Recent zbMATH articles in MSC 33https://zbmath.org/atom/cc/332022-11-17T18:59:28.764376ZUnknown authorWerkzeugMathieu-Fibonacci serieshttps://zbmath.org/1496.110312022-11-17T18:59:28.764376Z"Tomovski, Živorad"https://zbmath.org/authors/?q=ai:tomovski.zivorad"Gerhold, Stefan"https://zbmath.org/authors/?q=ai:gerhold.stefanThe purpose of this article is to give series representations for generalized Mathieu series via Lambert-type series. The proofs use formulas by \textit{H. M. Srivastava} and the first author [JIPAM, J. Inequal. Pure Appl. Math. 5, No. 2, Paper No. 45, 13 p. (2004; Zbl 1068.33032)] and Jacobi theta function formulas. Furthermore, integral representations of series of reciprocal Fibonacci numbers are given by using monotone sequences increasing to infinity. Similarly, integral representations for series of reciprocal Lucas numbers and generalized Mathieu series are derived. Finally, corresponding asymptotic expansions of Mathieu-Fibonacci series are proved by using Mellin transform.
Reviewer: Thomas Ernst (Uppsala)A new \(q\)-extension of the (H.2) congruence of Van Hamme for primes \(p\equiv 1\pmod{4}\)https://zbmath.org/1496.110332022-11-17T18:59:28.764376Z"Wang, Chen"https://zbmath.org/authors/?q=ai:wang.chen.1|wang.chenThis work enriches the impressive list of papers inspired by the famous congruence (H.2) from \textit {L. van Hamme} [Lect. Notes Pure Appl. Math. 192, 223--236 (1997; Zbl 0895.11051)]
\[
\sum^{(p-1)/2}_{k=0} \frac{\left( \frac{1}{2}\right)^3_k}{k!^3} \equiv \begin{cases} -\Gamma_p \left( \frac{1}{4}\right)^4 \quad \pmod{p^2} & \text{if } p\equiv 1 \pmod{4},\\
0 \pmod{p^2} & \text{if } p\equiv 3 \pmod{4}, \end{cases}
\]
where \(p\) is an odd prime, \((x)_k=x(x+1) \cdots (x+k-1)\) represents the Pochhammer symbol, and \(\Gamma_p (x)\) is the \(p\)-adic Gamma function.
More specifically, the author is motivated by the refinement of \textit {L. Long} and \textit {R. Ramakrishna} [Adv. Math. 290, 773--808 (2016; Zbl 1336.33018)]
\[
\sum^{(p-1)/2}_{k=0} \frac{\left( \frac{1}{2}\right)^3_k}{k!^3} \equiv \begin{cases} -\Gamma_p \left( \frac{1}{4}\right)^4 \quad \pmod{p^3} & \text{if } p\equiv 1 \pmod{4},\\
-\frac{p^2}{16} \Gamma_p \left( \frac{1}{4}\right)^4 \quad \pmod{p^3} & \text{if } p\equiv 3 \pmod{4}, \end{cases}
\]
whose \(q\)-extension of the case \(p\equiv 1 \pmod{4}\) is the main purpose of this paper.
Recalling a \(q\)-congruence found by \textit{V. J. W. Guo} [Result. Math. 76, No. 2, Paper No. 109, 12 p. (2021; Zbl 1470.11034)], the author establishes that, for \(n\equiv 1 \pmod{4}\) and \(n>1\),
\[
\sum_{k=0}^{(n-1)/2} \frac{(q;q^2)_k^2 (q^2;q^4)_k}{(q^2;q^2)_k^2 (q^4;q^4)_k}q^{2k} \equiv [n] \frac{(q^3;q^4)_{(n-1)/2}}{(q^5;q^4)_{(n-1)/2}}+[n]^3 \sum_{k=0}^{(n-3)/2} \frac{(1+q^{2k+1}) (q^3;q^4)_k}{[2k+1]^2 (q^5;q^4)_k}q^{2k+1} \pmod{\Phi_n(q)^3},
\]
where \([n]=(1-q^n)/(1-q)\) is the \(q\)-integer, \((x;q)_n\) is the \(q\)-Pochhammer symbol, and \(\Phi_n(q)\) is the \(n\)th cyclotomic polynomial in \(q\).
The author remarks that such achievement differs from a similar result due to \textit{C. Wei} [Result. Math. 76, No. 2, Paper No. 92, 9 p. (2021; Zbl 1470.33015)].
The author also proves two auxiliary theorems by employing the substitution \(q^n=1+t(q) \Phi_n(q)\), an identity about the \(n\)th harmonic numbers \(H_n^{(m)}\) of order \(m\) he computed via \texttt{SIGMA}, and additional congruences involving \(H_n^{(m)}\) supplied by himself in collaboration with \textit {H. Pan} [Math. Z. 300, No. 1, 161--177 (2022; Zbl 07463785)].
Reviewer: Enzo Bonacci (Latina)Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbershttps://zbmath.org/1496.110422022-11-17T18:59:28.764376Z"Qi, Feng"https://zbmath.org/authors/?q=ai:qi.feng"Dağlı, Muhammet Cihat"https://zbmath.org/authors/?q=ai:dagli.muhammet-cihat"Lim, Dongkyu"https://zbmath.org/authors/?q=ai:lim.dongkyuSummary: In this paper, with the aid of the Faà di Bruno formula and by virtue of properties of the Bell polynomials of the second kind, the authors define a kind of notion of degenerate Narumi numbers and polynomials, establish explicit formulas for degenerate Narumi numbers and polynomials, and derive explicit formulas for the Narumi numbers and polynomials and for (degenerate) Cauchy numbers.Evaluations of the Rogers-Ramanujan continued fraction by theta-function identitieshttps://zbmath.org/1496.110662022-11-17T18:59:28.764376Z"Paek, Dae Hyun"https://zbmath.org/authors/?q=ai:paek.dae-hyunSummary: In this paper, we use theta-function identities involving parameters \(l_{5,n}, l_{5,n}^{\prime}\), and \(l_{5,4n}^{\prime}\) to evaluate the Rogers-Ramanujan continued fractions \(R(e^{-2\pi \sqrt{n/20}})\) and \(S(e^{-\pi \sqrt{n/5}})\) for some positive rational numbers \(n\).The \({{\mathbb{F}}}_p\)-Selberg integralhttps://zbmath.org/1496.130082022-11-17T18:59:28.764376Z"Rimányi, Richárd"https://zbmath.org/authors/?q=ai:rimanyi.richard"Varchenko, Alexander"https://zbmath.org/authors/?q=ai:varchenko.alexander-nThe authors present three proofs of a formula for computing the Selberg integral over a finite field \(\mathbb{F}_p\) with \(p\) odd. They also present a formula for the \(\mathbb{F}_p\)-Aomata integral which is proved along with their first proof of the \(\mathbb{F}_p\)-Selberg integral formula. They include a nice visulaization of \(S_n(a,b,3)\) for \(n \in \{1,2,3\}\) and \(p=11\) which they used in their first two proofs of the \(\mathbb{F}_p\)-Selberg integral formula.
The solutions to KZ equations over \(\mathbb{C}\) are generalizations of the general Selberg integral. In the last section of the paper, they relate the \(\mathbb{F}_p\)-Selberg integrals to solutions of KZ equations over \(\mathbb{F}_p\).
Reviewer: Janet Vassilev (Albuquerque)On the mixed-twist construction and monodromy of associated Picard-Fuchs systemshttps://zbmath.org/1496.140382022-11-17T18:59:28.764376Z"Malmendier, Andreas"https://zbmath.org/authors/?q=ai:malmendier.andreas"Schultz, Michael T."https://zbmath.org/authors/?q=ai:schultz.michael-t\textit{C. F. Doran} and \textit{A. Malmendier} [Adv. Theor. Math. Phys. 23, No. 5, 1271--1359 (2019; Zbl 1478.14025)] have introduced an iterative mixed-twist construction (MTC) of families of Jacobian elliptic Calabi-Yau \(n\)-folds from a family of Jacobian elliptic Calabi-Yau \((n-1)\)-folds for all \(n\geq 2\). The authors of the present paper use this construction to obtain a multi-parameter family of \(K3\) surfaces of Picard rank \(\rho \geq 16\). Upon identifying a particular Jacobian elliptic fibration on its general member, they determine the lattice polarization and the Picard-Fuchs system for the family and construct a sequence of restrictions that lead to extensions of the polarization by two-elementary lattices. They prove that the Picard-Fuchs operators for the restricted families coincide with known resonant hypergeometric systems. For the one-parameter mirror families of deformed Fermat hypersurfaces they show that the MTC produces a non-resonant GKZ system for which a basis of solutions in the form of absolutely convergent Mellin-Barnes integrals exists whose monodromy is explicitly computed.
Reviewer: Vladimir P. Kostov (Nice)Hypergeometry, integrability and Lie theory. Virtual conference, Lorentz Center, Leiden, the Netherlands, December 7--11, 2020https://zbmath.org/1496.170012022-11-17T18:59:28.764376ZPublisher's description: This volume contains the proceedings of the virtual conference on Hypergeometry, Integrability and Lie Theory, held from December 7--11, 2020, which was dedicated to the 50th birthday of Jasper Stokman.
The papers represent recent developments in the areas of representation theory, quantum integrable systems and special functions of hypergeometric type.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Etingof, Pavel; Kazhdan, David}, Characteristic functions of \(p\)-adic integral operators, 1-27 [Zbl 07602313]
\textit{Garbali, Alexandr; Zinn-Justin, Paul}, Shuffle algebras, lattice paths and the commuting scheme, 29-68 [Zbl 07602314]
\textit{Kolb, Stefan}, The bar involution for quantum symmetric pairs -- hidden in plain sight, 69-77 [Zbl 07602315]
\textit{Koornwinder, Tom H.}, Charting the \(q\)-Askey scheme, 79-94 [Zbl 07602316]
\textit{Rains, Eric M.}, Filtered deformations of elliptic algebras, 95-154 [Zbl 07602317]
\textit{Regelskis, Vidas; Vlaar, Bart}, Pseudo-symmetric pairs for Kac-Moody algebras, 155-203 [Zbl 07602318]
\textit{Reshetikhin, N.; Stokman, J. V.}, Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains, 205-241 [Zbl 07602319]
\textit{Rösler, Margit; Voit, Michael}, Elementary symmetric polynomials and martingales for Heckman-Opdam processes, 243-262 [Zbl 07602320]
\textit{Schomerus, Volker}, Conformal hypergeometry and integrability, 263-285 [Zbl 07602321]
\textit{Varchenko, Alexander}, Determinant of \(\mathbb{F}_p\)-hypergeometric solutions under ample reduction, 287-307 [Zbl 07602322]
\textit{Varchenko, Alexander}, Notes on solutions of KZ equations modulo \(p^s\) and \(p\)-adic limit \(s\to\infty\), 309-347 [Zbl 07602323]Generalizations of truncated M-fractional derivative associated with \((p,k)\)-Mittag-Leffler function with classical propertieshttps://zbmath.org/1496.260042022-11-17T18:59:28.764376Z"Chand, Mehar"https://zbmath.org/authors/?q=ai:chand.mehar"Agarwal, Praveen"https://zbmath.org/authors/?q=ai:agarwal.praveenSummary: In the present chapter, we have generalized the truncated M-fractional derivative. This new differential operator denoted by \({}_{i,p}\mathscr{D}_{M, k, \alpha, \beta}^{\sigma, \gamma ,q},\) where the parameter \(\sigma\) associated with the order of the derivative is such that \(0 < \sigma < 1\) and \(M\) is the notation to designate that the function to be derived involves the truncated \((p, k)\)-Mittag-Leffler function. The operator \({}_{i,p}\mathscr{D}_{M, k, \alpha, \beta}^{\sigma, \gamma ,q}\) satisfies the properties of the integer-order calculus. We also present the respective fractional integral from which emerges, as a natural consequence, the result, which can be interpreted as an inverse property. Finally, we obtain the analytical solution of the M-fractional heat equation, linear fractional differential equation, and present a graphical analysis.
For the entire collection see [Zbl 1485.65002].Wilker and Huygens type inequalities for mixed trigonometric-hyperbolic functionshttps://zbmath.org/1496.260152022-11-17T18:59:28.764376Z"Bagul, Yogesh J."https://zbmath.org/authors/?q=ai:bagul.yogesh-j"Bhayo, Barkat A."https://zbmath.org/authors/?q=ai:bhayo.barkat-ali"Dhaigude, Ramkrishna M."https://zbmath.org/authors/?q=ai:dhaigude.ramkrishna-m"Raut, Vinay M."https://zbmath.org/authors/?q=ai:raut.vinay-mSome trigonometric-hyperbolic Wilker-type and Huygens-type inequalities are obtained.
The main results are given in the following two theorems.
Theorem. For \(0<x<\frac{\pi}{2}\) the following inequalities hold:
\[
2-\frac{x^2}{2} < \frac{x}{\sin x} + \left( \frac{\tanh x}{x}\right)^2 < 2 < 2+\frac{x^4}{180} < \left( \frac{\sin x}{x} \right)^2 + \frac{x}{\tanh x},
\]
\[
\frac{\sinh x}{x}+ \left( \frac{x}{\tan x}\right)^2 < 2-\frac{x^2}{6} < 2 < 2 +\frac{x^5}{45 \tan x} < \left(\frac{\sinh x}{x} \right)^2 + \frac{x}{\tan x}.
\]
Theorem. For \(0<x<\frac{\pi}{2}\) the following inequalities hold:
\[
2\frac{\sinh x}{x} + \frac{x}{\tan x} < 3 -\frac{x^4}{180} < 3 < 3+\frac{31x^4}{180} <2\frac{x}{\sinh x} + \frac{\tan x}{x},
\]
\[
\frac{\sinh x}{x} + 2\frac{x}{\tan x} < 3 -\frac{x^2}{2} < 3 < 3+\frac{x^2}{2} <\frac{x}{\sinh x} + 2\frac{\tan x}{x},
\]
\[
3 -\frac{x^4}{180} <2\frac{\sin x}{x} + \frac{x}{\tanh x} < 3 <2\frac{x}{\sin x} + \frac{\tanh x}{x} < 3+\frac{31x^4}{180},
\]
\[
3 -\frac{x^2}{2} <\frac{x}{\sin x} + 2\frac{\tanh x}{x} < 3 <\frac{\sin x}{x} + 2\frac{x}{\tanh x} < 3+\frac{x^2}{2}.
\]
Reviewer: Sanja Varošanec (Zagreb)Revisiting Fejér-Hermite-Hadamard type inequalities in fractal domain and applicationshttps://zbmath.org/1496.260362022-11-17T18:59:28.764376Z"Rashid, Saima"https://zbmath.org/authors/?q=ai:rashid.saima"Khalid, Aasma"https://zbmath.org/authors/?q=ai:khalid.aasma"Karaca, Yeliz"https://zbmath.org/authors/?q=ai:karaca.yeliz"Chu, Yu-Ming"https://zbmath.org/authors/?q=ai:chu.yumingOn a more accurate half-discrete Hilbert-type inequality involving hyperbolic functionshttps://zbmath.org/1496.260412022-11-17T18:59:28.764376Z"You, Minghui"https://zbmath.org/authors/?q=ai:you.minghui"Sun, Xia"https://zbmath.org/authors/?q=ai:sun.xia"Fan, Xiansheng"https://zbmath.org/authors/?q=ai:fan.xiansheng(no abstract)The second Hankel determinant for subclasses of bi-univalent functions associated with a nephroid domainhttps://zbmath.org/1496.300122022-11-17T18:59:28.764376Z"Srivastava, Hari Mohan"https://zbmath.org/authors/?q=ai:srivastava.hari-mohan"Murugusundaramoorthy, Gangadharan"https://zbmath.org/authors/?q=ai:murugusundaramoorthy.gangadharan"Bulboacă, Teodor"https://zbmath.org/authors/?q=ai:bulboaca.teodorSummary: In the present paper, we obtain the estimates for the first two initial Taylor-Maclaurin coefficients and for the upper bounds of the Fekete-Szegö functional for new subclasses of the class \(\Sigma\) of normalized analytic and bi-univalent functions, which are defined here with the aid of the Nephroid function. We also determine upper bounds of the functional \(|a_2a_4-a_3^2|\) for the functions that belong to these classes. A related open problem as well some potential directions for further researches are posed in the concluding section.Integral means and Yamashita's conjecture associated with the Janowski type \((j, k)\)-symmetric starlike functionshttps://zbmath.org/1496.300132022-11-17T18:59:28.764376Z"Srivastava, H. M."https://zbmath.org/authors/?q=ai:srivastava.hari-mohan"Prajapati, A."https://zbmath.org/authors/?q=ai:prajapati.anuja"Gochhayat, P."https://zbmath.org/authors/?q=ai:gochhayat.priyabratSummary: Let \(L_1(r, f)\) and \(\Delta(r, f)\) denote, respectively, the integral means and the area of the image of the subdisk
\[
\mathbb{D}_r:= \{z: z \in\mathbb{C}\text{ and }|z|<r; 0\leqq r<1\}
\]
of a function \(f\), which is analytic in \(\mathbb{D}\). For \(j=0, 1, 2, \dots, k-1\) (\(k = 1, 2, 3, \dots\)), \(A\in\mathbb{C}\); \(-1\leqq B \leqq 0\) with \(A\neq B\), we introduce the family of the Janowski type \((j, k)\)-symmetric starlike functions, which is denoted by \(\mathcal{ST}_{[j, k]}(A, B)\). Here, in this article, we first derive the bounds on \(L_1(r, f_{j, k})\) for every \(f_{j, k}\in\mathcal{ST}_{[j, k]}(A, B)\). The necessary coefficient condition for functions in the class \(\mathcal{ST}_{[j, k]}(A, B)\) is then presented. Our investigation leads us to get the sharp bounds on Yamashita's functional of the form \(\Delta \left(r, \frac{z}{f_{j, k}}\right)\). Finally, we provide the sharp estimate of the \(n\)th logarithmic coefficient.Sums of two-parameter deformations of multiple polylogarithmshttps://zbmath.org/1496.330012022-11-17T18:59:28.764376Z"Kato, Masaki"https://zbmath.org/authors/?q=ai:kato.masakiSummary: We introduce a generating function of sums of two-parameter deformations of multiple polylogarithms, denoted by \(\Phi_2(a;p,q)\), and study a \(q\)-difference equation satisfied by it. We show that this \(q\)-difference equation can be solved by expanding \(\Phi_2(a;p,q)\) into power series of the parameter \(p\) and then using the method of variation of constants. By letting \(p \rightarrow 0\) in the main theorem, we find that the generating function of sums of \(q\)-interpolated multiple zeta values can be written in terms of the \(q\)-hypergeometric function \(_3 \phi_2\), which is due to Li-Wakabayashi.The hypergeometric function, the confluent hypergeometric function and WKB solutionshttps://zbmath.org/1496.330022022-11-17T18:59:28.764376Z"Aoki, Takashi"https://zbmath.org/authors/?q=ai:aoki.takashi"Takahashi, Toshinori"https://zbmath.org/authors/?q=ai:takahashi.toshinori"Tanda, Mika"https://zbmath.org/authors/?q=ai:tanda.mikaTake the Gauss hypergeometric equation and instead of parameters \(a,b,c\) let us write \(a=\alpha_0+\alpha \eta, b=\beta_0+\beta\eta, c=\gamma_0+\gamma\eta\) and instead of the unknown function \(w\) let us take \(w=x^{-c/2}(1-x)^{(-1/2)(a+b-c+1)}\psi\). Then the equation is written in the form
\[
(-\frac{d^2}{dx^2}+\eta^2Q)\psi=0,\quad Q=\sum_{j=0}^N \eta^{-j}Q_j(x).
\]
To such an equation the WKB analysis can be applied. It gives formal solutions that using the Borel summation can be transformed to analytic solutions. In the paper under review a relation between these solutions and the classical solutions of the Gauss equation is established.
Also analogous results for the equation for the function \(F_{1,1}\) are obtained.
Reviewer: Dmitry Artamonov (Moskva)A new linear inversion formula for a class of hypergeometric polynomialshttps://zbmath.org/1496.330032022-11-17T18:59:28.764376Z"Nasri, Ridha"https://zbmath.org/authors/?q=ai:nasri.ridha"Simonian, Alain"https://zbmath.org/authors/?q=ai:simonian.alain-d"Guillemin, Fabrice"https://zbmath.org/authors/?q=ai:guillemin.fabrice-mSummary: Given complex parameters \(x,\nu,\alpha,\beta\) and \(\gamma\notin-\mathbb{N}\), consider the infinite lower triangular matrix \(\mathbf{A}(x,\nu;\alpha,\beta,\gamma)\) with elements
\[
A_{n,k}(x,\nu;\alpha,\beta,\gamma)=(-1)^k\binom{n+\alpha}{k+\alpha}\cdot F(k-n,-(\beta+n)\nu;-(\gamma+n);x)
\]
for \(1\leq k\leq n\), depending on the hypergeometric polynomials \(F(-n,\cdot;\cdot;x)\), \(n\in\mathbb{N}^*\). After stating a general criterion for the inversion of infinite matrices in terms of associated generating functions, we prove that the inverse matrix \(\mathbf{B}(x,\nu;\alpha,\beta,\gamma)=\mathbf{A}(x,\nu;\alpha,\beta,\gamma)^{-1}\) is given by
\begin{align*}
B_{n,k}(x,\nu;\alpha,\beta,\gamma)= & (-1)^k\binom{n+\alpha}{k+\alpha}\\
& \bigg[\frac{\gamma+k}{\beta+k}F(k-n,(\beta+k)\nu;\gamma+k;x)\\
& +\frac{\beta-\gamma}{\beta+k}F(k-n,(\beta+k)\nu;1+\gamma+k;x)\bigg]
\end{align*}
for \(1\leq k\leq n\), thus providing a new class of linear inversion formulas. Functional relations for the generating functions of related sequences \(S\) and \(T\), that is, \(T=\mathbf{A}(x,\nu;\alpha,\beta,\gamma)S\Longleftrightarrow S=\mathbf{B}(x,\nu;\alpha,\beta,\gamma)T\), are also provided.Integral formula for the Bessel function of the first kindhttps://zbmath.org/1496.330042022-11-17T18:59:28.764376Z"De Micheli, Enrico"https://zbmath.org/authors/?q=ai:de-micheli.enricoSummary: In this paper, we prove a new integral formula for the Bessel function of the first kind \(J_\mu (z)\). This formula generalizes to any \(\mu ,z\in{\mathbb{C}}\) the classical integral representations of Bessel and Poisson.A new class of double integralshttps://zbmath.org/1496.330052022-11-17T18:59:28.764376Z"Anil, Aravind K."https://zbmath.org/authors/?q=ai:anil.aravind-k"Prathima, J."https://zbmath.org/authors/?q=ai:prathima.j"Kim, Insuk"https://zbmath.org/authors/?q=ai:kim.insukSummary: In this paper we aim to establish a new class of six definite double integrals in terms of gamma functions. The results are obtained with the help of some definite integrals obtained recently by Kim and Edward equality. The results established in this paper are simple, interesting, easily established and may be useful potentially.When does a hypergeometric function \(_pF_q\) belong to the Laguerre-Pólya class \(LP^+\)?https://zbmath.org/1496.330062022-11-17T18:59:28.764376Z"Sokal, Alan D."https://zbmath.org/authors/?q=ai:sokal.alan-dSummary: I show that a hypergeometric function \(_pF_q(a_1, \ldots, a_p; b_1, \ldots, b_q; \cdot)\) with \(p \leq q\) belongs to the Laguerre-Pólya class \(L P^+\) for arbitrarily large \(b_{p + 1}, \ldots, b_q > 0\) if and only if, after a possible reordering, the differences \(a_i - b_i\) are nonnegative integers. This result arises as an easy corollary of the case \(p = q\) proven two decades ago by Ki and Kim. I also give explicit examples for the case \(_1F_2\).Bargmann-type transforms and modified harmonic oscillatorshttps://zbmath.org/1496.330072022-11-17T18:59:28.764376Z"Chihara, Hiroyuki"https://zbmath.org/authors/?q=ai:chihara.hiroyukiSummary: We study some complete orthonormal systems on the real line. These systems are determined by Bargmann-type transforms, which are Fourier integral operators with complex-valued quadratic phase functions. Each system consists of eigenfunctions for a second-order elliptic differential operator like the Hamiltonian of the harmonic oscillator. We also study the commutative case of a certain class of systems of second-order differential operators called the non-commutative harmonic oscillators. By using the diagonalization technique, we compute the eigenvalues and eigenfunctions for the commutative case of the non-commutative harmonic oscillators. Finally, we study a family of functions associated with an ellipse in the phase plane. We show that the family is a complete orthogonal system on the real line.Asymptotics of multiple orthogonal Hermite polynomials \(H_{n_1,n_2}(z,\alpha)\) determined by a third-order differential equationhttps://zbmath.org/1496.330082022-11-17T18:59:28.764376Z"Dobrokhotov, S. Yu."https://zbmath.org/authors/?q=ai:dobrokhotov.sergei-yu"Tsvetkova, A. V."https://zbmath.org/authors/?q=ai:tsvetkova.anna-vSummary: In the paper, we study the asymptotics of multiple orthogonal Hermite polynomials \(H_{n_1,n_2}(z,\alpha)\) that are determined by orthogonality relations with respect to two weights that are Gaussian exponents with shifted maxima. These polynomials can be defined using recurrence relations, and also, as shown by A. I. Aptekarev, A. Branquinho, and W. Van Assche, as certain solutions to a third-order differential equation. Starting from this differential equation, we obtain asymptotics of such polynomials as \(|n|=\sqrt{n_1^2+n_2^2} \rightarrow \infty\) in the form of the Airy function \(A_i\) and its derivative \(A_i'\) of a compound argument.On the \(L^2\)-norm of Gegenbauer polynomialshttps://zbmath.org/1496.330092022-11-17T18:59:28.764376Z"Ferizović, Damir"https://zbmath.org/authors/?q=ai:ferizovic.damir|ferizovic.damir.1Summary: Gegenbauer, also known as ultra-spherical, polynomials appear often in numerical analysis or interpolation. In the present text we find a recursive formula for and compute the asymptotic behavior of their \(L^2\)-norm.Limit theorems for Jacobi ensembles with large parametershttps://zbmath.org/1496.330102022-11-17T18:59:28.764376Z"Hermann, Kilian"https://zbmath.org/authors/?q=ai:hermann.kilian"Voit, Michael"https://zbmath.org/authors/?q=ai:voit.michaelSummary: Consider \(\beta\)-Jacobi ensembles on the alcoves
\[
A:=\{ x\in\mathbb{R}^N \mid -1\leq x_1\leq \cdots\leq x_N\leq 1\}
\]
with parameters \(k_1,k_2,k_3\geq 0\). In the freezing case \((k_1,k_2,k_3)=\kappa\cdot (a,b,1)\) with \(a,b>0\) fixed and \(\kappa\to\infty\), we derive a central limit theorem. The drift and covariance matrix of the limit are expressed via the zeros of classical Jacobi polynomials. We also determine the eigenvalues and eigenvectors of the covariance matrices. Our results are related to corresponding limits for \(\beta\)-Hermite and Laguerre ensembles for \(\beta\to\infty\).On some formulas for the Horn functions \(H_3(a, b; c; w, z)\), \(H^{(c)}_6(a; c; w, z)\) and Humbert function \(\Phi_3(b; c; w, z)\)https://zbmath.org/1496.330112022-11-17T18:59:28.764376Z"Brychkov, Yu. A."https://zbmath.org/authors/?q=ai:brychkov.yury-a"Savischenko, N. V."https://zbmath.org/authors/?q=ai:savischenko.nikolay-vSummary: Some new relations for the Horn function \(H_3(a, b; c; w, z)\), confluent Horn function \(H^{(c)}_6(a; c; w, z)\) and Humbert function\(\Phi_3(b; c; w, z)\) are obtained including differentiation and integration formulas, series and values for specific choice of parameters and variables. Some generating fuctions for various special functions are given in terms of these Horn functions.Properties of \(\psi\)-Mittag-Leffler fractional integralshttps://zbmath.org/1496.330122022-11-17T18:59:28.764376Z"Oliveira, D. S."https://zbmath.org/authors/?q=ai:de-souza-oliveira.fabiano|oliveira.daniela-s|oliveira.david-senaSummary: This paper aims to investigate properties of fractional integrals with three-parameters Mittag-Leffler function kernel. We prove that the Cauchy problem and the Volterra integral equation are equivalent. We find a closed-form to the solution of the Cauchy problem using successive approximations method and \(\psi\)-Caputo fractional derivative.Fractional Fourier transform to stability analysis of fractional difffferential equations with Prabhakar derivativeshttps://zbmath.org/1496.340132022-11-17T18:59:28.764376Z"Deepa, S."https://zbmath.org/authors/?q=ai:deepa.s-n"Ganesh, A."https://zbmath.org/authors/?q=ai:ganesh.anumanthappa"Ibrahimov, V."https://zbmath.org/authors/?q=ai:ibrahimov.vagif-r"Santra, S. S."https://zbmath.org/authors/?q=ai:santra.shyam-sundar"Govindan, V."https://zbmath.org/authors/?q=ai:govindan.vediyappan"Khedher, K. M."https://zbmath.org/authors/?q=ai:khedher.khaled-mohamed"Noeiaghdam, S."https://zbmath.org/authors/?q=ai:noeiaghdam.samadSummary: In this paper, the authors introduce the Prabhakar derivative associated with the generalised Mittag-Leffler function. Some properties of the Prabhakar integrals, Prabhakar derivatives and some of their extensions, like fractional Fourier transform of Prabhakar integrals and fractional Fourier transform of Prabhakar derivatives are introduced. This note aims to study the Mittag-Leffler-Hyers-Ulam stability of the linear and nonlinear fractional differential equations with the Prabhakar derivative. Furthermore, we give a brief definition of the Mittag-Leffler-Hyers-Ulam problem and a method for solving fractional differential equations using the fractional Fourier transform. We show that the fractional differential equations are Mittag-Leffler-Hyers-Ulam stable in the sense of Prabhakar derivatives.Indefinite integrals from Wronskians and related linear second-order differential equationshttps://zbmath.org/1496.340582022-11-17T18:59:28.764376Z"Conway, John T."https://zbmath.org/authors/?q=ai:conway.john-thomasSummary: Many indefinite integrals are derived for Bessel functions and associated Legendre functions from particular transformations of their differential equations which are closely linked to Wronskians. A large portion of the results for Bessel functions is known, but all the results for associated Legendre functions appear to be new. The method can be applied to many other special functions. All results have been checked by differentiation using Mathematica.On a regularisation of a nonlinear differential equation related to the non-homogeneous Airy equationhttps://zbmath.org/1496.341282022-11-17T18:59:28.764376Z"Filipuk, Galina"https://zbmath.org/authors/?q=ai:filipuk.galina-v"Kecker, Thomas"https://zbmath.org/authors/?q=ai:kecker.thomas"Zullo, Federico"https://zbmath.org/authors/?q=ai:zullo.federicoIn nonlinear ordinary differential equations (ODE), there may exist points where the existence theorem of Cauchy does not apply. Then, a classical procedure (blow-up) is to resolve all such exceptional points by considering a (hopefully finite) sequence of equivalent systems defined in different coordinates, chosen so as to preserving the structure of singularities.
The present paper is a pedagogical introduction to this technique, on a toy nonlinear second order ODE (2.1) whose general solution is known since it is built from a linear ODE (inhomogeneous Airy) by elimination of the constant term.
However, the authors fail to get out of a usual trap in this procedure, which is the choice of a two-dimensional system equivalent to the second order ODE. This choice is not unique, and their choice (2.7) leads, according to the authors, to an infinite sequence of blow-ups. Since the toy ODE is linearizable, the sequence should be finite, and the authors should consider a choice different from (2.7).
For the entire collection see [Zbl 1481.26002].
Reviewer: Robert Conte (Gif-sur-Yvette)Jacobi spectral discretization for nonlinear fractional generalized seventh-order KdV equations with convergence analysishttps://zbmath.org/1496.353432022-11-17T18:59:28.764376Z"Hafez, R. M."https://zbmath.org/authors/?q=ai:hafez.ramy-mahmoud"Youssri, Y. H."https://zbmath.org/authors/?q=ai:youssri.y-h(no abstract)Dispersive shock wave, generalized Laguerre polynomials, and asymptotic solitons of the focusing nonlinear Schrödinger equationhttps://zbmath.org/1496.353652022-11-17T18:59:28.764376Z"Kotlyarov, Vladimir"https://zbmath.org/authors/?q=ai:kotlyarov.vladimir-p"Minakov, Alexander"https://zbmath.org/authors/?q=ai:minakov.alexanderSummary: We consider dispersive shock waves of the focusing nonlinear Schrödinger equation generated by discontinuous initial conditions which are periodic or quasiperiodic on the left semiaxis and zero on the right semiaxis. As an initial function, we use a finite-gap potential of the Dirac operator given in an explicit form through hyperelliptic theta-functions. The aim of this paper is to study the long-time asymptotics of the solution of this problem in a vicinity of the leading edge, where a train of asymptotic solitons are generated. Such a problem was studied in the work of the diest author and \textit{E. Ya. Khruslov} [Teor. Mat. Fiz. 68, No. 2, 751--761 (1986; Zbl 0621.35092)] and the first author [Math. Notes 49, No. 1--2, 172--180 (1991; Zbl 0734.35126); translation from Mat. Zametki 49, No. 2, 84--94 (1991)] using Marchenko's inverse scattering techniques. We investigate this problem exceptionally using the Riemann-Hilbert (RH) problem techniques that allow us to obtain explicit formulas for asymptotic solitons themselves in contrast with the cited papers where asymptotic formulas are obtained only for the square of the absolute value of solution. Using transformations of the main RH problems, we arrive at a model problem corresponding to the parametrix at the end points of the continuous spectrum of the Zakharov-Shabat spectral problem. The parametrix problem is effectively solved in terms of the generalized Laguerre polynomials, which naturally appeared after appropriate scaling of the Riemann-Hilbert problem in small neighborhoods of the end points of the continuous spectrum. Further asymptotic analysis gives an explicit formula for solitons at the edge of dispersive waves. Thus, we give the complete description of the train of asymptotic solitons: not only bearing the envelope of each asymptotic soliton, but its oscillating structure is found explicitly. Besides, the second term of asymptotics describing an interaction between these solitons and oscillating background is also found. This gives the fine structure of the edge of dispersive shock waves.\par{\copyright 2019 American Institute of Physics}Numerical study for time fractional stochastic semi linear advection diffusion equationshttps://zbmath.org/1496.354682022-11-17T18:59:28.764376Z"Sweilam, N. H."https://zbmath.org/authors/?q=ai:sweilam.nasser-hassan"El-Sakout, D. M."https://zbmath.org/authors/?q=ai:elsakout.d-m"Muttardi, M. M."https://zbmath.org/authors/?q=ai:muttardi.m-mSummary: In this work, a stochastic fractional advection diffusion model with multiplicative noise is studied numerically. The Galerkin finite element method in space and finite difference in time are used, where the fractional derivative is in Caputo sense. The error analysis is investigated via Galerkin finite element method. In terms of the Mittag Leffler function, the mild solution is obtained. For the error estimates, the strong convergence for the semi and fully discrete schemes are proved in a semigroup structure. Finally, two numerical examples are given to confirm the theoretical results.Eigenfunctions of a discrete elliptic integrable particle model with hyperoctahedral symmetryhttps://zbmath.org/1496.370612022-11-17T18:59:28.764376Z"van Diejen, Jan Felipe"https://zbmath.org/authors/?q=ai:van-diejen.jan-felipe"Görbe, Tamás"https://zbmath.org/authors/?q=ai:gorbe.tamas-fThe main purpose of the authors is to carry out a finite-dimensional reduction of the eigenvalue problem for a second-order difference operator describing the quantum Hamiltonian of an elliptic Ruijsenaars type \(n\)-particle model on the circle with hyperoctahedral symmetry.
Reviewer: Mohammed El Aïdi (Bogotá)Link theorem and distributions of solutions to uncertain Liouville-Caputo difference equationshttps://zbmath.org/1496.390062022-11-17T18:59:28.764376Z"Srivastava, Hari Mohan"https://zbmath.org/authors/?q=ai:srivastava.hari-mohan"Mohammed, Pshtiwan Othman"https://zbmath.org/authors/?q=ai:mohammed.pshtiwan-othman"Guirao, Juan L. G."https://zbmath.org/authors/?q=ai:garcia-guirao.juan-luis"Hamed, Y. S."https://zbmath.org/authors/?q=ai:hamed.yasser-sSummary: We consider a class of initial fractional Liouville-Caputo difference equations (IFLCDEs) and its corresponding initial uncertain fractional Liouville-Caputo difference equations (IUFLCDEs). Next, we make comparisons between two unique solutions of the IFLCDEs by deriving an important theorem, namely the main theorem. Besides, we make comparisons between IUFLCDEs and their \(\varrho\)-paths by deriving another important theorem, namely the link theorem, which is obtained by the help of the main theorem. We consider a special case of the IUFLCDEs and its solution involving the discrete Mittag-Leffler. Also, we present the solution of its \(\varrho\)-paths via the solution of the special linear IUFLCDE. Furthermore, we derive the uniqueness of IUFLCDEs. Finally, we present some test examples of IUFLCDEs by using the uniqueness theorem and the link theorem to find a relation between the solutions for the IUFLCDEs of symmetrical uncertain variables and their \(\varrho\)-paths.Finite time stability of fractional delay difference systems: a discrete delayed Mittag-Leffler matrix function approachhttps://zbmath.org/1496.390132022-11-17T18:59:28.764376Z"Du, Feifei"https://zbmath.org/authors/?q=ai:du.feifei"Jia, Baoguo"https://zbmath.org/authors/?q=ai:jia.baoguoSummary: A discrete delayed Mittag-Leffler matrix function is developed in this paper. Based on this function, an explicit formula of the solution of fractional delay difference system (FDDS) is derived. Furthermore, a criterion on finite time stability (FTS) of FDDS with constant coefficients is obtained by use of this formula. However, it can't be directly used to investigate the FTS of FDDS with variable coefficients. To overcome this difficulty, a comparison theorem of FDDS is established to obtain a criterion of the FTS of FDDS with variable coefficients. Finally, a numerical example is given to show the effectiveness of the proposed results.Completeness conditions of systems of Bessel functions in weighted \(L^2\)-spaces in terms of entire functionshttps://zbmath.org/1496.420082022-11-17T18:59:28.764376Z"Khats', Ruslan"https://zbmath.org/authors/?q=ai:khats.r-vSummary: Let \(J_\nu\) be the Bessel function of the first kind of index \(\nu\geq 1/2\), \(p\in\mathbb{R}\) and \((\rho_k)_{k\in\mathbb{N}}\) be a sequence of distinct nonzero complex numbers. Sufficient conditions for the completeness of the system \(\left\{x^{-p-1}\sqrt{x\rho_k}J_\nu(x\rho_k): k\in\mathbb{N}\right\}\) in the weighted space \(L^2((0;1); x^{2p} dx)\) are found in terms of an entire function with the set of zeros coinciding with the sequence \((\rho_k)_{k\in\mathbb{N}}\).Simon's OPUC Hausdorff dimension conjecturehttps://zbmath.org/1496.420352022-11-17T18:59:28.764376Z"Damanik, David"https://zbmath.org/authors/?q=ai:damanik.david"Guo, Shuzheng"https://zbmath.org/authors/?q=ai:guo.shuzheng"Ong, Darren C."https://zbmath.org/authors/?q=ai:ong.darren-cSummary: We show that the Szegő matrices, associated with Verblunsky coefficients \(\{{\alpha}_n\}_{n\in{{\mathbb{Z}}}_+}\) obeying \(\sum_{n = 0}^\infty n^{\gamma} |{\alpha}_n|^2 < \infty\) for some \({\gamma} \in (0,1)\), are bounded for values \(z \in \partial{\mathbb{D}}\) outside a set of Hausdorff dimension no more than \(1 - {\gamma}\). In particular, the singular part of the associated probability measure on the unit circle is supported by a set of Hausdorff dimension no more than \(1-{\gamma}\). This proves the OPUC Hausdorff dimension conjecture of \textit{B. Simon} [Orthogonal polynomials on the unit circle. Part 1: Classical theory. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1082.42020)].Computational and theoretical aspects of Romanovski-Bessel polynomials and their applications in spectral approximationshttps://zbmath.org/1496.420372022-11-17T18:59:28.764376Z"Zaky, Mahmoud A."https://zbmath.org/authors/?q=ai:zaky.mahmoud-a"Abo-Gabal, Howayda"https://zbmath.org/authors/?q=ai:abo-gabal.howayda"Hafez, Ramy M."https://zbmath.org/authors/?q=ai:hafez.ramy-mahmoud"Doha, Eid H."https://zbmath.org/authors/?q=ai:doha.eid-hThe paper under review presents the main properties of a finite class of orthogonal polynomials with respect to the inverse gamma distribution over the positive real line called Romanovski-Bessel polynomials. More precisely, it introduces the related Romanovski-Bessel-Gauss-type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in weighted Sobolev space. It also addresses the relationship between such kinds of finite orthogonal polynomials and other classes of finite and infinite orthogonal polynomials.
Reviewer: M. Abdessadek Saib (Tebessa)An uncertainty principle for spectral projections on rank one symmetric spaces of noncompact typehttps://zbmath.org/1496.430052022-11-17T18:59:28.764376Z"Ganguly, Pritam"https://zbmath.org/authors/?q=ai:ganguly.pritam"Thangavelu, Sundaram"https://zbmath.org/authors/?q=ai:thangavelu.sundaramThe authors present a weaker version of Chernoff's theorem for Bessel and Jacobi operators. This result is used to prove a refined version of Ingham's theorem for the Helgason Fourier transform on rank one Riemannian symmetric spaces of noncompact type. The authors also prove an Ingham type uncertainty principle for the generalized spectral projections associated to the Laplace-Beltrami operator. Similar Ingham type results for the generalized spectral projections associated to Dunkl Laplacian are also discussed.
Reviewer: Ashish Bansal (Delhi)On Hankel type integral transform associated with Whittaker and hypergeometric functionshttps://zbmath.org/1496.440052022-11-17T18:59:28.764376Z"Ghayasuddin, Mohd"https://zbmath.org/authors/?q=ai:ghayasuddin.mohd"Khan, Waseem Ahmad"https://zbmath.org/authors/?q=ai:khan.waseem-ahmad"Mishra, Lakshmi Narayan"https://zbmath.org/authors/?q=ai:mishra.lakshmi-narayanSummary: In the present research note, we establish a Hankel type integral transform involving the product of Whittaker function \(W_{k,m}\) and hypergeometric function \(_1F_2\). By using the result of \textit{A. Erdélyi} et al. [Tables of integral transforms. Vol. I. New York: McGraw-Hill Book Co. (1954; Zbl 0055.36401)], we express this transform into Srivastava triple hypergeometric series \(F^{(3)} [x,y,z]\). Some special cases of our main transform are also indicated.Ramanujan's master theorem and two formulas for the zero-order Hankel transformhttps://zbmath.org/1496.440062022-11-17T18:59:28.764376Z"Kisselev, A. V."https://zbmath.org/authors/?q=ai:kisselev.a-vThe author invokes the Ramanujan's Master Theorem [\textit{G. H. Hardy}, Ramanujan. Twelve lectures on subjects suggested by his life and work. Reprint. New York: Chelsea Publishing Company (1959; Zbl 0086.26202)] to establish `two formulas for Hankel transforms of order zero for even functions using the inverse Mellin transform'. By defining that
\[
A\left( q \right)=\mathcal{H}_0(f)(q)= \int_0^\infty J_0\left(qx \right)f\left( x \right)x \,dx,\tag{1}
\]
for an even function \(f\), written as \(f (x) = g(x^2)\), the author first proves the following theorem:
Theorem 2.1 The Hankel transform of order zero (1), with \(f (x) = g(x^2)\) and \(q > 0\)
may be expressed in the form
\[
A\left( q \right) = \frac{1}{\pi iq^2}\int_{\alpha -i\infty}^{\alpha+i\infty}\bar{g}^{\left(s \right)}\left(0 \right)\Gamma \left(s+1\right)\left(\frac{q^2}{4}\right)^{-s}\mathrm{d}s
\]
for \(-1<\alpha <0\), provided
\begin{itemize}
\item[(a)] \(g(z)\) is a regular function with Taylor series expansion about \(z = 0\) of the form
\[
g\left( z \right) = \sum\limits_{m = 0}^\infty\frac{\bar{g}^{\left( m \right)}\left(0 \right)}{m!}\left(-z\right)^m;
\]
\item[(b)] \(g(z) = O(z^{-d})\) as \(z\to \infty\), for \(d > \frac{1}{4}\);
\item[(c)] \(\bar{g}^{\left( s \right)}\left( 0 \right)\) is a regular (single-valued) function defined on the half-plane
\[
H\left( \delta \right) = \left\{s \in \mathbb{C}:\mathrm{Re}s\ge - \delta \right\}
\]
for some \(\frac{1}{4} < \delta < 1\) and satisfies the growth condition
\[
\left|\bar{g}^{\left(s \right)}\left(0 \right)\right| < Ce^{Pv+ A\left| w \right|}
\]
for some \(A < \frac{\pi}{2}\) and all \(s = v + iw \in H(\delta)\).
\end{itemize}
The applications of this theorem are then discussed by the author where he also deduces a new improper parametric integral of the Bessel function in Example 2.7.
The following generalization of Theorem 2.1 is then proven by the author:
Theorem 2.9 The Hankel transform (1) of the function \(f (x) = h(x^4)\) and \(q > 0\) may be expressed in the form
\[
A\left( q \right) = \frac{1}{iq^2\sqrt \pi}\int_{\alpha - i\infty}^{\alpha+i\infty}\bar{h}\left(s \right)\frac{\Gamma \left(2s +1 \right)}{\Gamma\left(\frac{1}{2}- s\right)}2^{6s+ 1}q^{- 4s}\, \mathrm{d}s,
\]
for \(\frac{-1}{2} < \alpha < 0\), provided that
\begin{itemize}
\item[(a)] \(h(z)\) is a regular function and its Taylor series at \(z = 0\) has the form
\[
h\left( z \right) = \sum\limits_{m = 0}^\infty\frac{\bar{h}^{\left( m \right)}\left( 0 \right)}{m!}\left(- z\right)^m;
\]
\item[(b)] \(h(z) = O(z^{-d})\), as \(z \to \infty\), for \(d > \frac{1}{8}\);
\item[(c)] \(\bar{h}^{\left( s \right)}\left( 0 \right)\) is a regular (single-valued) function defined on a half-plane
\[
H\left( \delta \right) = \left\{s \in \mathbb{C} :\mathrm{Re}s\ge-\delta \right\}
\]
for some \(\frac{1}{8} < \delta < \frac{1}{2}\) and satisfies the growth condition
\[
\left|\bar{g}^{\left(s \right)}\left(0\right)\right|<Ce^{Pv + A\left| w \right|}
\]
for some \(A < \frac{\pi}{2}\) and all \(s = v + iw \in H(\delta)\).
\end{itemize}
Reviewer: Lalit Mohan Upadhyaya (Dehradun)Perspectives on general left-definite theoryhttps://zbmath.org/1496.470402022-11-17T18:59:28.764376Z"Frymark, Dale"https://zbmath.org/authors/?q=ai:frymark.dale"Liaw, Constanze"https://zbmath.org/authors/?q=ai:liaw.constanzeSummary: In 2002, \textit{L. L. Littlejohn} and \textit{R. Wellman} [J. Differ. Equations 181, No. 2, 280--339 (2002; Zbl 1008.47029)] developed a celebrated general left-definite theory for semi-bounded self-adjoint operators with many applications to differential operators. The theory starts with a semi-bounded self-adjoint operator and constructs a continuum of related Hilbert spaces and self-adjoint operators that are intimately related with powers of the initial operator. The development spurred a flurry of activity in the field that is still ongoing today. The main goal of this expository (with the exception of Proposition~1) manuscript is to compare and contrast the complementary theories of general left-definite theory, the Birman-Krein-Vishik (BKV) theory of self-adjoint extensions and singular perturbation theory. In this way, we hope to encourage interest in left-definite theory as well as point out directions of potential growth where the fields are interconnected. We include several related open questions to further these goals.
For the entire collection see [Zbl 1479.47003].A multivariate version of Hammer's inequality and its consequences in numerical integrationhttps://zbmath.org/1496.650392022-11-17T18:59:28.764376Z"Guessab, Allal"https://zbmath.org/authors/?q=ai:guessab.allal"Semisalov, Boris"https://zbmath.org/authors/?q=ai:semisalov.boris-vladimirovichSummary: According to Hammer's inequality [\textit{P. C. Hammer}, Math. Mag. 31, 193--195 (1958; Zbl 0085.11402)], which is a refined version of the famous Hermite-Hadamard inequality, the midpoint rule is always more accurate than the trapezoidal rule for any convex function defined on some real numbers interval \([a,b]\). In this paper, we consider some properties of a multivariate extension of this result to an arbitrary convex polytope. The proof is based on the use of Green formula. In doing so, we will prove an inequality recently conjectured in [the authors, BIT 58, No. 3, 613--660 (2018; Zbl 1496.65040)] about a natural multivariate version of the classical trapezoidal rule. Our proof is based on a generalization of Hammer's inequality in a multivariate setting. It also provides a way to construct new ``extended'' cubature formulas, which give a reasonably good approximation to integrals in which they have been tested. We particularly pay attention to the explicit expressions of the best possible constants appearing in the error estimates for these cubature formulas.Numerical integration using integrals over hyperplane sections of simplices in a triangulation of a polytopehttps://zbmath.org/1496.650402022-11-17T18:59:28.764376Z"Guessab, Allal"https://zbmath.org/authors/?q=ai:guessab.allal"Semisalov, Boris"https://zbmath.org/authors/?q=ai:semisalov.boris-vladimirovichSummary: In this paper, we consider the problem of approximating a definite integral of a given function \(f\) when, rather than its values at some points, a number of integrals of \(f\) over some hyperplane sections of simplices in a triangulation of a polytope \(P\) in \(\mathbb{R}^d\) are only available. We present several new families of ``extended'' integration formulas, all of which are a weighted sum of integrals over some hyperplane sections of simplices, and which contain in a special case of our result multivariate analogues of the midpoint rule, the trapezoidal rule and the Simpson's rule. Along with an efficient algorithm for their implementations, several illustrative numerical examples are provided comparing these cubature formulas among themselves. The paper also presents the best possible explicit constants for their approximation errors. We perform numerical tests which allow the comparison of the new cubature formulas. Finally, we will discuss a conjecture suggested by the numerical results.Non-polynomial spectral-Galerkin method for time-fractional diffusion equation on unbounded domainhttps://zbmath.org/1496.651752022-11-17T18:59:28.764376Z"Darvishi, H."https://zbmath.org/authors/?q=ai:darvishi.h"Kerayechian, A."https://zbmath.org/authors/?q=ai:kerayechian.asghar"Gachpazan, M."https://zbmath.org/authors/?q=ai:gachpazan.morteza(no abstract)Effective numerical technique for solving variable order integro-differential equationshttps://zbmath.org/1496.651772022-11-17T18:59:28.764376Z"El-Gindy, Taha M."https://zbmath.org/authors/?q=ai:el-gindy.taha-m"Ahmed, Hoda F."https://zbmath.org/authors/?q=ai:ahmed.hoda-f"Melad, Marina B."https://zbmath.org/authors/?q=ai:melad.marina-bSummary: In this article, an effective numerical technique for solving the variable order Fredholm-Volterra integro-differential equations (VO-FV-IDEs), systems of VO-FV-IDEs and variable order Volterra partial integro-differential equations (VO-V-PIDEs) is given. The suggested technique is built on the combination of the spectral collocation method with some types of operational matrices of the variable order fractional differentiation and integration of the shifted fractional Gegenbauer polynomials (SFGPs). The proposed technique reduces the considered problems to systems of algebraic equations that are straightforward to solve. The error bound estimation of using SFGPs is discussed. Finally, the suggested technique's authenticity and efficacy are tested via presenting several numerical applications. Comparisons between the outcomes achieved by implementing the proposed method with other numerical methods in the existing literature are held, the obtained numerical results of these applications reveal the high precision and performance of the proposed method.Inclusion method of optimal constant with quadratic convergence for \(H_0^1\)-projection error estimates and its applicationshttps://zbmath.org/1496.652212022-11-17T18:59:28.764376Z"Kinoshita, Takehiko"https://zbmath.org/authors/?q=ai:kinoshita.takehiko"Watanabe, Yoshitaka"https://zbmath.org/authors/?q=ai:watanabe.yoshitaka"Yamamoto, Nobito"https://zbmath.org/authors/?q=ai:yamamoto.nobito"Nakao, Mitsuhiro T."https://zbmath.org/authors/?q=ai:nakao.mitsuhiro-tSummary: We present an interval inclusion method for optimal constants of second-order error estimates of \(H_0^1\)-projections to finite-degree polynomial spaces. These constants can be applied to error estimates of the Lagrange-type finite element method. Moreover, the proposed a priori error estimates are applicable to residual iteration techniques for the verification of solutions to nonlinear elliptic equations. Some numerical examples by the finite element method will be shown for comparison with other approaches, which confirm us the actual usefulness of the results in this paper for the numerical verification method for PDEs.Modulated wave and modulation instability gain brought by the cross-phase modulation in birefringent fibers having anti-cubic nonlinearityhttps://zbmath.org/1496.810472022-11-17T18:59:28.764376Z"Abbagari, Souleymanou"https://zbmath.org/authors/?q=ai:abbagari.souleymanou"Saliou, Youssoufa"https://zbmath.org/authors/?q=ai:saliou.youssoufa"Houwe, Alphonse"https://zbmath.org/authors/?q=ai:houwe.alphonse"Akinyemi, Lanre"https://zbmath.org/authors/?q=ai:akinyemi.lanre"Inc, Mustafa"https://zbmath.org/authors/?q=ai:inc.mustafa"Bouetou, Thomas B."https://zbmath.org/authors/?q=ai:bouetou-bouetou.thomasSummary: In this paper, we investigate the modulated wave and W-shaped profile in birefringent fibers having the anti-cubic nonlinearity terms. We use the traveling wave hypothesis to show out the velocity of the soliton and the constraint relation on the anti-cubic nonlinear terms. We use the Jacobi elliptic function solutions to point out two types of combined solutions. After some assumption on the modulus of the Jacobi elliptic function, we have shown out the combined bright-bright soliton and dark-dark soliton-like solutions. We use the linearizing algorithm to analyze the modulation instability (MI) growth rate. We have shown that the anti-cubic nonlinear terms and cross-phase modulation (XPM) can increase MI bands and the amplitude of the MI growth rate. To corroborate the prediction made on analytical results, we use the numerical investigation to show the propagation of the modulated wave and W-shaped profile in terms of cell index. We exhibited through the numerical results that the modulated wave can conserve high energy during its propagation in birefringent fibers. The obtained results will certainly open new perspectives in optical fibers during the transmission of huge data.Bound state solutions and thermodynamic properties of modified exponential screened plus Yukawa potentialhttps://zbmath.org/1496.810482022-11-17T18:59:28.764376Z"Antia, Akaninyene D."https://zbmath.org/authors/?q=ai:antia.akaninyene-d"Okon, Ituen B."https://zbmath.org/authors/?q=ai:okon.ituen-b"Isonguyo, Cecilia N."https://zbmath.org/authors/?q=ai:isonguyo.cecilia-n"Akankpo, Akaninyene O."https://zbmath.org/authors/?q=ai:akankpo.akaninyene-o"Eyo, Nsemeke E."https://zbmath.org/authors/?q=ai:eyo.nsemeke-eSummary: In this research paper, the approximate bound state solutions and thermodynamic properties of Schrödinger equation with modified exponential screened plus Yukawa potential (MESPYP) were obtained with the help Greene-Aldrich approximation to evaluate the centrifugal term. The Nikiforov-Uvarov (NU) method was used to obtain the analytical solutions. The numerical bound state solutions of five selected diatomic molecules, namely mercury hydride (HgH), zinc hydride (ZnH), cadmium hydride (CdH), hydrogen bromide (HBr) and hydrogen fluoride (HF) molecules were also obtained. We obtained the energy eigenvalues for these molecules using the resulting energy eigenequation and total unnormalized wave function expressed in terms of associated Jacobi polynomial. The resulting energy eigenequation was presented in a closed form and applied to study partition function (Z) and other thermodynamic properties of the system such as vibrational mean energy (U), vibrational specific heat capacity (C), vibrational entropy (S) and vibrational free energy (F). The numerical bound state solutions were obtained from the resulting energy eigenequation for some orbital angular quantum number. The results obtained from the thermodynamic properties are in excellent agreement with the existing literature. All numerical computations were carried out using spectroscopic constants of the selected diatomic molecules with the help of MATLAB 10.0 version. The numerical bound state solutions obtained increases with an increase in quantum state.Evolution of energy and magnetic moment of a quantum charged particle in power-decaying magnetic fieldshttps://zbmath.org/1496.810522022-11-17T18:59:28.764376Z"Dodonov, V. V."https://zbmath.org/authors/?q=ai:dodonov.victor-v"Horovits, M. B."https://zbmath.org/authors/?q=ai:horovits.m-bSummary: We consider a quantum spinless nonrelativistic charged particle moving in the \(xy\) plane under the action of a homogeneous time-dependent magnetic field \(B(t) = B_0(1 + t/t_0)^{-1-g}\), directed along the \(z\)-axis and described by means of the vector potential \(\mathbf{A}(t) = B(t)[-y, x]/2\). Assuming that the particle was initially in the thermal equilibrium state with a negligible coupling to a reservoir, we obtain exact formulas for the time-dependent mean values of the energy and magnetic moment in terms of the Bessel functions. Different regimes of the evolution are discovered and illustrated in several figures. The energy goes asymptotically to a finite value if \(g > 0\) (``fast'' decay), while it goes asymptotically to zero if \(g \leq 0\) (``slow'' decay). The dependence on parameter \(t_0\) practically disappears when \(1 + g\) is close to zero value (``superslow'' decay). The mean magnetic moment goes to zero for \(g > 1\), while it grows unlimitedly if \(g < 1\).Exact solutions of an asymmetric double well potentialhttps://zbmath.org/1496.810552022-11-17T18:59:28.764376Z"Sun, Guo-Hua"https://zbmath.org/authors/?q=ai:sun.guohua"Dong, Qian"https://zbmath.org/authors/?q=ai:dong.qian"Bezerra, V. B."https://zbmath.org/authors/?q=ai:bezerra.valdir-b"Dong, Shi-Hai"https://zbmath.org/authors/?q=ai:dong.shihaiSummary: The analytical solutions of an asymmetric double well potential \(V(x)=a\, x^2-b\, x^3+c\, x^4\) are found to be a triconfluent Heun function \(H_T(\alpha , \beta , \gamma ; z)\). It should be emphasized that these potential parameters are taken arbitrarily without any restriction on them. The wave functions which depend on the potential parameters are shrunk toward to the origin for given \(b\) and \(c\) when the parameter \(a\) increases, while they are moved far from the origin and toward to the left when the parameter \(b\) increases for given \(a\) and \(c\). Also, when the parameter \(c\) increases for given \(a\) and \(b\) they have the similar property to the case when the parameter \(a\) increases.Differential recurrences for the distribution of the trace of the \(\beta\)-Jacobi ensemblehttps://zbmath.org/1496.810912022-11-17T18:59:28.764376Z"Forrester, Peter J."https://zbmath.org/authors/?q=ai:forrester.peter-j"Kumar, Santosh"https://zbmath.org/authors/?q=ai:kumar.santosh.2|kumar.santosh.1|kumar.santosh.4|kumar.santosh|kumar.santosh.3Summary: Examples of the \(\beta\)-Jacobi ensemble in random matrix theory specify the joint distribution of the transmission eigenvalues in scattering problems. For this application, the trace is of relevance as determining the conductance. Earlier, in the case \(\beta = 1\), the trace statistic was isolated in studies of covariance matrices in multivariate statistics. There, Davis showed that for \(\beta = 1\) the trace statistic could be characterised by \((N + 1) \times (N + 1)\) matrix differential equations, now understood for general \(\beta > 0\) as part of the theory of Selberg correlation integrals. However the characterisation provided was incomplete, as the connection problem of determining the linear combination of Frobenius type solutions that correspond to the statistic was not solved. We solve this connection problem for Jacobi parameters \(b\) and Dyson index \(\beta\) non-negative integers. The distribution then has the functional form of a series of piecewise power functions times a polynomial, and our characterisation gives a recurrence for the computation of the polynomials. For all \(\beta > 0\) we express the Fourier-Laplace transform of the trace statistic in terms of a generalised hypergeometric function based on Jack polynomials.Super Hirota bilinear equations for the super modified BKP hierarchyhttps://zbmath.org/1496.811122022-11-17T18:59:28.764376Z"Chen, Huizhan"https://zbmath.org/authors/?q=ai:chen.huizhanSummary: In this paper, the super modified BKP (SmBKP) hierarchy is constructed from the perspective of the neutral free superfermions by using highest weight representations of the infinite-dimensional Lie superalgebra \(\mathfrak{b}_{\infty|\infty}(\mathfrak{g})\). Based upon this, the corresponding super Hirota bilinear identity of the SmBKP hierarchy is obtained by using the super Boson-Fermion correspondence of type B, and some specific examples of super Hirota bilinear equations are given. The super bilinear identity with respect to super wave and adjoint wave functions is also constructed. At last, we also give a class of solutions other than group orbit by the neutral free superfermions.Stationary optical solitons with complex Ginzburg-Landau equation having nonlinear chromatic dispersion and Kudryashov's refractive index structureshttps://zbmath.org/1496.811142022-11-17T18:59:28.764376Z"Ekici, Mehmet"https://zbmath.org/authors/?q=ai:ekici.mehmetSummary: This work obtains stationary optical solitons for complex Ginzburg-Landau equation that is structured with Kudryashov's self-phase modulation structures. The chromatic dispersion is also taken to be nonlinear. Six forms of nonlinear refractive index are considered. The adopted integration scheme is the generalized \(G^\prime/G\)-expansion approach that yields solutions in terms of Jacobi's elliptic functions. The limiting approach, when applied with the modulus of ellipticity leads to the stationary optical solitons that finally emerge from the model.