absolutely monotone, then all its derivatives are absolutely monotone, too. It is known that the following functions \(f: [0,1)\to \mathbb{R}\) are absolutely monotone:\par\(f(z)=-\ln(1-z)\);\par \(f(z)=\frac{1}{1-z}\);\par \(f(0)=0\) and \(f(z)=1+\frac{z}{\ln(1-z)}\) for \(z\in (0,1)\).\par In the paper under review, the author completes this list by showing that the map \[f(z)=\ln\left( 1-\frac{\ln(1-z)}{c}\right)\] is absolutely monotone if and only if \(c\ge 1\). Consequently, all derivatives of \(f\) are absolutely monotone if \(c\ge 1\). The motivation for studying such functions comes from the theory of continuous-time branching processes.
Reviewer: Tomasz Natkaniec (Gdańsk)Two monotonic functions defined by two derivatives of a function involving trigamma functionhttps://zbmath.org/1538.260262024-08-14T19:23:59.529552Z"Qi, Feng"https://zbmath.org/authors/?q=ai:qi.fengSummary: In the paper, by virtue of the convolution theorem for the Laplace transforms, with the help of monotonicity and logarithmic concavity of a function involving exponential function, and by means of analytic techniques, the author presents necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic or monotonic.A generalized Fejér-Hadamard inequality for harmonically convex functions via generalized fractional integral operator and related resultshttps://zbmath.org/1538.260912024-08-14T19:23:59.529552Z"Kang, Shin Min"https://zbmath.org/authors/?q=ai:kang.shin-min"Abbas, Ghulam"https://zbmath.org/authors/?q=ai:abbas.ghulam"Farid, Ghulam"https://zbmath.org/authors/?q=ai:farid.ghulam"Nazeer, Waqas"https://zbmath.org/authors/?q=ai:nazeer.waqas(no abstract)The Dirichlet problem for rectangle and new identities for elliptic integrals and functionshttps://zbmath.org/1538.300182024-08-14T19:23:59.529552Z"Alekseeva, Elena Sergeevna"https://zbmath.org/authors/?q=ai:alekseeva.elena-sergeevna"Rassadin, Aleksandr Èduardovich"https://zbmath.org/authors/?q=ai:rassadin.aleksandr-eduardovichSummary: In the paper, results of comparison of two different methods of exact solution of the Dirichlet problem for rectangle are presented, namely, method of conformal mapping and method of variables' separation. By means of this procedure normal derivative of Green's function for rectangular domain was expressed via Jacobian elliptic functions. Under approaching to rectangle's boundaries these formulas give new representations of the Dirac delta function. Moreover in the framework of suggested ideology a number of identities for the complete elliptic integral of the first kind were obtained. These formulas may be applied to summation of both numerical and functional series; also they may be useful for analytic number theory.Some geometric properties of certain families of \(q\)-Bessel functionshttps://zbmath.org/1538.300222024-08-14T19:23:59.529552Z"Aktaş, İbrahim"https://zbmath.org/authors/?q=ai:aktas.ibrahim"Din, Muhey U"https://zbmath.org/authors/?q=ai:din.muhey-uSummary: In this paper, we are mainly interested in finding sufficient conditions for the \(q\)-close-to-convexity of certain families of \(q\)-Bessel functions with respect to certain functions in the open unit disk. The strong convexity and strong starlikeness of the same functions are also the part of our investigation.Unified approach to univalency of the Dziok-Srivastava and fractional calculus operatorshttps://zbmath.org/1538.300472024-08-14T19:23:59.529552Z"Kiryakova, Virginia"https://zbmath.org/authors/?q=ai:kiryakova.virginia-sSummary: In the Geometric Function Theory (GFT) much attention is paid to various linear integral operators mapping the class \(S\) of the univalent functions and its subclasses into themselves. In \textit{Yu. E. Khokhlov}, Ukr. Math. J. 37, 188--192 (1985; Zbl 0589.30021); translation from Ukr. mat. Zh. 37, No. 2, 220--226 (1985)], \textit{Y. E. Hohlov} [PLISKA, Stud. Math. Bulg. 10, 87--92 (1989; Zbl 0829.30007)] Hohlov obtained sufficient conditions that guarantee such mappings for the operator defined by means of Hadamard product with the Gauss hypergeometric function. In our earlier papers as \textit{V. Kiryakova} et al. [Fract. Calc. Appl. Anal. 1, No. 1, 79--104 (1998; Zbl 0951.30012)],
\textit{V. Kiryakova} and \textit{M. Saigo} [C. R. Acad. Bulg. Sci. 58, No. 10, 1127--1134 (2005; Zbl 1088.30006)], \textit{V. Kiryakova} [Fract. Calc. Appl. Anal. 9, No. 2, 159--176 (2006; Zbl 1138.30007)], [\textit{V. Kiryakova}, in: GFTA'2010 proceedings volume. Proceedings of the international symposium on geometric function theory and applications, Sofia, Bulgaria, August 27--31, 2010. Dedicated to the 70th anniversary of H. M. Srivastava. Sofia: Bulgarian Academy of Sciences, Institute of Mathematics and Informatics. 29--40 (2010; Zbl 1218.30040)], etc., we extended his method to the operators of the Generalized Fractional Calculus (GFC, \textit{V. Kiryakova} [Generalized fractional calculus and applications. Harlow: Longman Scientific \& Technical; New York: John Wiley \& Sons (1994; Zbl 0882.26003)]). These operators have product functions of the forms \(_{m+1}F_m\) and \(_{m+1}\Psi_m\) and integral representations by means of the Meijer \(G\)- and Fox \(H\)-functions. \par It happens that the used techniques can be extended to propose sufficient conditions that guarantee mapping of the univalent, respectively of the convex functions, into univalent functions in the case of the more general Dziok-Srivastava operator from \textit{J. Dzio} and \textit{H. M. Srivastava} [Appl. Math. Comput. 103, No. 1, 1--13 (1999)], defined as a Hadamard product with an arbitrary generalized hypergeometric function \(_pF_q\). We suggest similar conditions also for its extension involving the Wright \(_p\Psi_q\)-function and called the Srivastava-Wright operator,
\textit{H. M. Srivastava} [Appl. Anal. Discrete Math. 1, No. 1, 56--71 (2007; Zbl 1224.30084)]. \par Since the Dziok-Srivastava operaror includes the above-mentioned GFC operators and many their special cases (operators of the classical FC), from the results proposed here one can derive univalence criteria for many named operators in the GFT, as the operators of Hohlov, Carlson and Shaffer, Saigo, Libera, Bernardi, Erdélyi-Kober, etc., by giving particular values to the orders \(p\le q+1\) of the generalized hypergeometric functions and to their parameters.Certain subclass of bi-univalent functions related to Horadam polynomials associated with \(q \)-derivativehttps://zbmath.org/1538.300592024-08-14T19:23:59.529552Z"Nandini, P."https://zbmath.org/authors/?q=ai:nandini.p"Latha, S."https://zbmath.org/authors/?q=ai:latha.satyanarayana|latha.s-r|latha.sridar|latha.s-k|latha.sridharSummary: In this paper, by making use of \(q\)-derivative, we define a new subclass of analytic and bi-univalent functions related to Horadam polynomials. For functions belonging to this class, we derive coefficient inequalities and the Fekete-Szegö inequalities. We also provide relevant connections of our results with those considered in earlier investigations.A note on an integral by Grigorii Mikhailovich Fichtenholzhttps://zbmath.org/1538.301522024-08-14T19:23:59.529552Z"Reynolds, Robert"https://zbmath.org/authors/?q=ai:reynolds.robert"Stauffer, Allan"https://zbmath.org/authors/?q=ai:stauffer.allanSummary: In this manuscript, the authors derived a definite integral involving the logarithmic function, function of powers and polynomials in terms of the Lerch function. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.Inclusion relations of various subclasses of harmonic univalent mappings and \(k\)-uniformly harmonic starlike functionshttps://zbmath.org/1538.310012024-08-14T19:23:59.529552Z"Porwal, Saurabh"https://zbmath.org/authors/?q=ai:porwal.saurabh"Magesh, Nanjundan"https://zbmath.org/authors/?q=ai:magesh.nanjundanSummary: The purpose of the present paper is to obtain inclusion relations between various subclasses of harmonic univalent mappings by applying a convolution operator involving generalized Wright functions. To be more precise, we investigate such connections with Goodman-Rønning-type harmonic univalent functions, \(k\)-uniformly harmonic convex functions and \(k\)-uniformly harmonic starlike functions in the open unit disc \(\mathbb{U}\). Some of our results generalize and correct the results of \textit{S. Maharana} and \textit{S. K. Sahoo} [Complex Var. Elliptic Equ. 66, No. 10, 1619--1641 (2021; Zbl 1477.31003)].On the construction of the elementary trigonometric functionshttps://zbmath.org/1538.330012024-08-14T19:23:59.529552Z"Medvegyev, Péter"https://zbmath.org/authors/?q=ai:medvegyev.peterSummary: The article discusses the construction of the elementary trigonometric functions. It discusses several approaches, but the main message is that to construct the trigonometric functions one needs to follow the same approach as one should use during the construction of the real exponential function, but one should use complex numbers. The key point is that one must construct the complex square root function to define the trigonometric functions on the binary numbers and then one should use some convexity argument to prove their differentiability. This approach, as the power series approach, based on the intimate relation between the trigonometric and the exponential functions.Extended hyperbolic function and its propertieshttps://zbmath.org/1538.330022024-08-14T19:23:59.529552Z"Menon, Mudita"https://zbmath.org/authors/?q=ai:menon.mudita"Mittal, Ekta"https://zbmath.org/authors/?q=ai:mittal.ekta"Gupta, Rajni"https://zbmath.org/authors/?q=ai:gupta.rajniSummary: Aim of this paper is to introduce extended hyperbolic function by using a modified extension of beta function [\textit{M. Shadab} et al., Far East J. Math. Sci. (FJMS) 103, 235--251 (2018; \url{doi:10.17654/MS103010235})] and to establish new properties like integral representation, Mellin transform and many more. Furthermore, we apply Prabhakar fractional integral operator, Caputo-Fabrizio operator and Atangana-Baleanu operator on it. Other than this, we present a graphical representation of the extended hyperbolic function with different values of \(\alpha\) also a graphical comparison between Caputo-Fabrizio operator and Atangana-Baleanu operator of hyperbolic function for different values of \(r\).Hypergeometric form of \((1+x^2) \frac{ib}{2} \exp (b \tan^{-1} x)\) and its applicationshttps://zbmath.org/1538.330032024-08-14T19:23:59.529552Z"Qureshi, M. I."https://zbmath.org/authors/?q=ai:qureshi.mohammad-idris|qureshi.mohd-idris"Bhat, Aarif Hussain"https://zbmath.org/authors/?q=ai:bhat.aarif-hussain"Majid, Javid"https://zbmath.org/authors/?q=ai:majid.javidSummary: In this article, we obtain hypergeometric forms (not available in the literature) of some composite functions like:
\[
\begin{aligned}
(1-y^2)^{\frac{d}{2}} &\exp (d \tanh^{-1} y), & \quad (1+x^2)^{\frac{g}{2}} &\cos (g \tan^{-1} x), &\quad (1+x^2)^{\frac{g}{2}} & \sin (g \tan^{-1} x), \\
(1+x^2)^{\frac{ik}{2}} &\cosh (k \tan^{-1} x), &\quad (1+x^2)^{\frac{ik}{2}} &\sinh (k \tan^{-1} x), &\quad (1-y^2)^{\frac{g}{2}} &\cosh (g \tanh^{-1} y), \\
(1-y^2)^{\frac{g}{2}} &\sinh (g \tanh^{-1} y), &\quad (1-y^2)^{\frac{ik}{2}} &\cos (k \tanh^{-1} y), &\quad (1-y^2)^{\frac{ik}{2}} &\sin (k \tanh^{-1}y),
\end{aligned}
\]
by using Leibniz theorem for successive differentiation and Maclaurin's series expansion. Some applications are also discussed.Successive integration of certain mathematical functions using hypergeometric approachhttps://zbmath.org/1538.330042024-08-14T19:23:59.529552Z"ul Rahman Shah, Tafaz"https://zbmath.org/authors/?q=ai:shah.tafaz-ul-rahman"Qureshi, M. I."https://zbmath.org/authors/?q=ai:qureshi.mohd-idris|qureshi.mohammad-idris"Majid, Javid"https://zbmath.org/authors/?q=ai:majid.javidSummary: In this article, we obtain successive integration of some typical mathematical functions like:
\(-\frac{4}{z} \ln \left(\frac{^+\sqrt{(1-z)}}{2}\right)\); \((z)^{\frac{1}{2}}\sin^{-1} \sqrt{(z)} + \sqrt{(1-z)}\); \((z)^{-\frac{1}{2}} \sin^{-1} \sqrt{(z)} + \sqrt{(1-z)}\); \(\frac{4}{z} \left[1-\sqrt{(1-z)} +\ln \left(\frac{1+\sqrt{(1-z)}}{2}\right)\right]\) and \(\frac{4}{z^2} \left[2 \sqrt{(1-z)} -2+z-2z\ln \left(\frac{1+\sqrt{(1-z)}}{2}\right)\right]\) by using the approach of the hypergeometric functions as the successive integration of these functions can not be performed by any other mathematical technique.On a unified Oberhettinger-type integral involving the product of Bessel functions and Srivastava polynomialshttps://zbmath.org/1538.330052024-08-14T19:23:59.529552Z"Pandey, S. C."https://zbmath.org/authors/?q=ai:pandey.shared-c"Chaudhary, K."https://zbmath.org/authors/?q=ai:chaudhary.kuldip-k|chaudhary.kuldipkumar-k|chaudhary.ketan|chaudhary.kamal|chaudhary.kuldeepSummary: The present paper is devoted to derive a generalized Oberhettinger-type integral formula. The derived form of the integral involves a finite product of the Srivastava polynomials with the first-kind Bessel functions. The outcomes are obtained in terms of the Srivastava and Daoust functions. Some of the significant particular cases are also determined.Multi-indexed Whittaker function and its propertieshttps://zbmath.org/1538.330062024-08-14T19:23:59.529552Z"Panwar, Savita"https://zbmath.org/authors/?q=ai:panwar.savita"Rai, Prakriti"https://zbmath.org/authors/?q=ai:rai.prakritiSummary: In this paper, we have introduced the multi-indexed Whittaker function (\(3m\)-parameter) by using the extended confluent hypergeometric function which is defined in terms of multi-indexed (\(3m\)-parameter) Mittag-Leffler function. We derive some properties of multi-indexed (\(3m\)-parameter) Whittaker function such as its integral representations, derivative formula and Hankel transform.Certain general double-series identities with their applicationshttps://zbmath.org/1538.330072024-08-14T19:23:59.529552Z"Chaudhary, Wali Mohd."https://zbmath.org/authors/?q=ai:chaudhary.wali-mohd"Qureshi, M. I."https://zbmath.org/authors/?q=ai:qureshi.mohammad-idris"Kashif Khan, M."https://zbmath.org/authors/?q=ai:kashif-khan.mSummary: The aim of this paper is to obtain some general double-series identities involving bounded sequence of complex numbers with suitable convergence conditions. We then employ these identities to establish several quadratic transformations for the product of exponential function and Goursat's functions \({}_2F_2 [a,m+d; 2a\pm j,d; 2z]\) and also transformations for the Kampé de Fériet's function \(F^{G:2;0}_{H:2;0} [2z,-z]\).Certain supercongruences deriving from hypergeometric series identitieshttps://zbmath.org/1538.330082024-08-14T19:23:59.529552Z"Jana, Arijit"https://zbmath.org/authors/?q=ai:jana.arijitSummary: In this paper, we deduce some supercongruences for sums involving third power of certain rising factorials using hypergeometric series identities and evaluations. In particular, we first relate a truncated hypergeometric sum with the coefficients of the modular form of weight 3. Further, we confirm certain supercongruence conjectures related to truncated hypergeometric series.Two further methods for deriving four results contiguous to Kummer's second theoremhttps://zbmath.org/1538.330092024-08-14T19:23:59.529552Z"Kim, Insuk"https://zbmath.org/authors/?q=ai:kim.insuk"Kim, Joohyung"https://zbmath.org/authors/?q=ai:kim.joohyungSummary: In the theory of generalized hypergeometric function, transformation and summation formulas play a key role. In particular, in one of the Kummer's transformation formulas, \textit{Y. Sup Kim} et al. [Zh. Vychisl. Mat. Mat. Fiz. 50, No. 1, 407--422 (2010; Zbl 1224.33001)] have obtained ten contiguous results in the form of a single result with the help of generalization of Gauss's second summation theorem obtained earlier by \textit{J. L. Lavoie} et al. [Indian J. Math. 34, No. 1, 23--32 (1992; Zbl 0793.33005); Math. Comput. 62, No. 205, 267--276 (1994; Zbl 0793.33006); J. Comput. Appl. Math. 72, No. 2, 293--300 (1996; Zbl 0853.33005)]. In this paper, we aim at presenting four of such results by the technique of contiguous function relations and integral method developed by \textit{T. M. MacRobert} [Functions of a complex variable. 4th rev. ed. New York: St (1954; Zbl 0056.29102)].On a new class of summation formulas involving generalized hypergeometric functionshttps://zbmath.org/1538.330102024-08-14T19:23:59.529552Z"Lim, Dongkyu"https://zbmath.org/authors/?q=ai:lim.dongkyu"Kulkarni, Vidha"https://zbmath.org/authors/?q=ai:kulkarni.vidha"Vyas, Yashoverdhan"https://zbmath.org/authors/?q=ai:vyas.yashoverdhan"Rathie, Arjun K."https://zbmath.org/authors/?q=ai:rathie.arjun-kumarSummary: In the theory of generalized hypergeometric series, classical summation theorems such as those of Gauss, Gauss's second, Kummer and Bailey for the series \(_3F_2\); Watson, Dixon, and Whipple for the series \(_3F_2\) and others play an important role. In [Math. Sci., Springer 9, No. 4, 215--223 (2015; Zbl 1405.33007)], \textit{M. R. Eslahchi} and \textit{M. Masjed-Jamei} applied the above-mentioned classical summation theorems in a very general hypergeometric identity available in the literature and obtained several interesting summation formulas involving generalized hypergeometric functions. In [Int. J. Math. Math. Sci. 2010, Article ID 309503, 26 p. (2010; Zbl 1210.33012)], \textit{Y. S. Kim} et al. established the extensions of the above-mentioned classical summation theorems together with a few more extended summation theorems. This paper aims to establish several new and interesting summation formulas involving generalized hypergeometric functions. This is achieved by applying the above-mentioned extended summation theorems in a very general hypergeometric identity available in the literature. The result obtained earlier by Eslahchi and Masjed-Jamei [loc. cit.] follows special cases of our main findings. The results established in the paper are simple, interesting, easily established, and may be potentially useful.On a family of bivariate orthogonal functionshttps://zbmath.org/1538.330112024-08-14T19:23:59.529552Z"Güldoğan Lekesiz, Esra"https://zbmath.org/authors/?q=ai:guldogan-lekesiz.esraSummary: In this paper we investigate a family of bivariate orthogonal functions arising as a generalization of Koornwinder polynomials in two variables. General properties like recurrence relations and partial differential equations are introduced. Some special cases are considered and a limit relation of these functions is studied. As a consequence, a new class of bivariate orthogonal polynomials is presented.Bivariate \(k\)-Mittag-Leffler functions with 2D-\(k\)-Laguerre-Konhauser polynomials and corresponding \(k\)-fractional operatorshttps://zbmath.org/1538.330122024-08-14T19:23:59.529552Z"Kürt, Cemaliye"https://zbmath.org/authors/?q=ai:kurt.cemaliye"Özarslan, Mehmet Ali"https://zbmath.org/authors/?q=ai:ozarslan.mehmet-aliSummary: In this paper, we first introduce new class of 2D-\(k\)-Laguerre-Konhauser polynomials, \(_{\delta }L_{k,n}^{(\alpha ,\beta)}(x,y)\), which generalizes the 2D-Laguerre-Konhauser polynomials (see [the authors, Appl. Math. Comput. 347, 631--644 (2019; Zbl 1428.33035)]). Then, we define a new family of bivariate \textit{k}-Mittag-Leffler functions \(E_{k,\alpha ,\beta ,\delta }^{(\gamma)}(x,y)\) and establish the \textit{k}-Riemann-Liouville double fractional integral and derivative of the functions \(E_{k,\alpha ,\beta ,\delta }^{(\gamma)}(x,y)\). Moreover, we introduce an integral operator \(_{k}\varepsilon_{\alpha ,\beta ,\delta ;\omega_{1},\omega_{2};a^{+},c^{+}}^{(\gamma)}\) which contains the bivariate \(k\)-Mittag-Leffler functions \(E_{k,\alpha ,\beta ,\delta }^{(\gamma)}(x,y)\) in the kernel and investigate the semigroup property of this operator. Finally, the left inverse operator of the integral operator \(_{k}\varepsilon_{\alpha ,\beta ,\delta ;\omega_{1},\omega_{2};a^{+},c^{+}}^{(\gamma)}\) is constructed.Bilateral generating function for \(P^\alpha_n (x;h)\) and \(Q^\alpha_n (x;h)\)https://zbmath.org/1538.330132024-08-14T19:23:59.529552Z"Mishra, Lakshmi Narayan"https://zbmath.org/authors/?q=ai:mishra.lakshmi-narayan"Singh, Rakesh Kumar"https://zbmath.org/authors/?q=ai:singh.rakesh-kumar"Tiwari, Shiv Kant"https://zbmath.org/authors/?q=ai:tiwari.shiv-kantSummary: Our aimed has been made to find out forms of two interesting properties of the biorthogonal polynomials sets P\(P^\alpha_n (x;h)\) and \(Q^\alpha_n (x;h)\). In this paper, we have generalized the generating functions of biorthogonal polynomials suggested by the Laguerre polynomials published in [\textit{J. D. E. Konhauser}, Pac. J. Math. 21, 303--314 (1967; Zbl 0156.07401)] and some more results of transformation of certain bilinear generating functions published in [\textit{L. Carlitz}, Ann. Mat. Pura Appl. (4) 86, 155--168 (1970; Zbl 0201.07102)] with the help of some applicable results of the hand book [\textit{M. Abramowitz} (ed.) and \textit{I. A. Stegun} (ed.), Handbook of mathematical functions with formulas, graphs and mathematical tables. Washington: U (1964; Zbl 0171.38503)] .Certain quadruple series equations involving Laguerre polynomialshttps://zbmath.org/1538.330142024-08-14T19:23:59.529552Z"Phadte, C. N."https://zbmath.org/authors/?q=ai:phadte.c-n"Tamba, N."https://zbmath.org/authors/?q=ai:tamba.n"Valaulikar, Y. S."https://zbmath.org/authors/?q=ai:valaulikar.yeshwant-shivraiSummary: \textit{H. M. Srivastava} [Nederl. Akad. Wet., Proc., Ser. A 75, 53--61 (1972; Zbl 0216.36301); J. Math. Phys. 23, 357 (1982; Zbl 0464.45005)] has solved dual series equations involving Bateman-k functions and Jacobi polynomials. \textit{H. M. Srivastava} [Gaṇita 43, No. 1--2, 75--84 (1992; Zbl 0831.45004)] has obtained more results for the Konhauser-biorhogonal set. \textit{J. S. Lowndes} [Pac. J. Math. 25, 123--127 (1968; Zbl 0159.09302); ibid. 29, 167--173 (1969; Zbl 0175.35903)], \textit{H. M. Srivastava} [ibid. 30, 525--527 (1969; Zbl 0179.10102)], \textit{J. S. Lowndes} and \textit{H. M. Srivastava} [J. Math. Anal. Appl. 150, No. 1, 181--187 (1990; Zbl 0687.45004)], \textit{H. M. Srivastava} [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 67, 395--401 (1980; Zbl 0423.45008)], \textit{H. M. Srivastava} and \textit{R. Panda} [Indag. Math. 40, 502--514 (1978; Zbl 0402.42016)] have obtained the solution of dual series equations involving Jacobi and Laguerre polynomials and also solved triple series equations involving Laguerre polynomials. \textit{B. M. Singh} et al. [Ukr. Mat. Zh. 62, No. 2, 231--237 (2010; Zbl 1224.33007)] have find out the solution of triple series equations involving Laguerre polynomials in a closed form. \textit{K. Narain} [``Certain quadruple series equations'', Sci. Res. J. 1, No. 2, 26--30 (2013), \url{http://www.scirj.org/papers-0913/scirj-0204.pdf}; ``Certain suadruple series equations with Jacobi polynomials as Kernels'', Int. J. Math. Res. 11, No. 1, 55--58 (2019), \url{http://www.scirj.org/papers-0913/scirj-0204.pdf}], \textit{R. K. Mudaliar} and \textit{K. Narain} [``Certain quadruple integral equations, Glob. J. Pure Appl. Math. 12, No. 3, 2867--2875 (2016), \url{http://www.ripublication.com/gjpam16/gjpamv12n4_10.pdf}] have solved Certain dual and quadruple series equations involving generalized Laguerre polynomials and Jacobi polynomials as kernels. In the present paper, an exact solution has been obtained for the quadruple series equations involving Laguerre polynomials by \textit{B. Noble} [Proc. Camb. Philos. Soc. 59, 363--371 (1963; Zbl 0115.28402)] modified multiplying factor technique.Fractional calculus and families of generalized Legendre-Laguerre-Appell polynomialshttps://zbmath.org/1538.330152024-08-14T19:23:59.529552Z"Wani, Shahid Ahmad"https://zbmath.org/authors/?q=ai:wani.shahid-ahmad"Mallah, Ishfaq Ahmad"https://zbmath.org/authors/?q=ai:mallah.ishfaq-ahmad"Ganie, Javid Ahmad"https://zbmath.org/authors/?q=ai:ganie.javid-ahmadSummary: In this article, new families of the generalized Legendre-Laguerre-Appell polynomials are introduced using a combination of operational definitions and integral representations. The integral transformations and the appropriate operational rules are used to obtain the explicit summation equations, determinant definitions, and recurrence relations for the generalised Legendre-Laguerre-Appell polynomials. For the generalized Legendre-Laguerre-Bernoulli, Legendre-Laguerre-Euler, and Legendre-Laguerre-Genocchi polynomials, an equivalent investigation of these findings is offered. Additionally, a number of identities for these polynomials are derived by using suitable operational definitions.Some results involving the \(_pR_q(\alpha, \beta; z)\) functionhttps://zbmath.org/1538.330162024-08-14T19:23:59.529552Z"Thakkar, Yogesh M."https://zbmath.org/authors/?q=ai:thakkar.yogesh-m"Shukla, Ajay"https://zbmath.org/authors/?q=ai:shukla.ajay-kumarSummary: The main aim of this paper is to discuss some classical properties of the \(_p R_q(\alpha, \beta; z)\) function such as integrals involving \(_pR_q(\alpha, \beta; z)\) function and its product with some algebraic functions and higher Tanscendental function viz, Hermite polynomial, Legendre polynomial, Legendre function, Jacobi polynomial, Galue type Struve function, six summation formulas of \(_p R_q(\alpha, \beta; z)\) function and relation between \(_pR_q(\alpha, \beta; z)\) and \(_pR_q(\alpha, \beta;- z)\) functions.On some applications of the new extended hypergeometric functionhttps://zbmath.org/1538.330172024-08-14T19:23:59.529552Z"Kaurangini, M. L."https://zbmath.org/authors/?q=ai:kaurangini.muhammad-l"Chaudhary, M. P."https://zbmath.org/authors/?q=ai:chaudhary.mahendra-pal"Ata, E."https://zbmath.org/authors/?q=ai:ata.enes"Abubakar, U. M."https://zbmath.org/authors/?q=ai:abubakar.u-m"Kiymaz, I. O."https://zbmath.org/authors/?q=ai:kiymaz.i-onurSummary: The fundamental goal of this article is to investigate applications of the new extended hypergeometric function that contained two Fox-Wright functions in its kernel to generating function. Moreover, the new extended Riemann-Liouville fractional derivative operator with some of its properties is also studied.To the properties of one Fox functionhttps://zbmath.org/1538.330182024-08-14T19:23:59.529552Z"Khushtova, F. G."https://zbmath.org/authors/?q=ai:khushtova.fatima-gidovnaSummary: The paper considers a particular case of a special Fox function with four parameters, which arises in the theory of boundary value problems for parabolic equations with a Bessel operator and a fractional time derivative. The research objective is to obtain some recurrence relations, formulas for differentiation and integral transformation of the function under consideration. The results are obtained through representation of the considered function in terms of the Mellin-Barnes integral. The function asymptotic expansions for large and small values of the argument are also used. Employing the integral representation and some wellknown formulas for the Euler gamma function, recurrent relations are obtained connecting functions with different parameters, as well as a function with its first-order derivative. A formula for differentiation of the \(n\)th order is obtained. The paper studies an improper integral of the first kind that includes the considered function with two dependent of the four parameters. We show that the improper integral can be written out in terms of the well-known special Macdonald function. With special values of the parameters of the considered function we obtain some well-known elementary and special functions. The results of the study are theoretical and applicable in the study of boundary value problems for degenerate parabolic equations with fractional time derivatives.Evaluation of some Gröbner-Hofreiter-type integrals using hypergeometric approach with applicationshttps://zbmath.org/1538.330192024-08-14T19:23:59.529552Z"Qureshi, Mohammad Idris"https://zbmath.org/authors/?q=ai:qureshi.mohammad-idris"Bhat, Aarif Hussain"https://zbmath.org/authors/?q=ai:bhat.aarif-hussain"Majid, Javid"https://zbmath.org/authors/?q=ai:majid.javidSummary: Gröbner-Hofreiter-type integrals were evaluated by the use of applicable contour integrals and Cauchy's residue theorem. In this paper, we obtain the solutions of Gröbner-Hofreiter-type integrals and other associated integrals with suitable convergence conditions by using hypergeometric approach. Some applications of Gröbner-Hofreiter-type integrals are also obtained in the form of Weber-Anger-type functions.Certain integrals of product of Mittag-Leffler function, \(M\)-series and \(I\)-function of two variableshttps://zbmath.org/1538.330202024-08-14T19:23:59.529552Z"Sacha, Dheerandra Shanker"https://zbmath.org/authors/?q=ai:sacha.dheerandra-shanker"Singh, Giriraj"https://zbmath.org/authors/?q=ai:singh.girirajSummary: The object of this paper is to establish certain unified integrals associated with \(I\)-function of two variables. First, we have evaluated integrals whose integrand is the product of generalized Mittag-Leffler function, generalized \(M\)-series and \(I\)-function of two variables. Moreover, the integrand of the last integral is the product of generalized Mittag-Leffler function, generalized \(M\)-series, \(H\)-function of one variables and \(I\)-function of two variables. We have evaluated this integral by means of Mellin transform of \(H\)-function of one variables. In consequence of general nature of \(I\)-function of two variables, some special cases also have been considered.Analytical expressions for the exact curved surface area of a hemiellipsoid via Mellin-Barnes type contour integrationhttps://zbmath.org/1538.330212024-08-14T19:23:59.529552Z"Pathan, M. A."https://zbmath.org/authors/?q=ai:pathan.mahmood-ahmad"Qureshi, M. I."https://zbmath.org/authors/?q=ai:qureshi.mohd-idris|qureshi.mohammad-idris"Majid, Javid"https://zbmath.org/authors/?q=ai:majid.javidSummary: In this article, we aim at obtaining the analytical expressions (not previously found and not recorded in the literature) for the exact curved surface area of a hemiellpsoid in terms of Appell's double hypergeometric function of first kind. The derivation is based on Mellin-Barnes type contour integral representations of generalized hypergeometric function \(_pF_q(z)\), Meijer's \(G\)-function and analytic continuation formula for Gauss function. Moreover, we obtain some special cases related to ellipsoid, Prolate spheroid and Oblate spheroid. The closed forms for the exact curved surface area of a hemiellipsoid are also verified numerically by using \textit{Mathematica Program}.Certain identities involving general double hypergeometric functions of Extonhttps://zbmath.org/1538.330222024-08-14T19:23:59.529552Z"Arora, Ashish"https://zbmath.org/authors/?q=ai:arora.ashish"Saxena, Saurabh"https://zbmath.org/authors/?q=ai:saxena.saurabh"Prakash, Ashish"https://zbmath.org/authors/?q=ai:prakash.ashishSummary: This paper deals with the generalizations and unifications of identities by using series iteration technique in association with hypergeometric function \({}_AF_B\) of single variable and hypergeometric function of double variables given by Appell, in the form of identities in association with Exton's hypergeometric function \(H\), \(G\) of double variables.Relations and identities via contour integral representations involving Hurwitz-Lerch zeta type functions for two variable Srivasatava-Daoust functionshttps://zbmath.org/1538.330232024-08-14T19:23:59.529552Z"Kumar, Hemant"https://zbmath.org/authors/?q=ai:kumar.hemant"Chandel, R. C. Singh"https://zbmath.org/authors/?q=ai:chandel.r-c-singhSummary: In this paper, we derive certain relations of the series of two variable Srivastava-Daoust functions with some known Mittag-Leffler and hypergeometric functions of two variables. Again, by these functions we obtain certain identities with other integral representations also. Finally, on application of these contour integral formulae of respective Srivastava-Daoust functions, we determine certain identities of the integrals involving the Hurwitz-Lerch zeta type functions.Generating functions of \((p, q)\)-analogue of aleph-function satisfying Truesdell's ascending and descending \(F_{p, q}\)-equationhttps://zbmath.org/1538.330242024-08-14T19:23:59.529552Z"Bhat, Altaf A."https://zbmath.org/authors/?q=ai:bhat.altaf-ahmad"Bhat, M. Younus"https://zbmath.org/authors/?q=ai:bhat.mohammad-younus"Maqbool, H."https://zbmath.org/authors/?q=ai:maqbool.humaira"Jain, D. K."https://zbmath.org/authors/?q=ai:jain.deepak-kumarSummary: In this paper we have obtained various forms of \((p, q)\)-analogue of Aleph-Function satisfying Truesdell's ascending and descending \(F_{p, q}\)-equation. These structures have been employed to arrive at certain generating functions for \((p, q)\)-analogue of Aleph-Function. Some specific instances of these outcomes as far as \((p, q)\)-analogue of I-function, H-function and G-functions have likewise been obtained.Some \(q\)-supercongruences from the Bailey transformationhttps://zbmath.org/1538.330252024-08-14T19:23:59.529552Z"Guo, Victor J. W."https://zbmath.org/authors/?q=ai:guo.victor-j-wSummary: Using the Bailey transformation formula together with the 'creative microscoping' method (recently introduced by \textit{V. J. W. Guo} and \textit{W. Zudilin} [J. Comb. Theory, Ser. A 178, Article ID 105362, 38 p. (2021; Zbl 1473.11046)]), we give \(q\)-analogues of two supercongruences of \textit{B. He} [Proc. Am. Math. Soc. 143, No. 12, 5173--5180 (2015; Zbl 1395.11010)] \[\sum_{k=0}^{(p-1)/2}(-1)^k \frac{\left(\frac{1}{2}\right)_k^3}{k!^3} \equiv (-1)^{(p-1)/2}p\sum_{k=0}^{(p-1)/2}\frac{\left(\frac{1}{2}\right)_k^3}{k! \left(\frac{3}{4}\right)_k \left(\frac{5}{4}\right)_k}\;\; (\bmod \; p^2),\] \[\sum_{k=0}^{(p-1)/2}(-1)^k \frac{\left(\frac{1}{2}\right)_k^2}{k!^2} \equiv (-1)^{(p-1)/2}p\sum_{k=0}^{(p-1)/2}\frac{\left(\frac{1}{2}\right)_k}{k! (4k+1)}\;\; (\bmod \; p^2)\] where \(p\) is an odd prime. One of our results implies that He's second supercongruence is still true modulo \(p^3\) for \(p \equiv 3 \; (\bmod \; 4)\). We also give two similar \(q\)-supercongruences.Basic bilateral hypergeometric function \(_2\Psi_2\) and continued fractionshttps://zbmath.org/1538.330262024-08-14T19:23:59.529552Z"Pant, G. S."https://zbmath.org/authors/?q=ai:pant.g-s"Chand, K. B."https://zbmath.org/authors/?q=ai:chand.k-b"Pathak, Manoj Kumar"https://zbmath.org/authors/?q=ai:pathak.manoj-kumarSummary: In this paper certain continued fractions of the ratios of \(_2\Psi_2\)-series have been established.A note on Heine's transformationhttps://zbmath.org/1538.330272024-08-14T19:23:59.529552Z"Singh, Satya Prakash"https://zbmath.org/authors/?q=ai:singh.satya-prakash.3|singh.satya-prakash.2|singh.satya-prakash"Rawat, Akash"https://zbmath.org/authors/?q=ai:rawat.akashSummary: In this paper, making use of \(q\)-binomial theorem different generalizations of Heine's first transformation have been discussed.On certain basic hypergeometric series identitieshttps://zbmath.org/1538.330282024-08-14T19:23:59.529552Z"Singh, Satya Prakash"https://zbmath.org/authors/?q=ai:singh.satya-prakash.2|singh.satya-prakash|singh.satya-prakash.3"Yadav, Vijay"https://zbmath.org/authors/?q=ai:yadav.vijay-kumar|vijay.yadavSummary: In this paper, making use of an identity, certain Rogers-Ramanujan type identities have been established.Andrews' type WP-Bailey lemma and its applicationshttps://zbmath.org/1538.330292024-08-14T19:23:59.529552Z"Vyas, Yashoverdhan"https://zbmath.org/authors/?q=ai:vyas.yashoverdhan"Pathak, Shivani"https://zbmath.org/authors/?q=ai:pathak.shivani"Fatawat, Kalpana"https://zbmath.org/authors/?q=ai:fatawat.kalpanaSummary: Over the years, the study of Bailey transform, Bailey lemma, Bailey pair, their variants and their applications are the major subjects of interest. Of course, it is due to the efficiency of the Bailey transform and lemma in producing many ordinary and q-hypergeometric identities, multiple series summation and transformation formulas, and the Rogers-Ramanujan type identities. Andrews investigated a WP-Bailey lemma and the pairs with the help of Bailey transform and used it to derive well-known summations and multiple series transformations. In this research paper, we investigate an Andrews' type WP-Bailey lemma and the pairs with the help of First Bailey Type Transform due to Joshi and Vyas. The investigated Andrews' type WP-Bailey lemma is then applied to obtain terminating multiple \(q\)-hypergeometric identities and construct the WP-Bailey type chains and a binary tree.
The paper is motivated by the observation that the basic (or \(q\)-) series and basic (or \(q\)-) polynomials, especially the basic (or \(q\)-) gamma and q-hypergeometric functions and basic (or \(q\)-) hypergeometric polynomials, are applicable particularly in several diverse areas including number theory, theory of partitions and combinatorial analysis as well
as in the study of combinatorial generating functions.Proofs of Guo and Schlosser's two conjectureshttps://zbmath.org/1538.330302024-08-14T19:23:59.529552Z"Xu, Chang"https://zbmath.org/authors/?q=ai:xu.chang"Wang, Xiaoxia"https://zbmath.org/authors/?q=ai:wang.xiaoxiaSummary: Many congruences and \(q\)-congruences have recently been established. In this paper, we confirm two \(q\)-supercongruences which were conjectured by \textit{V. J. W. Guo} and \textit{M. J. Schlosser} [Result. Math. 75, No. 4, Paper No. 155, 12 p. (2020; Zbl 1469.11032)].Meromorphic continuations of multiple \(q\)-hypergeometric functionshttps://zbmath.org/1538.330312024-08-14T19:23:59.529552Z"Ernst, Thomas"https://zbmath.org/authors/?q=ai:ernst.thomasSummary: The purpose of this article is to compute meromorphic continuations of several multiple \(q\)-hypergeometric functions simply by using the formula for meromorphic continuation of \(_2\varphi_1\). This leads to \(q\)-analogues of analytic continuation formulas by Appell and Kampé de Fériet, Exton and Srivastava. All these formulas are proved in detail. For each of the multiple functions, we write down the canonical system of \(q\)-difference equations, and by the meromorphic continuation formulas, we automatically get other solutions to these systems in the neighbourhood of infinity. We recall that for \(q\)-hypergeometric functions, branching only occurs at the `principal' regular singular points 0 and \(\infty\).Semi integration of some algebraic functionshttps://zbmath.org/1538.330322024-08-14T19:23:59.529552Z"Qureshi, M. I."https://zbmath.org/authors/?q=ai:qureshi.mohd-idris|qureshi.mohammad-idris"Majid, Javid"https://zbmath.org/authors/?q=ai:majid.javidSummary: The aim of this article is to obtain the semi-integrals of certain functions in terms of Complete Elliptic integrals of different kinds by making use of Euler's linear transformation, Pfaff-Kummer linear transformation and series manipulation technique.On the semi-differentials of some complete elliptic integrals and their differenceshttps://zbmath.org/1538.330332024-08-14T19:23:59.529552Z"Qureshi, M. I."https://zbmath.org/authors/?q=ai:qureshi.mohammad-idris|qureshi.mohd-idris"Majid, Javid"https://zbmath.org/authors/?q=ai:majid.javid"Ara, Jahan"https://zbmath.org/authors/?q=ai:ara.jahanSummary: In this article we aim at obtaining the semi-differentials of Complete Elliptic integrals of different kinds and their differences in terms of algebraic functions by using series manipulation technique and Pfaff-Kümmer linear transformation.Sharp power-type Heronian and Lehmer means inequalities for the complete elliptic integralshttps://zbmath.org/1538.330342024-08-14T19:23:59.529552Z"Zhao, Tie-hong"https://zbmath.org/authors/?q=ai:zhao.tiehong"Chu, Yu-ming"https://zbmath.org/authors/?q=ai:chu.yumingSummary: In the article, we prove that the inequalities
\[
H_p(\mathscr{K}(r), \mathscr{E}(r)) > \frac{\pi}{2},\quad L_q(\mathscr{K}(r), \mathscr{E}(r)) > \frac{\pi}{2}
\]
hold for all \(r\in(0, 1)\) if and only if \(p \geq -3/4\) and \(q \geq -3/4\), where \(H_p(a, b)\) and \(L_q(a, b)\) are respectively the \(p\)-th power-type Heronian mean and \(q\)-th Lehmer mean of \(a\) and \(b\), and \(\mathscr{K}(r)\) and \(\mathscr{E}(r)\) are respectively the complete elliptic integrals of the first and second kinds.Series in Mittag-Leffler functions: geometry of convergencehttps://zbmath.org/1538.330352024-08-14T19:23:59.529552Z"Paneva-Konovska, Jordanka"https://zbmath.org/authors/?q=ai:paneva-konovska.jordanka-dSummary: We consider series, defined by means of the Mittag-Leffler functions, find the domains of convergence and study the behaviour on the boundaries of these domains. We give analogues of the classical theorems for the power series like Cauchy-Hadamard, Abel as well as Fatou type theorems. The asymptotic formulae for the Mittag-Leffler functions in the cases of ``large'' values of indices that are used in the proofs of the convergence theorems for the considered series are also provided.Finite and infinite integral formulas associated with a family of incomplete \(I\)-functionshttps://zbmath.org/1538.330362024-08-14T19:23:59.529552Z"Jain, Prachi"https://zbmath.org/authors/?q=ai:jain.prachi"Gupta, Neena"https://zbmath.org/authors/?q=ai:gupta.neena.1|gupta.neena"Gupta, Arvind"https://zbmath.org/authors/?q=ai:gupta.arvind-kumar"Saxena, V. P."https://zbmath.org/authors/?q=ai:saxena.vinod-prakashSummary: In recent years, research focuses on the integral representations of several kinds of special functions. In this paper, first we establish the integral representation of incomplete \(I\)-functions. Further, we find out some special cases of these integrals. Finally, we derived certain integrals involving a product of incomplete \(I\)-function and some other special functions.Two series which generalize Dirichlet's lambda and Riemann's zeta functions at positive integer argumentshttps://zbmath.org/1538.330372024-08-14T19:23:59.529552Z"Markov, Lubomir"https://zbmath.org/authors/?q=ai:markov.lubomir-pSummary: The series \(\sum^\infty_{k = 0}\frac{G_N(k)}{(2k+1)^r}\) and \(\sum^\infty_{k = 1}\frac{H_N(k)}{k^r}\) are considered, where \(G_N(k)\) and \(H_N(k)\) are the Borwein-Chamberland sums appeared in the expansions of integer powers of the arcsine reported in the paper [\textit{J. M. Borwein} and \textit{M. Chamberland}, Int. J. Math. Math. Sci. 2007, Article ID 19381, 10 p. (2007; Zbl 1137.33300)]. For \(3 \leq r \in\mathbb{N}\), representations for these series in terms of zeta values are derived, extending a theorem proved in the paper [\textit{J. A. Ewell}, Can. Math. Bull. 34, No. 1, 60--66 (1991; Zbl 0731.11048)]. Several corollaries (especially for the case \(r = 3\)) are obtained, extending some known representations, including Euler's famous rapidly converging series for \(\zeta(3)\). The technique can be applied to the case \(r = 2\) and it yields generalizations of the formulas \(\sum^\infty_{k = 0} \frac{1}{(2k+1)^2} = \frac{\pi^2}{8}\) and \(\sum^\infty_{k = 1}\frac{1}{k^2} = \frac{\pi^2}{6}\).Extension of Mathieu series and alternating Mathieu series involving the Neumann function \(Y_{\nu}\)https://zbmath.org/1538.330382024-08-14T19:23:59.529552Z"Parmar, Rakesh K."https://zbmath.org/authors/?q=ai:parmar.rakesh-kumar"Milovanović, Gradimir V."https://zbmath.org/authors/?q=ai:milovanovic.gradimir-v"Pogány, Tibor K."https://zbmath.org/authors/?q=ai:pogany.tibor-kThe extended Mathieu series \(\mathbb{S}(r)\) and its alternating variant \(\tilde{\mathbb{S}}(r)\) are introduced respectively via the following formulas:
\[(\mathbb{S}_{\mu,\nu}(r)=\sqrt{\frac{\pi}{2r}}\int_{0}^{\infty} \frac{x^{\mu-1}}{e^{x}-1}Y_{\nu}(rx)dx\]
for \(r>0\), \(\mu >0\) and \(\mu+\nu\geq1\); and
\[\tilde{\mathbb{S}}_{\mu,\nu}(r)=\sqrt{\frac{\pi}{2r}}\int_{0}^{\infty}\frac{x^{\mu-1}}{e^{x}+1}Y_{\nu}(rx)dx\]
for \(r>0,\mu>0\) and \(\mu+\nu\geq0.\) Here \(Y_{\nu}\) denotes the Bessel function of the second kind. The usual Mathieu series \(\mathcal{S}(r)\) and its alternating variant \(\tilde{\mathcal{S}}(r)\) are retrieved for \(\mu=5/2\) and \(\nu=-1/2.\) Several results connecting the extended Mathieu series with other classes of special functions like the polylogarithms, the Riemann zeta function, the Dirichlet eta function, and the associated Legendre function of the second kind are presented.\par The paper also includes the extended versions of the complete Butzer-Flocke-Hauss Omega functions and some functional bounding inequalities for the extended Mathieu series.
Reviewer: Constantin Niculescu (Craiova)Some summation formulas for double hypergeometric functions of Srivastava-daoust having \(\pm 1\) argumentshttps://zbmath.org/1538.330392024-08-14T19:23:59.529552Z"Qureshi, M. I."https://zbmath.org/authors/?q=ai:qureshi.mohammad-idris"Bhat, Bilal Ahmad"https://zbmath.org/authors/?q=ai:bhat.bilal-ahmad"Shah, Tafaz Ul Rahman"https://zbmath.org/authors/?q=ai:shah.tafaz-ul-rahmanSummary: In this paper, by using the hypergeometric approach we evaluate four definite integrals having the integrand (in the form of real powers of \(\tan \theta\)), in order to get various summation formulas for Srivastava-Daoust double hypergeometric functions containing the arguments \(\pm 1\).Five series equations involving generalized Bateman \(k\)-functionshttps://zbmath.org/1538.330402024-08-14T19:23:59.529552Z"Shrivastava, Omkar Lal"https://zbmath.org/authors/?q=ai:lal-shrivastava.omkar"Narain, Kuldeep"https://zbmath.org/authors/?q=ai:narain.kuldeep"Shrivastava, Sumita"https://zbmath.org/authors/?q=ai:shrivastava.sumitaSummary: In this paper, the solution of five series equations involving generalized Bateman \(k\)-functions is obtained by reducing them to Fredholm integral equation of the second kind. The solution presented in
this paper is obtained by employing the techniques of \textit{M. Lal} and \textit{K. Narain} [Acta Cienc. Indica, Math. 15, No. 4, 363--366 (1989; Zbl 0729.45005)] involving generalized Bateman \(k\)-functions by reducing them to the solution of a Fredholm integral equation of second kind with different boundary conditions. Thus we have seen that Bateman \(k\)-functions are having interesting properties to solve double, triple, quadruple and five series equations as special functions. These solutions are very
useful in Mathematical and Quantum Physics, Aero and Fluid Dynamics and Thermodynamics.\(\aleph\)-function and persistence of equilibrium states in biological systemshttps://zbmath.org/1538.330412024-08-14T19:23:59.529552Z"Shukla, Harish"https://zbmath.org/authors/?q=ai:shukla.harish"Dubey, Divya"https://zbmath.org/authors/?q=ai:dubey.divya"Shrivastava, Rajeev"https://zbmath.org/authors/?q=ai:shrivastava.rajeev-kumar|shrivastava.rajeev-p"Mishra, P. P."https://zbmath.org/authors/?q=ai:mishra.poonam-prakash|mishra.prayag-prasad|mishra.prem-prakashSummary: In an attempt to give extensions of the results in the theory and applications of special functions we derive a proper integral whose integrand contains Aleph function having general argument and used it in finding out a solution of a non-linear differential equation \(\partial u/\partial t = D(\partial^2 u /\partial x^2) +f(u)\) elated to a reaction-diffusion problems in biological models by means of linearization procedure.Fractional differential equations associated with generalized fractional operators in \(F_{p,\nu}\) spacehttps://zbmath.org/1538.340182024-08-14T19:23:59.529552Z"Bansal, Manish Kumar"https://zbmath.org/authors/?q=ai:bansal.manish-kumar"Harjule, Priyanka"https://zbmath.org/authors/?q=ai:harjule.priyanka"Kumar, Devendra"https://zbmath.org/authors/?q=ai:kumar.devendra.3Summary: The key aim of the present work is to study the fractional differential equations (FDEs) pertaining to generalized fractional operators in \(F_{p,\nu}\) space. First, we derive the Laplace transform of the generalized fractional operators in terms of generalized modified Bessel function type transform \(\mathbb{L}_{\alpha,\beta}^{(\sigma)}\) in \(F_{p,\nu}\) space. The results obtained are used to solve the FDEs involving constant as well as variable coefficients in \(F_{p,\nu}\) space. Due to the general nature of M-S-M integral operators many new and useful special cases of the key results can be obtained.Constructing reliable approximations of the random fractional Hermite equation: solution, moments and densityhttps://zbmath.org/1538.340192024-08-14T19:23:59.529552Z"Burgos, Clara"https://zbmath.org/authors/?q=ai:burgos.clara"Caraballo, Tomás"https://zbmath.org/authors/?q=ai:caraballo.tomas"Cortés, Juan Carlos"https://zbmath.org/authors/?q=ai:cortes.juan-carlos"Villafuerte, Laura"https://zbmath.org/authors/?q=ai:villafuerte.laura"Villanueva, Rafael Jacinto"https://zbmath.org/authors/?q=ai:villanueva.rafael-jacintoSummary: We extend the study of the random Hermite second-order ordinary differential equation to the fractional setting. We first construct a random generalized power series that solves the equation in the mean square sense under mild hypotheses on the random inputs (coefficients and initial conditions). From this representation of the solution, which is a parametric stochastic process, reliable approximations of the mean and the variance are explicitly given. Then, we take advantage of the random variable transformation technique to go further and construct convergent approximations of the first probability density function of the solution. Finally, several numerically simulations are carried out to illustrate the broad applicability of our theoretical findings.Existence and stability results for stochastic fractional neutral differential equations with Gaussian noise and Lévy noisehttps://zbmath.org/1538.340362024-08-14T19:23:59.529552Z"Umamaheswari, P."https://zbmath.org/authors/?q=ai:umamaheswari.p"Balachandran, K."https://zbmath.org/authors/?q=ai:balachandran.krishnan"Annapoorani, N."https://zbmath.org/authors/?q=ai:annapoorani.natarajan"Kim, Daewook"https://zbmath.org/authors/?q=ai:kim.daewookSummary: In this paper we prove the existence and uniqueness of solution of stochastic fractional neutral differential equations with Gaussian noise or Lévy noises by using the Picar-Lindelöf successive approximation scheme. Further stability results of nonlinear stochastic fractional dynamical system with Gaussian and Lévy noises are established. Examples are provided to illustrate the theoretical results.On the partial stability of nonlinear impulsive Caputo fractional systemshttps://zbmath.org/1538.342152024-08-14T19:23:59.529552Z"Ghanmi, Boulbaba"https://zbmath.org/authors/?q=ai:ghanmi.boulbaba"Ghnimi, Saifeddine"https://zbmath.org/authors/?q=ai:ghnimi.saifeddineSummary: In this work, stability with respect to part of the variables of nonlinear impulsive Caputo fractional differential equations is investigated. Some effective sufficient conditions of stability, uniform stability, asymptotic uniform stability and Mittag Leffler stability. The approach presented is based on the specially introduced piecewise continuous Lyapunov functions. Furthermore, some numerical examples are given to show the effectiveness of our obtained theoretical results.Relatively exact controllability of fractional stochastic delay system driven by Lévy noisehttps://zbmath.org/1538.343002024-08-14T19:23:59.529552Z"Huang, Jizhao"https://zbmath.org/authors/?q=ai:huang.jizhao"Luo, Danfeng"https://zbmath.org/authors/?q=ai:luo.danfengSummary: In this article, we consider the relatively exact controllability of fractional stochastic delay system (FSDS) driven by Lévy noise. First, we derive the solution of linear FSDS via delayed matrix functions of Mittag-Leffler (M-L). Subsequently, by virtue of the controllability Grammian matrix, we explore the relatively exact controllability of linear FSDS. In addition, with the aid of Jensen's inequality, Hölder's inequality, and Itô's isometry, the existence and uniqueness of the considered nonlinear FSDS are investigated by employing Banach contraction principle. Thereafter, the relatively exact controllability of nonlinear FSDS is discussed. Finally, the theoretical results are supported through examples.
{{\copyright} 2023 John Wiley \& Sons, Ltd.}Exact solutions and finite time stability for higher fractional-order differential equations with pure delayhttps://zbmath.org/1538.343092024-08-14T19:23:59.529552Z"Liu, Li"https://zbmath.org/authors/?q=ai:liu.li.7"Dong, Qixiang"https://zbmath.org/authors/?q=ai:dong.qixiang"Li, Gang"https://zbmath.org/authors/?q=ai:li.gang.8Summary: We obtain the exact solutions to the higher fractional-order nonhomogeneous delayed differential equations with Caputo-type fractional derivative by using a set of newly defined generalized delayed Mittag-Leffler matrix functions. The Laplace transform and inductive construction are taken up as the major solving approaches. Thereafter, we consider some special cases and prove that the new exact solutions are suitable for the delayed differential equations with arbitrary order \(\alpha >0\). Additionally, we propose some criteria on the finite time stability of the higher fractional-order delay differential equations. Finally, an illustrative example is presented to test the correctness of the theoretical results.
{{\copyright} 2022 John Wiley \& Sons, Ltd.}The Cauchy problem for the delay differential equation with Dzhrbashyan-Nersesyan fractional derivativehttps://zbmath.org/1538.343102024-08-14T19:23:59.529552Z"Mazhgikhova, Madina Gumarovna"https://zbmath.org/authors/?q=ai:mazhgikhova.madina-gumarovnaSummary: In recent, the number of works devoted to the study of problems for fractional order differential equations is growing noticeably. The interest of researchers is due to the fact that the number of areas of science in which equations containing fractional derivatives are used varies from biology and medicine to control theory, engineering, finance, as well as optics, physics, and so on. The inclusion of delay in the fractional order equation significantly affects the course of the process described by this equation, since the unknown function is given for different values of the argument, which includes a history effect into the equation. Therefore, mathematical models containing a fractional operator and a delay argument are more accurate than models containing integer derivatives. In this paper, we study the Cauchy problem for a linear ordinary delay differential equation with the Dzhrbashyan - Nersesyan fractional differentiation operator, which is generalizing the Riemann-Liouville and Gerasimov-Caputo fractional operators. The results of the work are obtained using the methods of the theory of integer and fractional calculus, methods of the theory of delay differential equations, the method of special functions. In this paper proves a theorem on the validity of an analogue of the Lagrange formula. It is also proved that the special function \(W_{\gamma_m}(t)\), which is defined in terms of the generalized Mittag-Leffler function (or the Prabhakar function), satisfies the equation and conditions associated with the one under study, and is the fundamental solution of the considered equation. The main result is that the existence and uniqueness theorem to the initial value problem is proved. The solution to the problem is written out in terms of the special function \(W_\nu(t)\).Periodic and solitary wave solutions for the one-dimensional cubic nonlinear Schrödinger modelhttps://zbmath.org/1538.353382024-08-14T19:23:59.529552Z"Bica, Ion"https://zbmath.org/authors/?q=ai:bica.ion"Mucalica, Ana"https://zbmath.org/authors/?q=ai:mucalica.anaSummary: Using a similar approach as \textit{D. J. Korteweg} and \textit{G. de Vries} [Phil. Mag. (5) XXXIX, 422--443 (1895; JFM 26.0881.02)], we obtain periodic solutions expressed in terms of the Jacobi elliptic function cn, \textit{M. Abramowitz} and \textit{I. A. Stegun} [A Wiley-Interscience Publication. Selected Government Publications. New York: John Wiley \& Sons, Inc; Washington, D. C.: National Bureau of Standards. xiv, 1046 p. (1984; Zbl 0643.33001)], for the self-focusing and defocusing one-dimensional cubic nonlinear Schrödinger equations. We will show that solitary wave solutions are recovered through a limiting process after the elliptic modulus of the Jacobi elliptic function cn that describes the periodic solutions for the self-focusing nonlinear Schrödinger model.Regularized Prabhakar derivative for partial differential equationshttps://zbmath.org/1538.354212024-08-14T19:23:59.529552Z"Bokhari, Ahmed"https://zbmath.org/authors/?q=ai:bokhari.ahmed"Baleanu, Dumitru"https://zbmath.org/authors/?q=ai:baleanu.dumitru-i"Belgacem, Rachid"https://zbmath.org/authors/?q=ai:belgacem.rachidSummary: Prabhakar fractional operator was applied recently for studying the dynamics of complex systems from several branches of sciences and engineering. In this manuscript, we discuss the regularized Prabhakar derivative applied to fractional partial differential equations using the Sumudu homotopy analysis method(PSHAM). Three illustrative examples are investigated to confirm our main results.Invariant subspaces and exact solutions: \((1+1)\) and \((2+1)\)-dimensional generalized time-fractional thin-film equationshttps://zbmath.org/1538.354442024-08-14T19:23:59.529552Z"Prakash, P."https://zbmath.org/authors/?q=ai:prakash.periasamy|prakash.pradyot|prakash.pandey-prem|prakash.prathibha|prakash.prem|prakash.p-v|prakash.pankaj"Thomas, Reetha"https://zbmath.org/authors/?q=ai:thomas.reetha"Bakkyaraj, T."https://zbmath.org/authors/?q=ai:bakkyaraj.thangarasuSummary: We investigate the applicability and efficiency of the invariant subspace method to \((2 + 1)\)-dimensional time-fractional nonlinear PDEs. We show how to find various types of invariant subspaces and reductions for the \((1 + 1)\) and \((2 + 1)\)-dimensional generalized nonlinear time-fractional thin-film equations which arise from the motion of liquid film on a solid surface under the influence of surface tension. We construct several kinds of exact solutions for the above-mentioned equations depending on arbitrary functions as either a combination of trigonometric, polynomial, Mittag-Leffler, and exponential type functions or any of these forms. Also, we demonstrate the applicability of the invariant subspace method to solve the initial and boundary value problem of nonlinear time-fractional PDEs for the first time and illustrate it through the physically important generalized time-fractional thin-film and linear time-fractional heat equations. Finally, we present some of the obtained exact solutions graphically.Solution of time-fractional equations via Sumudu-Adomian decomposition methodhttps://zbmath.org/1538.354502024-08-14T19:23:59.529552Z"Tarate, Shivaji Ashok"https://zbmath.org/authors/?q=ai:tarate.shivaji-ashok"Bhadane, Ashok P."https://zbmath.org/authors/?q=ai:bhadane.ashok-p"Gaikwad, Shrikisan B."https://zbmath.org/authors/?q=ai:gaikwad.shrikisan-b"Kshirsagar, Kishor Ashok"https://zbmath.org/authors/?q=ai:kshirsagar.kishor-ashokSummary: This paper investigates the semi-analytical solutions of linear and non-linear Time Fractional Klein-Gordon equations with appropriate initial conditions to apply the New Sumudu-Adomian Decomposition method (NSADM). This paper shows the semi-analytical as well as a graphical interpretation of the solution by using mathematical software ``Mathematica Wolform'' and considering Caputo's sense derivatives to semi-analytical results obtained by NSADM.Modified weighted \((0, 1, 3)\)-interpolationhttps://zbmath.org/1538.410022024-08-14T19:23:59.529552Z"Kumar, Shrawan"https://zbmath.org/authors/?q=ai:kumar.shrawan"Mathur, Neha"https://zbmath.org/authors/?q=ai:mathur.nehaSummary: The aim of this paper is to give the existence, uniqueness, and explicit representation of the weighted \((0, 1, 3)\)-interpolation polynomials on the roots all classical orthogonal polynomials.An analysis of \((0, 1, 2; 0)\) polynomial interpolation including interpolation on boundary points of interval \([-1, 1]\)https://zbmath.org/1538.410082024-08-14T19:23:59.529552Z"Singh, Yamini"https://zbmath.org/authors/?q=ai:singh.yamini"Srivastava, R."https://zbmath.org/authors/?q=ai:srivastava.r-n-lal|srivastava.rahul|srivastava.rakesh|srivastava.rajshree|srivastava.ravindra-kumar|srivastava.rajula|srivastava.rajiv-kumar|srivastava.romi|srivastava.rashmi|srivastava.ritu|srivastava.rajendra-kumar|srivastava.r-c|srivastava.richa|srivastava.r-b-l|srivastava.rohan|srivastava.rajesh-kumar|srivastava.rajeev|srivastava.rajendra-p|srivastava.rohit|srivastava.radhendushka|srivastava.r-mohan|srivastava.ravi-k|srivastava.r-s-l|srivastava.ramesh-c|srivastava.ranjana|srivastava.ranjan|srivastava.rekha|srivastava.renu|srivastava.r-j|srivastava.reetaSummary: In this paper, we survey an interpolation on polynomials with Hermite conditions on the zeros of ultraspherical polynomials at interval \([-1,1]\). Our aim is to demonstrate the existence, uniqueness, explicit representation and convergence theorem of the interpolatory polynomials, which are the zeros of the polynomials \(P_n^{(k)} (x)\) and \(P_{n-1}^{(k+1)} (x)\) respectively, where \(P_n^{(k)} (x)\) is the ultraspherical polynomial of degree \(n\).Fourier coefficients for Laguerre-Sobolev type orthogonal polynomialshttps://zbmath.org/1538.420162024-08-14T19:23:59.529552Z"Molano, Alejandro"https://zbmath.org/authors/?q=ai:molano.alejandroSummary: In this paper, the authors take the first step in the study of constructive methods by using Sobolev polynomials.Harmonic analysis associated with the Heckman-Opdam-Jacobi operator on \(\mathbb{R}^{d+1}\)https://zbmath.org/1538.420282024-08-14T19:23:59.529552Z"Bahba, Fida"https://zbmath.org/authors/?q=ai:bahba.fida"Ghabi, Rabiaa"https://zbmath.org/authors/?q=ai:ghabi.rabiaaSummary: In this paper we consider the Heckman-Opdam-Jacobi operator \(\Delta_{HJ}\) on \(\mathbb{R}^{d+1}\). We define the Heckman-Opdam-Jacobi intertwining operator \(V_{HJ}\), which turns out to be the transmutation operator between \(\Delta_{HJ}\) and the Laplacian \(\Delta_{d+1}\). Next we construct \(^tV_{HJ}\) the dual of this intertwining operator. We exploit these operators to develop a new harmonic analysis corresponding to \(\Delta_{HJ}\).Real Paley-Wiener theorems for the special relativistic space-time Fourier transformhttps://zbmath.org/1538.420312024-08-14T19:23:59.529552Z"Fei, Minggang"https://zbmath.org/authors/?q=ai:fei.minggang"Li, Qiu"https://zbmath.org/authors/?q=ai:li.qiuSummary: In this paper, we consider a special relativistic space-time Fourier transform (SFT) in Clifford algebra \(\mathrm{Cl}_{(3,1)}\), which was introduced recently from a mathematical point of view by \textit{E. Hitzer} [Math. Methods Appl. Sci. 42, No. 7, 2244--2255 (2019; Zbl 1414.42008)]. Two versions of real Paley-Wiener theorems for the SFT are established.Fourier transforms of some special functions in terms of orthogonal polynomials on the simplex and continuous Hahn polynomialshttps://zbmath.org/1538.420322024-08-14T19:23:59.529552Z"Güldoğan Lekesiz, Esra"https://zbmath.org/authors/?q=ai:guldogan-lekesiz.esra"Aktaş, Rabia"https://zbmath.org/authors/?q=ai:aktas.rabia"Area, Iván"https://zbmath.org/authors/?q=ai:area.ivanSummary: In this paper, Fourier transform of multivariate orthogonal polynomials on the simplex is presented. A new family of multivariate orthogonal functions is obtained using the Parseval's identity and several recurrence relations are derived.Perturbations of Laguerre-Hahn class linear functionals by Dirac delta derivativeshttps://zbmath.org/1538.420522024-08-14T19:23:59.529552Z"Dueñas, Herbert"https://zbmath.org/authors/?q=ai:duenas.herbert"Garza, Luis E."https://zbmath.org/authors/?q=ai:garza.luis-eSummary: We analyze perturbations of linear functionals (both on the real line and on the unit circle) that belong to the Laguerre-Hahn class. In particular, we obtain an expression for the Stieltjes and Caratheodory functions associated with the perturbed functionals, and we show that the Laguerre-Hahn class is preserved. We also discuss the invariance of the class under the Szegő transformation.The Hausdorff-Young inequality and Freud weightshttps://zbmath.org/1538.420552024-08-14T19:23:59.529552Z"Calderón, C. P."https://zbmath.org/authors/?q=ai:calderon.calixto-p"Torchinsky, A."https://zbmath.org/authors/?q=ai:torchinsky.alberto\textit{Z. Ditzian} proved in [J. Math. Anal. Appl. 398, 582--587 (2013; Zbl 1351.42035)] analogues of the Hausdorff-Young inequality for Freud weights. In particular, he proved that the Freud coefficients belong to a weighted \( \ell^q \) space, where \( q=p,\) the conjugate index to \( p\). Here the authors establish sharpened versions of those results in the sense of replacing \( p^\prime \) by \( q<p^\prime \). They consider expansions in Freud functions and in Freud polynomials, find Lorentz and Orlicz space estimates, and \( n \)-dimensional expansions.
Reviewer: Alexei Lukashov (Moskva)A transformation involving basic multivariable I-function of Prathimahttps://zbmath.org/1538.440042024-08-14T19:23:59.529552Z"Sahni, Nidhi"https://zbmath.org/authors/?q=ai:sahni.nidhi"Kumar, Dinesh"https://zbmath.org/authors/?q=ai:kumar.dinesh"Ayant, F. Y."https://zbmath.org/authors/?q=ai:ayant.frederic-y"Singh, Sunil"https://zbmath.org/authors/?q=ai:singh.sunilSummary: In this document an expansion formula for \(q\)-analogue of multivariable I-function have been given by the applications of the \(q\)-Leibniz law for the \(q\)-derivatives of multiplication of two functions. Expansion formulas concerning the basic I-function of two variables, \(q\)-analogue H function of two variables, basic analogue Meijer G-function of two variables, basic I-function of one variable, basic analogue H-function of one variables, basic analogue Meijer G-function of one variables were given as special cases of the main formula.Triple series equations involving generalized Laguerre polynomialshttps://zbmath.org/1538.450012024-08-14T19:23:59.529552Z"Lal Shrivastava, Omkar"https://zbmath.org/authors/?q=ai:lal-shrivastava.omkar"Narain, Kuldeep"https://zbmath.org/authors/?q=ai:narain.kuldeep"Shrivastava, Sumita"https://zbmath.org/authors/?q=ai:shrivastava.sumitaSummary: In this paper, the solutions of two sets of triple series equations involving generalized Laguerre polynomials have been obtained by reducing them to a Fredholm integral equation of second kind. In each case, the problem is reduced to the solution of a Fredholm integral equation of the second kind. We consider certain triple series equations involving generalized Laguerre polynomials which are generalization of those considerd by Lawndes-Srivastava. Connected to this work solutions have been considered by
\textit{I. N. Sneddon} [Mixed boundary value problems in potential theory. (1966; Zbl 0139.28801)], \textit{J. S. Lowndes} and \textit{H. M. Srivastava} [J. Math. Anal. Appl. 150, No. 1, 181--187 (1990; Zbl 0687.45004)], \textit{A. P. Dwivedi} and \textit{T. N. Trivedi} [Indian J. Pure Appl. Math. 7, 951--960 (1976; Zbl 0407.33008)], \textit{B. M. Singh} et al. [Ukr. Mat. Zh. 62, No. 2, 231--237 (2010; Zbl 1224.33007)], \textit{K. Narain} [``Certain simultaneous triple series equations involving Laguerre polynomials'', Math. Theory Model. 12, No. 3, 129--131 (2013), \url{https://www.iiste.org/Journals/index.php/MTM/article/view/8444}], \textit{H. M. Srivastava} and \textit{R. Panda} [Indag. Math. 40, 502--514 (1978; Zbl 0402.42016)], etc.Exact solutions of one nonlinear countable-dimensional system of integro-differential equationshttps://zbmath.org/1538.450042024-08-14T19:23:59.529552Z"Rassadin, Aleksandr Èduardovich"https://zbmath.org/authors/?q=ai:rassadin.aleksandr-eduardovichSummary: In the present paper, a nonlinear countable-dimensional system of integro-differential equations is investigated, whose vector of unknowns is a countable set of functions of two variables. These variables are interpreted as spatial coordinate and time. The nonlinearity of this system is constructed from two simultaneous convolutions: first convolution is in the sense of functional analysis and the second one is in the sense of linear space of double-sided sequences. The initial condition for this system is a double-sided sequence of functions of one variable defined on the entire real axis. The system itself can be written as a single abstract equation in the linear space of double-sided sequences. As the system may be resolved with respect to the time derivative, it may be presented as a dynamical system. The solution of this abstract equation can be interpreted as an approximation of the solution of a nonlinear integro-differential equation, whose unknown function depends not only on time, but also on two spatial variables. General representation for exact solution of system under study is obtained in the paper. Also two kinds of particular examples of exact solutions are presented. The first demonstrates oscillatory spatio-temporal behavior, and the second one shows monotone in time behavior. In the paper typical graphs of the first components of these solutions are plotted. Moreover, it is demonstrated that using some procedure one can generate countable set of new exact system's solutions from previously found solutions. From radio engineering point of view this procedure just coincides with procedure of upsampling in digital signal processing.A feedforward neural network based on Legendre polynomial for solving linear Fredholm integro-differential equationshttps://zbmath.org/1538.450242024-08-14T19:23:59.529552Z"Shao, Xinping"https://zbmath.org/authors/?q=ai:shao.xinping"Yang, Liupan"https://zbmath.org/authors/?q=ai:yang.liupan"Guo, Anqi"https://zbmath.org/authors/?q=ai:guo.anqiSummary: This paper presents a method for solving linear Fredholm integro-differential equations using a feedforward neural network based on Legendre polynomials. Firstly, the Legendre polynomials are used to approximate the unknown function and the kernel function in the equations. Secondly, The roots of Legendre polynomials are obtained by using Newton iteration method to obtain Gaussian integration points, and the obtained Gaussian integration points are used as the input nodes of the neural network, and the corresponding weight is learned by the gradient descent method to obtain an approximate solution. Finally, through the numerical value Case analysis to verify the effectiveness of the method.On the generalized Fresnel sine integrals and convolutionhttps://zbmath.org/1538.460482024-08-14T19:23:59.529552Z"Lazarova, Limonka"https://zbmath.org/authors/?q=ai:lazarova.limonka-koceva"Jolevska-Tuneska, Biljana"https://zbmath.org/authors/?q=ai:jolevska-tuneska.biljanaSummary: The generalized Fresnel sine integral \(S_k(x)\) and its associated functions \(S_{k+}(x)\), \(S_{k-}(x)\) are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of the generalized Fresnel sine integral and its associated functions are then found.Asymptotic behavior of the Laplace transform near the originhttps://zbmath.org/1538.460512024-08-14T19:23:59.529552Z"Nemzer, Dennis"https://zbmath.org/authors/?q=ai:nemzer.dennisSummary: The unilateral Laplace transform is extended to a space of generalized functions \(\mathcal{B}_L\) which contains the space of transformable distributions supported on the interval \([0,\infty)\). The Mittag-Leffler functions are found to be useful in comparing the asymptotic behavior of an element of \(\mathcal{B}_L\) at infinity to the asymptotic behavior of its transform at a singularity.Norm estimates for the pseudo-differential operator involving fractional Hankel-like transform on \(\mathcal{S}\)-type spaceshttps://zbmath.org/1538.470792024-08-14T19:23:59.529552Z"Mahato, Kanailal"https://zbmath.org/authors/?q=ai:mahato.kanailal"Pasawan, Durgesh"https://zbmath.org/authors/?q=ai:pasawan.durgeshSummary: In this article, we focus on the continuity as well as boundedness properties of fractional Hankel-like transform and pseudo-differential operator associated with a fractional Hankel-like transform on some suitably designed Gelfand-Shilov spaces of type \(\mathcal{S}\). Moreover, we have further investigated on certain class of ultradifferentiable function spaces for the above integral transform and operator.The deltoid curve and triangle transformationshttps://zbmath.org/1538.510162024-08-14T19:23:59.529552Z"Rieck, Michael Q."https://zbmath.org/authors/?q=ai:rieck.michael-qSummary: Deltoid curves appear as consequences of certain procedures in triangle geometry. The best known of these is the construction based on Simson lines, described by Steiner. This is carefully related, in this article, to a less known construction. The standard deltoid in the complex plane and its tangent lines are principle objects of study in this report. It is known that each point in the interior of this curve is the orthocenter of a triangle with distinct vertices on the unit circle, whose product is one. (If instead the point is on the deltoid, then at least two of the vertices coalesce, resulting in a degenerate triangle.) \par When the vertices are all raised to some specified integer power, a new (possibly degenerate) triangle results. By varying the triangle, one may thus consider the map taking the original triangle's orthocenter to the resulting triangle's orthocenter. Such maps are the other principle objects of study here. The points that are mapped to the deltoid lie on easily described curves. By varying the power involved in the map, a pleasing family of curves results, which includes a trifolium curve. The points that are mapped instead to the origin are described as the points of intersection of certain tangents to the deltoid.Asymptotic behaviour of \(d\)-variate absorption distributions, \(d = 1, 2\)https://zbmath.org/1538.600112024-08-14T19:23:59.529552Z"Vamvakari, Malvina"https://zbmath.org/authors/?q=ai:vamvakari.malvina-gSummary: In this work we study the asymptotic behavior of univariate and bivariate absorption discrete \(q\)-distributions. Specifically, the pointwise convergence of the univariate absorption distribution to a deformed Gaussian one and that of the bivariate absorption to a bivariate deformed Gaussian one, are provided.An extended version of hyper Poisson distribution and some of its applicationshttps://zbmath.org/1538.600272024-08-14T19:23:59.529552Z"Satheesh Kumar, C."https://zbmath.org/authors/?q=ai:kumar.c-satheesh"Unnikrishnan Nair, B."https://zbmath.org/authors/?q=ai:unnikrishnan-nair.b(no abstract)Some notes on the four-parameter Kies distributionhttps://zbmath.org/1538.600332024-08-14T19:23:59.529552Z"Zaevski, Tsvetelin S."https://zbmath.org/authors/?q=ai:zaevski.tsvetelin-stefanov"Kyurkchiev, Nikolay"https://zbmath.org/authors/?q=ai:kyurkchiev.nikolay-vIn the paper under review, the four-parameter Kies distribution is investigated. The authors propose the correct statement of the results for the moments and the mean residual life function. First, some results on the hypergeometric functions are presented. In Proposition 3.1, an alternative form of the mean residual life function is obtained. Proposition 3.2 presents the correct statement of the moments and the mean residual life function. Finally, the moment generating function of a four-parameter Kies distributed random variable is obtained.
Reviewer: Angela Slavova (Sofia)Functional convergence of Berry's nodal lengths: approximate tightness and total disorderhttps://zbmath.org/1538.600732024-08-14T19:23:59.529552Z"Notarnicola, Massimo"https://zbmath.org/authors/?q=ai:notarnicola.massimo"Peccati, Giovanni"https://zbmath.org/authors/?q=ai:peccati.giovanni"Vidotto, Anna"https://zbmath.org/authors/?q=ai:vidotto.annaSummary: We consider \textit{M. V. Berry}'s random planar wave model [J. Phys. A, Math. Gen. 10, 2083--2091 (1977; Zbl 0377.70014)], and prove spatial functional limit theorems -- in the high-energy limit -- for discretized and truncated versions of the random field obtained by restricting its nodal length to rectangular domains. Our analysis is crucially based on a detailed study of the projection of nodal lengths onto the so-called \textit{second Wiener chaos}, whose high-energy fluctuations are given by a Gaussian \textit{total disorder field} indexed by polygonal curves. Such an exact characterization is then combined with moment estimates for suprema of stationary Gaussian random fields, and with a tightness criterion by \textit{Y. Davydov} and \textit{R. Zitikis} [Ann. Inst. Stat. Math. 60, No. 2, 345--365 (2008; Zbl 1333.60098)].Characterization theorems for the \(B\)-\(q\)-binomial and the \(q\)-Poisson distributionshttps://zbmath.org/1538.621672024-08-14T19:23:59.529552Z"Bouzida, Imed"https://zbmath.org/authors/?q=ai:bouzida.imedSummary: In this paper, the \(q\)-binomial and the \(q\)-hypergeometric distributions are redefined and re-introduced in the compact form. The redefined distributions are named \(B\)-\(q\)-binomial and \(B\)-\(q\)-hypergeometric. Furthermore, the generalization of the well-known Patil and Seshadri characterization is reported in the \(q\)-calculus. The characterizations of \(B\)-\(q\)-binomial and \(B\)-\(q\)-hypergeometric distributions are presented by using a conditional \(q\)-distribution. A necessary and sufficient condition identifying the \(q\)-Poisson distribution is outlined.Numerical solutions of Hadamard fractional differential equations by generalized Legendre functionshttps://zbmath.org/1538.651762024-08-14T19:23:59.529552Z"Istafa, Ghafirlia"https://zbmath.org/authors/?q=ai:istafa.ghafirlia"Rehman, Mujeeb ur"https://zbmath.org/authors/?q=ai:ur-rehman.mujeebSummary: This paper aims to develop a numerical method for the solutions of Hadamard fractional differential equations. We introduced Hadamard fractional Legendre functions by modifying classical Legendre polynomials and used them to approximate the numerical solutions of linear and nonlinear Hadamard fractional differential equations. The solution is approached by approximating the appropriate terms in Hadamard fractional differential equations using Hadamard fractional Legendre functions and converting the problem to a system of algebraic equations. Quasilinearization technique is employed to linearize the nonlinear Hadamard fractional differential equations. An upper bound for approximation error is derived. This method provides reasonably accurate results for a relatively smaller order of Hadamard fractional Legendre functions.
{{\copyright} 2022 John Wiley \& Sons, Ltd.}A new method of solving the best approximate solution for a nonlinear fractional equationhttps://zbmath.org/1538.653102024-08-14T19:23:59.529552Z"Du, Hong"https://zbmath.org/authors/?q=ai:du.hong"Yang, Xinyue"https://zbmath.org/authors/?q=ai:yang.xinyue"Chen, Zhong"https://zbmath.org/authors/?q=ai:chen.zhongSummary: A new method of solving the best approximate solution for nonlinear fractional equations with smooth and nonsmooth solutions in reproducing kernel space is proposed in the paper. The nonlinear equation outlines some important equations, such as fractional diffusion-wave equation, nonlinear Klein-Gordon equation and time-fractional sine-Gordon equation. By constructing orthonormal bases in reproducing kernel space using Legendre orthonormal polynomials and Jacobi fractional orthonormal polynomials, the best approximate solution is obtained by searching the minimum of residue in the sense of \(\|\cdot\|_C\). Numerical experiments verify that the method has higher accuracy.Shifted Jacobi collocation scheme for multidimensional time-fractional order telegraph equationhttps://zbmath.org/1538.654162024-08-14T19:23:59.529552Z"Hafez, R. M."https://zbmath.org/authors/?q=ai:hafez.ramy-mahmoud"Youssri, Y. H."https://zbmath.org/authors/?q=ai:youssri.y-hSummary: We propose a numerical scheme to solve a general class of time-fractional order telegraph equation in multidimensions using collocation points nodes and approximating the solution using double shifted Jacobi polynomials. The main characteristic behind this approach is to investigate a time-space collocation approximation for temporal and spatial discretizations. The applica bility and accuracy of the present technique have been examined by the given numerical examples in this paper. By means of these numerical examples, we ensure that the present technique is simple, applicable, and accurate.A novel shifted Jacobi operational matrix method for nonlinear multi-terms delay differential equations of fractional variable-order with periodic and anti-periodic conditionshttps://zbmath.org/1538.654192024-08-14T19:23:59.529552Z"Khodabandelo, Hamid R."https://zbmath.org/authors/?q=ai:khodabandelo.hamid-r"Shivanian, Elyas"https://zbmath.org/authors/?q=ai:shivanian.elyas"Abbasbandy, Saeid"https://zbmath.org/authors/?q=ai:abbasbandy.saeidSummary: This work presents the generalized nonlinear multi-terms fractional variable-order delay differential equation with periodic and anti-periodic conditions. In this paper, a novel shifted Jacobi operational matrix technique is introduced to solve a class of these equations mentioned, so that the main problem becomes a system of algebraic equations that we can solve numerically. The suggested technique is successfully developed for the aforementioned problem. Comprehensive numerical tests are provided to demonstrate the generality, efficiency, accuracy of presented scheme and the flexibility of this technique. The numerical experiments compared it with the true solution, indicating the validity and efficiency of this scheme. Note that the procedure is easy to implement and this technique will be considered as a generalization of many numerical schemes. Furthermore, the error and its bound are estimated.
{{\copyright} 2022 John Wiley \& Sons, Ltd.}Modified Legendre rational spectral method for Burgers equation on the whole linehttps://zbmath.org/1538.654322024-08-14T19:23:59.529552Z"Tian, Liutao"https://zbmath.org/authors/?q=ai:tian.liutao"Jiao, Yujian"https://zbmath.org/authors/?q=ai:jiao.yujianSummary: In this paper, we propose a spectral method for the Burgers equation using the modified Legendre rational functions, and prove its generalized stability and convergence. Numerical results demonstrate the efficiency of the new approach.Composite spectral method for the Neumann problem of the Burgers equation on the half linehttps://zbmath.org/1538.654362024-08-14T19:23:59.529552Z"Wang, Tian-jun"https://zbmath.org/authors/?q=ai:wang.tianjun"Chai, Guo"https://zbmath.org/authors/?q=ai:chai.guoSummary: A composite spectral method that exactly satisfies the homogeneous Neumann boundary condition is presented for the Burgers equation on the half line. The composite method composes strengths of traditional single methods. Some new composite quasi-orthogonal approximation results are established. The composite spectral schemes are provided for a linear model problem and the Burgers equation with the Neumann boundary condition. Its convergence and stability are strictly proved. Numerical results are given to show the efficiency of the approach and agree well with theory analysis.A numerical method for solving systems of hypersingular integro-differential equationshttps://zbmath.org/1538.656102024-08-14T19:23:59.529552Z"De Bonis, Maria Carmela"https://zbmath.org/authors/?q=ai:de-bonis.maria-carmela"Mennouni, Abdelaziz"https://zbmath.org/authors/?q=ai:mennouni.abdelaziz"Occorsio, Donatella"https://zbmath.org/authors/?q=ai:occorsio.donatellaSummary: This paper is concerned with a collocation-quadrature method for solving systems of Prandtl's integro-differential equations based on de la Vallée Poussin filtered interpolation at Chebyshev nodes. We prove stability and convergence in Hölder-Zygmund spaces of locally continuous functions. Some numerical tests are presented to examine the method's efficacy.Bicomplex neural networks with hypergeometric activation functionshttps://zbmath.org/1538.680582024-08-14T19:23:59.529552Z"Vieira, Nelson"https://zbmath.org/authors/?q=ai:vieira.nelsonSummary: Bicomplex convolutional neural networks (BCCNN) are a natural extension of the quaternion convolutional neural networks for the bicomplex case. As it happens with the quaternionic case, BCCNN has the capability of learning and modelling external dependencies that exist between neighbour features of an input vector and internal latent dependencies within the feature. This property arises from the fact that, under certain circumstances, it is possible to deal with the bicomplex number in a component-wise way. In this paper, we present a BCCNN, and we apply it to a classification task involving the colourized version of the well-known dataset MNIST. Besides the novelty of considering bicomplex numbers, our CNN considers an activation function a Bessel-type function. As we see, our results present better results compared with the one where the classical ReLU activation function is considered.Quaternionic convolutional neural networks with trainable Bessel activation functionshttps://zbmath.org/1538.680592024-08-14T19:23:59.529552Z"Vieira, Nelson"https://zbmath.org/authors/?q=ai:vieira.nelsonSummary: Quaternionic convolutional neural networks (QCNN) possess the ability to capture both external dependencies between neighboring features and internal latent dependencies within features of an input vector. In this study, we employ QCNN with activation functions based on Bessel-type functions with trainable parameters, for performing classification tasks. Our experimental results demonstrate that this activation function outperforms the traditional \textit{ReLU} activation function. Throughout our simulations, we explore various network architectures. The use of activation functions with trainable parameters offers several advantages, including enhanced flexibility, adaptability, improved learning, customized model behavior, and automatic feature extraction.The bending contact problem of elastic rectangular plate with rigid inclusionhttps://zbmath.org/1538.740952024-08-14T19:23:59.529552Z"Pachulia, Bachuki"https://zbmath.org/authors/?q=ai:pachulia.bachukiSummary: In the article bending problem of elastic rectangular plate with rigid inclusion is considered. The problem is formulated in the form of the integral equation, with respect to the jump of lateral force. Using the method of orthogonal polynomial, the integral equation is reduced to the infinite system of linear algebraic equations. Quasi-regularity of this system is proved.The inapplicability of variational methods in the cohesive crack problem for initially rigid traction-separation relation and its solution using integral equationshttps://zbmath.org/1538.741282024-08-14T19:23:59.529552Z"Singh, Gaurav"https://zbmath.org/authors/?q=ai:singh.gauravSummary: A cohesive crack problem with initially rigid traction-separation relation has been considered here. It has been shown that the total potential energy is not differentiable at the crack-tip. This makes the application of a variational operator over this total potential energy theoretically incorrect in the general sense. It further implies that variational methods are not applicable in this situation. One way to overcome this problem is to introduce a penalty of initial stiffness in the traction-separation relationship. However, this means modifying the material property of a solid. To avoid variational operator, an integral equation form of this problem has been numerically solved. The results thus obtained conform to the present understanding of cohesive stresses. This method is theoretically more accurate to solve a cohesive crack problem whenever the total potential energy is non-differentiable.
{\copyright} 2023 Wiley-VCH GmbH.On self-organization structure for fluid dynamical systems via solitary waveshttps://zbmath.org/1538.760372024-08-14T19:23:59.529552Z"Tenkam, H. M."https://zbmath.org/authors/?q=ai:tenkam.h-m"Doungmo Goufo, E. F."https://zbmath.org/authors/?q=ai:doungmo-goufo.emile-franc"Kumar, S."https://zbmath.org/authors/?q=ai:kumar.sunil.3Summary: The process of self-organization occurs and is used in many aspects of life with applications found in domains of biological, physical and machining systems. Finding ways to create this kind of processes has attracted the interest of many scientists around the world. We combine in this paper some mathematical concepts to model and generate the self-organization process happening in wave motion. We make use of the Harry Dym system together with the fractal and fractional operators. The resulting model is solved numerically and its stability results are provided. Numerical simulations show the combined system involved in a self-organization dynamic with the replication of the initial objects and the formation of subsequent fractal patterns which vary with the fractional operator. The results prove that we are in the presence of a system capable of artificially structuring fractals using mathematical concepts, numerical techniques, codes and simulations.Analytical algorithms for direct and inverse problems pertaining to the electromagnetic excitation of a layered medium by \(N\) dipoleshttps://zbmath.org/1538.780102024-08-14T19:23:59.529552Z"Kalogeropoulos, Andreas"https://zbmath.org/authors/?q=ai:kalogeropoulos.andreas"Tsitsas, Nikolaos L."https://zbmath.org/authors/?q=ai:tsitsas.nikolaos-lSummary: The direct problem of exciting a multilayered spherical medium by a distribution of arbitrarily located (internal and external) \( N \) electric dipoles is considered. A formulation based on excitation operators is adopted. The solution of the direct problem is obtained by means of an overall superposition method that combines elements of the T-matrix approach, the Sommerfeld's method, and Green's functions' techniques. Approximations are extracted for the obtained far-field patterns in the long-wavelength (low-frequency) regime. A set of source-localization inverse problems is considered. Analytical algorithms are devised in which nonlinear systems are obtained from specifically selected low-frequency far-field measurements. These systems are subsequently solved explicitly including all special cases. The developed algorithms are noninvasive and can determine accurately the characteristics of internal sources.
{\copyright} 2023 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley \& Sons Ltd.A qualitative analysis of bifocusing method for a real-time anomaly detection in microwave imaginghttps://zbmath.org/1538.780112024-08-14T19:23:59.529552Z"Kang, Sangwoo"https://zbmath.org/authors/?q=ai:kang.sangwoo"Park, Won-Kwang"https://zbmath.org/authors/?q=ai:park.won-kwang"Son, Seong-Ho"https://zbmath.org/authors/?q=ai:son.seong-hoSummary: In this contribution, we consider the bifocusing method (BFM) for identifying small anomalies from scattering parameter data in microwave imaging. To this end, we design an imaging function of BFM and perform a qualitative analysis for the imaging function by establishing a relationship with the infinite series of Bessel functions of integer order, applied frequency, material properties, and the antenna arrangement. The revealed mathematical structure of the imaging function and simulation results with synthetic data show why BFM is fast and effective for identifying unknown anomalies in microwave imaging.The semiclassical limit of a quantum Zeno dynamicshttps://zbmath.org/1538.810242024-08-14T19:23:59.529552Z"Cunden, Fabio Deelan"https://zbmath.org/authors/?q=ai:cunden.fabio-deelan"Facchi, Paolo"https://zbmath.org/authors/?q=ai:facchi.paolo"Ligabò, Marilena"https://zbmath.org/authors/?q=ai:ligabo.marilenaSummary: Motivated by a quantum Zeno dynamics in a cavity quantum electrodynamics setting, we study the asymptotics of a family of symbols corresponding to a truncated momentum operator, in the semiclassical limit of vanishing Planck constant \(\hbar \rightarrow 0\) and large quantum number \(N\rightarrow \infty\), with \(\hbar N\) kept fixed. In a suitable topology, the limit is the discontinuous symbol \(p\chi_D (x,p)\) where \(\chi_D\) is the characteristic function of the classically permitted region \(D\) in phase space. A refined analysis shows that the symbol is asymptotically close to the function \(p\chi_D^{(N)}(x,p)\), where \(\chi_D^{(N)}\) is a smooth version of \(\chi_D\) related to the integrated Airy function. We also discuss the limit from a dynamical point of view.Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearityhttps://zbmath.org/1538.810282024-08-14T19:23:59.529552Z"Finco, Domenico"https://zbmath.org/authors/?q=ai:finco.domenico"Tentarelli, Lorenzo"https://zbmath.org/authors/?q=ai:tentarelli.lorenzo"Teta, Alessandro"https://zbmath.org/authors/?q=ai:teta.alessandroSummary: We discuss \textit{strong} local and global well-posedness for the three-dimensional NLS equation with nonlinearity concentrated on \(\mathbb{S}^2\). Precisely, local well-posedness is proved for any \(C^2\) power-nonlinearity, while global well-posedness is obtained either for small data or in the defocusing case under some growth assumptions. With respect to point-concentrated NLS models, widely studied in the literature, here the dimension of the support of the nonlinearity does not allow a direct extension of the known techniques and calls for new ideas.
{{\copyright} 2023 IOP Publishing Ltd \& London Mathematical Society}Exactly solvable model of the one-dimensional confined harmonic oscillatorhttps://zbmath.org/1538.810302024-08-14T19:23:59.529552Z"Jafarov, Elchin I."https://zbmath.org/authors/?q=ai:jafarov.elchin-i"Jafarova, Aynura M."https://zbmath.org/authors/?q=ai:jafarova.aynura-m"Oste, Roy"https://zbmath.org/authors/?q=ai:oste.roySummary: An exactly solvable model of the one-dimensional quantum harmonic oscillator confined in a box with infinite walls is constructed. We have found explicit expressions of the non-equidistant energy spectrum as well as stationary states wavefunctions in both
momentum and position configuration spaces. It is shown that they are expressed through continuous \(q\)-Hermite polynomials. We have also found an explicit expression for the kernel of the finite-continuous Fourier transform between these two representation spaces.Spectrum and orthogonality of the Bethe ansatz for the periodic \(q\)-difference Toda chain on \(\mathbb{Z}_{m+1}\)https://zbmath.org/1538.810312024-08-14T19:23:59.529552Z"van Diejen, Jan Felipe"https://zbmath.org/authors/?q=ai:van-diejen.jan-felipeSummary: By means of a \(q\)-boson-\(q\)-Toda correspondence pointed out by \textit{A. Duval} and \textit{V. Pasquier} [J. Phys. A, Math. Theor. 49, No. 15, Article ID 154006, 25 p. (2016; Zbl 1351.37257)], the \(n\)-particle hamiltonian of the periodic quantum relativistic Toda chain on \(\mathbb{Z}_{m+1}\) is mapped to the hamiltonian of a previously studied lattice discretization of the Lieb-Liniger model (which encodes the dynamics of \(m+1\) \(q\)-bosons on \(\mathbb{Z}_{n+1}\) in the center-of-mass frame). The map in question makes it possible to retrieve quantum integrals and an orthogonal eigenbasis of Bethe Ansatz wave functions given by Hall-Littlewood polynomials for the pertinent periodic \(q\)-difference Toda chain from the corresponding quantum integrals and eigenbasis for the lattice Lieb-Liniger model. This approach entails the spectrum of the periodic \(q\)-difference Toda chain in terms of the critical points of associated Yang-Yang type Morse functions and links the diagonalization via the algebraic Bethe Ansatz performed by A. Duval and V. Pasquier [loc. cit.] directly to the spectral analysis of the lattice Lieb-Liniger model.Bethe ansatz diagonalization of the Heun-Racah operatorhttps://zbmath.org/1538.820102024-08-14T19:23:59.529552Z"Bernard, Pierre-Antoine"https://zbmath.org/authors/?q=ai:bernard.pierre-antoine"Carcone, Gauvain"https://zbmath.org/authors/?q=ai:carcone.gauvain"Crampé, Nicolas"https://zbmath.org/authors/?q=ai:crampe.nicolas"Vinet, Luc"https://zbmath.org/authors/?q=ai:vinet.lucSummary: The diagonalization of the Heun-Racah operator is studied with the help of the modified algebraic Bethe ansatz. This operator is the most general bilinear expression in two generators of the Racah algebra. A presentation of this algebra is given in terms of dynamical operators and allows the construction of Bethe vectors for the Heun-Racah operator. The associated Bethe equations are derived for both the homogeneous and inhomogeneous cases.Numerical simulation of bifurcational curves in long non-homogeneous Josephson junctionshttps://zbmath.org/1538.820732024-08-14T19:23:59.529552Z"Atanasova, P. Kh."https://zbmath.org/authors/?q=ai:atanasova.pavlina-kh"Bojadjiev, T. L."https://zbmath.org/authors/?q=ai:bojadjiev.t-lSummary: The bifurcations of the solutions of multiparametric nonlinear boundary problem in Physics of Josephson junctions (JJ) are investigated numerically. Two cases are considered: JJ with overlap geometry and with in-line geometry. In order to study the stability of the solution of the nonlinear boundary problem with respect to small ``space-time'' perturbations a Sturm-Liouville problem generated from this solution is considered. The bifurcational points are calculated by the continuous analogue of the method of Newton. A good coincidence of the numerical results and experimental data is received. Some numerical results and an investigation of the critical curves are illustrated graphically.