Recent zbMATH articles in MSC 33https://zbmath.org/atom/cc/332021-01-08T12:24:00+00:00WerkzeugHarmonic polynomials via differentiation.https://zbmath.org/1449.460312021-01-08T12:24:00+00:00"Estrada, Ricardo"https://zbmath.org/authors/?q=ai:estrada.ricardoSummary: It is well-known that if \(p\) is a homogeneous polynomial of degree \(k\) in \(n\) variables, \(p \in {\mathcal{P}_k}\), then the ordinary derivative \(p\left(\nabla \right)\left({{r^{2 - n}}} \right)\) has the form \({A_{n, k}}\mathcal{Y}\left(x \right){r^{2 - n - 2k}}\) where \({A_{n, k}}\) is a constant and \(\mathcal{Y}\) is a harmonic homogeneous polynomial of degree \(k\), \(\mathcal{Y} \in {\mathcal{H}_k}\), actually the projection of \(p\) onto \({\mathcal{H}_k}\). Here we study the distributional derivative \(p\left({\bar \nabla} \right)\left({{r^{2 - n}}} \right)\) and show that the ordinary part is still a multiple of \(\mathcal{Y}\), but that the delta part is independent of \(\mathcal{Y}\), that is, it depends only on \(p - \mathcal{Y}\). We also show that the exponent \(2 - n\) is special in the sense that the corresponding results for \(p\left(\nabla \right)\left({{r^\alpha}} \right)\) do not hold if \(\alpha \ne 2 - n\). Furthermore, we establish that harmonic polynomials appear as multiples of \({r^{2 - n - 2k - 2k'}}\) when \(p\left(\nabla \right)\) is applied to harmonic multipoles of the form \(\mathcal{Y}'\left(x \right){r^{2 - n - 2k'}}\) for some \(\mathcal{Y}' \in {\mathcal{H}_k}\).Trigonometric functions in elementary mathematics.https://zbmath.org/1449.330012021-01-08T12:24:00+00:00"Smýkalová, Radka"https://zbmath.org/authors/?q=ai:smykalova.radkaThe book discusses various elementary aspects and applications of trigonometric functions. Most of the text is accessible to high-school students, but it will be useful to teachers as well. Despite being elementary, it contains a significant amount of advanced material, including olympiad-level problems. The contents are as follows:
Chapter 1 deals with the history of trigonometric functions, beginning with the roots or trigonometry in the ancient Greece, proceeding to the developments in the medieval India and Arabic lands, and concluding with Euler's analytical approach to trigonometric functions.
The next three chapters provide a modern and systematic treatment of high-school trigonometry and trigonometric functions. Chapter 2 begins with trigonometry in right triangles, and includes geometric derivations of the trigonometric addition formulas. It covers some slightly more advanced topics, such as the construction of a regular pentagon, or expressing the values of trigonometric functions for special angle values in terms of radicals. Chapters 3 proceeds to high-school trigonometry of general triangles, and extends the definitions of trigonometric functions to arguments from the interval \((90^\circ,180^\circ)\). It covers the well-known laws of sines and cosines, and the less familiar law of tangents and Mollweide's formula. Finally, Chapter 4 uses the unit circle to introduce trigonometric functions for an arbitrary real argument, discusses the validity of addition formulas in this more general setting, and devotes considerable space to the solution of trigonometric equations and inequalities.
Chapter 5 contains an overview of many advanced (but still elementary) identities and inequalities involving trigonometric functions. The final Chapter 6 describes some applications, such as the solution of algebraic equations using trigonometric substitutions, derivations of various identities and inequalities using the trigonometric form of complex numbers, and the appearance of trigonometric functions in cartography (with emphasis on the Mercator projection).
Reviewer: Antonín Slavík (Praha)Solution in explicit form of non-local problem for differential equation with partial fractional derivative of Riemann-Liouville.https://zbmath.org/1449.353202021-01-08T12:24:00+00:00"Saĭganova, Svetlana Aleksandrovna"https://zbmath.org/authors/?q=ai:saiganova.svetlana-aleksandrovnaSummary: A non-local problem for a mixed type equation with partial fractional derivative of Riemann-Liouville is studied, boundary condition of which contains generalized operator of fractional integro-differentiation. Unique solution of the problem is then proved.Classical orthogonal polynomials and some new properties for their centroids of zeroes.https://zbmath.org/1449.330102021-01-08T12:24:00+00:00"Aloui, B."https://zbmath.org/authors/?q=ai:aloui.baghdadi"Chammam, W."https://zbmath.org/authors/?q=ai:chammam.wathekThe aims of the paper is to highlight new properties of the centroid of the zeros of a polynomial. As a illustration, they apply these techniques to \(O\)-classical orthogonal polynomials, where \(O\) is the derivative operator \(D\) or the \(q\)-derivative \(H\).
Reviewer: Francisco Pérez Acosta (La Laguna)Chebyshev type inequalities for conformable fractional integrals.https://zbmath.org/1449.260072021-01-08T12:24:00+00:00"Set, Erhan"https://zbmath.org/authors/?q=ai:set.erhan"Akdemir, Ahmet Ocak"https://zbmath.org/authors/?q=ai:akdemir.ahmet-ocak"Mumcu, Ilker"https://zbmath.org/authors/?q=ai:mumcu.ilkerSummary: In this article, firstly some necessary definitions and results involving fractional integrals are given. Secondly, a new identity involving conformable fractional integrals is given. Then, by using this identity, we establish new Chebyshev inequalities for the Chebyshev functional via conformable fractional integral.Integral representations and asymptotic expansion formulas of Mittag-Leffler-type function of two variables.https://zbmath.org/1449.330222021-01-08T12:24:00+00:00"Yashagin, Nikolaĭ Sergeevich"https://zbmath.org/authors/?q=ai:yashagin.nikolai-sergeevichSummary: A special function generalizing Mittag-Leffler-type function for two variables is considered. Integral representations for this function in different variation range of arguments for a certain value of parameters are obtained. Asymptotic formulas and asymptotic properties of this function for large arguments are established. Theorems for these formulas and these properties are provided.On two bivariate kinds of \((p,q)\)-Bernoulli polynomials.https://zbmath.org/1449.330162021-01-08T12:24:00+00:00"Sadjang, P. Njionou"https://zbmath.org/authors/?q=ai:sadjang.p-njionou|sadjang.patrick-njionou"Duran, Ugur"https://zbmath.org/authors/?q=ai:duran.ugurSummary: The main aim of this paper is to introduce and investigate \((p,q)\)-extensions of two bivariate kinds of Bernoulli polynomials and numbers. We firstly examine several \((p,q)\)-analogues of the Taylor expansions of products of some trigonometric functions and determine their coefficients which are also analyzed in detail. Then, we introduce two bivariate kinds of \((p,q)\)-Bernoulli polynomials and acquired multifarious formulas and relations including connection formulas, recurrence formulas, correlations with aforementioned coefficients, partial \((p,q)\)-differential equations and \((p,q)\)-integral representations.Bounding the convex combination of arithmetic and integral means in terms of one-parameter harmonic and geometric means.https://zbmath.org/1449.260532021-01-08T12:24:00+00:00"Qian, Wei-Mao"https://zbmath.org/authors/?q=ai:qian.weimao"Zhang, Wen"https://zbmath.org/authors/?q=ai:zhang.wen.3"Chu, Yu-Ming"https://zbmath.org/authors/?q=ai:chu.yumingSummary: In the article, we find the best possible parameters \(\lambda_{1}\), \(\mu_{1}\), \(\lambda_{2}\) and \(\mu_{2}\) on the interval \([0,1/2]\) such that the double inequalities \[H(a, b; \lambda_{1})<\alpha A(a,b)+(1-\alpha)T(a,b)<H(a, b; \mu_{1}),\] \[G(a, b; \lambda_{2})<\alpha A(a,b)+(1-\alpha)T(a,b)<G(a, b; \mu_{2})\] hold for all \(\alpha\in [0,1]\) and \(a, b>0\) with \(a\neq b\), where \(A(a,b)=(a+b)/2\), \(T(a,b)=2\int_{0}^{\pi/2}a^{\cos^{2}\theta}b^{\sin^{2}\theta}d\theta/\pi\), \(H(a, b; \lambda)=2[\lambda a+(1-\lambda)b][\lambda b+(1-\lambda)a]/(a+b)\), \(G(a, b; \mu)=\sqrt{[\mu a +(1-\mu)b][\mu b+(1-\mu)a]}\) are the arithmetic, integral, one-parameter harmonic and one-parameter geometric means of \(a\) and \(b\), respectively.Sharp bounds for Toader-type mean in terms of harmonic, geometric, centroidal and contra-harmonic means.https://zbmath.org/1449.260512021-01-08T12:24:00+00:00"He, Xiaohong"https://zbmath.org/authors/?q=ai:he.xiaohong"Xu, Huizuo"https://zbmath.org/authors/?q=ai:xu.huizuo"Qian, Weimao"https://zbmath.org/authors/?q=ai:qian.weimaoSummary: In this paper, we present the best possible parameters \({\alpha_1}, {\alpha_2}, {\alpha_3}, {\alpha_4}, {\beta_1}, {\beta_2}, {\beta_3}, {\beta_4} \in (0,1)\) such that the double inequalities \[\begin{array}{l}{\alpha_1}E (a, b) + (1 - {\alpha_1})G (a, b) < T[ A (a, b), G (a, b)] < {\beta_1}E (a, b) + (1 - {\beta_1})G (a, b), \\ {\alpha_2}E (a, b) + (1 - {\alpha_2})H (a, b) < T[A (a, b), G (a, b)] < {\beta_2}E (a, b) + (1 - {\beta_2})H (a, b), \\ {\alpha_3}C (a, b) + (1 - {\alpha_3})G (a, b) < T[A (a, b), G (a, b)] < {\beta_3}C (a, b) + (1 - {\beta_3})G (a, b), \\ {\alpha_4}C (a, b) + (1 - {\alpha_4})H (a, b) < T[A (a, b), G (a, b)] < {\beta_4}C (a, b) + (1 - {\beta_4})H (a, b)\end{array}\] hold for all \(a, b > 0\) with \(a \ne b\). As an application, we establish a new bound for the complete elliptic integral of second kind, where \[\begin{array}{l}H (a, b) = \frac{2ab}{a + b}, \;\;G (a, b) = \sqrt{ab}, \;\;E (a, b) = \frac{2 ({a^2} + ab + {b^2})}{3 (a + b)},\\ C (a, b) = \frac{{a^2} + {b^2}}{a + b}, \;\;T (a, b) = \frac{2}{\pi}\int_0^{\pi/2}\sqrt {{a^2}{\mathrm{cos}}^2t + {b^2}{\mathrm{sin}}^2t}{\mathrm{d}}t\end{array}\] are the harmonic, geometric, centroidal, contra-harmonic and Toader means of two numbers \(a\) and \(b\), respectively.A kernel-based technique to solve three-dimensional linear Fredholm integral equations of the second kind over general domains.https://zbmath.org/1449.653582021-01-08T12:24:00+00:00"Esmaeili, Hamid"https://zbmath.org/authors/?q=ai:esmaeili.hamid"Moazami, Davoud"https://zbmath.org/authors/?q=ai:moazami.davoudSummary: In this article, we study a kernel-based method to solve three-dimensional linear Fredholm integral equations of the second kind over general domains. The radial kernels are utilized as a basis in the discrete collocation method to reduce the solution of linear integral equations to that of a linear system of algebraic equations. Integrals appeared in the scheme are approximately computed by the Gauss-Legendre and Monte Carlo quadrature rules. The method does not require any background mesh or cell structures, so it is mesh free and accordingly independent of the domain geometry. Thus, for the three-dimensional linear Fredholm integral equation, an irregular domain can be considered. The convergence analysis is also given for the method. Finally, numerical examples are presented to show the efficiency and accuracy of the technique.Problem with shift for the third-order equation with discontinuous coefficients.https://zbmath.org/1449.353262021-01-08T12:24:00+00:00"Repin, Oleg Aleksandrovich"https://zbmath.org/authors/?q=ai:repin.oleg-aleksandrovich"Kumykova, Svetlana Kanshubievna"https://zbmath.org/authors/?q=ai:kumykova.svetlana-kanshubievnaSummary: The unique solvability of boundary value problem with Saigo operators for the third-order equation with multiple characteristics was investigated. The uniqueness theorem with constraints of inequality type on the known functions and different orders of generalized fractional integro-differentiation was proved. The existence of solution is equivalently reduced to the solvability of Fredholm integral equation of the second kind.Cubic transmuted uniform distribution: an alternative to beta and Kumaraswamy distributions.https://zbmath.org/1449.600202021-01-08T12:24:00+00:00"Rahman, Md. Mahabubur"https://zbmath.org/authors/?q=ai:rahman.md-mahabubur"Al-Zahrani, Bander"https://zbmath.org/authors/?q=ai:al-zahrani.bander-m"Shahbaz, Saman Hanif"https://zbmath.org/authors/?q=ai:shahbaz.saman"Shahbaz, Muhammad Qaiser"https://zbmath.org/authors/?q=ai:shahbaz.muhammad-qaiserSummary: In this article, a new cubic transmuted (CT) family of distributions has been proposed by adding one more parameter. We have introduced cubic transmuted uniform (CTU) distribution by using the proposed class. We have also provided a detail description of the statistical properties of the proposed CTU distribution along with its estimation and real-life application.On the nonlinear \(varPsi\)-Hilfer fractional differential equations.https://zbmath.org/1449.340232021-01-08T12:24:00+00:00"Kucche, Kishor D."https://zbmath.org/authors/?q=ai:kucche.kishor-d"Mali, Ashwini D."https://zbmath.org/authors/?q=ai:mali.ashwini-d"Sousa, J. Vanterler da C."https://zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.joseSummary: We consider the nonlinear Cauchy problem for \(\varPsi\)-Hilfer fractional differential equations and investigate the existence, interval of existence and uniqueness of solution in the weighted space of functions. The continuous dependence of solutions on initial conditions is proved via Weissinger fixed point theorem. Picard's successive approximation method has been developed to solve the nonlinear Cauchy problem for differential equations with \(\varPsi\)-Hilfer fractional derivative and an estimation has been obtained for the error bound. Further, by Picard's successive approximation, we derive the representation formulae for the solution of linear Cauchy problem for \(\varPsi\)-Hilfer fractional differential equation with constant coefficient and variable coefficient in terms of Mittag-Leffler function and generalized (Kilbas-Saigo) Mittag-Leffler function respectively.Biorthogonal systems of solutions of the Helmholtz equation in a cylindrical coordinate system.https://zbmath.org/1449.351922021-01-08T12:24:00+00:00"Sukhorol's'kyĭ, M. A."https://zbmath.org/authors/?q=ai:sukhorolskyi.m-a"Dostoĭna, V. V."https://zbmath.org/authors/?q=ai:dostoina.veronika-v"Veselovs'ka, O. V."https://zbmath.org/authors/?q=ai:veselovska.olga-vSummary: We constructed a system of solutions of the Helmholtz equation in cylindrical coordinates in the form of homogeneous polynomials by two biorthogonal systems of functions.Some new results for Chebyshev matrix polynomials of first kind.https://zbmath.org/1449.330132021-01-08T12:24:00+00:00"Shehata, Ayman"https://zbmath.org/authors/?q=ai:shehata.aymanSummary: The main aim of the present paper is to investigate some new relations and generating matrix functions for Chebyshev matrix polynomials of the first kind. Some consequences of our main results are also discussed.Fully diagonalized spectral methods for solving Neumann boundary value problems.https://zbmath.org/1449.653352021-01-08T12:24:00+00:00"Liu, Fujun"https://zbmath.org/authors/?q=ai:liu.fujun"Lu, Jing"https://zbmath.org/authors/?q=ai:lu.jingSummary: The fully diagonalized spectral methods using generalized Laguerre functions are proposed for the second-order elliptic problems with Neumann boundary conditions on the half line. Some Fourier-like Sobolev orthogonal basis functions are constructed for the diagonalized Laguerre spectral methods of Neumann boundary value problems. Numerical experiments demonstrate the effectiveness and the spectral accuracy.A note on sums of a class of series.https://zbmath.org/1449.330062021-01-08T12:24:00+00:00"Jun, Sungtae"https://zbmath.org/authors/?q=ai:jun.sungtae"Milovanović, Gradimir V."https://zbmath.org/authors/?q=ai:milovanovic.gradimir-v"Kim, Insuk"https://zbmath.org/authors/?q=ai:kim.insuk"Rathie, Arjun K."https://zbmath.org/authors/?q=ai:rathie.arjun-kumarSummary: The aim of this note is to provide sums of a unified class of series of the form \[S_i(a)=\sum_{k=0}^{\infty} (-1)^{k} \binom{a-i}{k} \frac{1}{2^{k}(a+k+1)}\] in the most general form for any \(i\in\mathbb{Z}\). For each \(\nu\in\mathbb{N}\), in four cases when \(i=\pm 2\nu\) and \(i=\pm(2\nu-1)\), simple explicit expressions for \(S_i(a)\) are obtained, e.g. \[S_{2\nu}(a)=\frac{2^{2\nu-1-a}}{(a-2\nu+1)_\nu}\left[\frac{\sqrt{\pi}\, \Gamma (a+1)}{\Gamma \left(a+\frac{3}{2}-\nu\right)}-P_{\nu-1}(a)\right],\] where \(P_\nu(a)\) is an algebraic polynomial in \(a\) of degree \(\nu\).
For \(i=1\) and \(a=n\) \((\in \mathbb{N})\), we recover the well known sum of the series due to \textit{M. Vowe} and \textit{H.-J. Seiffert} [Elem. Math. 42, No. 4, 111--112 (1987; Zbl 1253.33007)]. Several other known results due to \textit{H. M. Srivastava} [Proc. Japan Acad., Ser. A 65, No. 1, 8--11 (1989; Zbl 0653.33004)] and \textit{Y. S. Kim} et al. [Commun. Korean Math. Soc. 27, No. 4, 745--751 (2012; Zbl 1253.33004)] can be considered as special cases of our result.On the generalized Gauss hypergeometric function.https://zbmath.org/1449.330052021-01-08T12:24:00+00:00"Virchenko, N. A."https://zbmath.org/authors/?q=ai:virchenko.nina-aSummary: In this work the \((\tau, \beta)\)-hypergeometric Gauss function is considered, the basic properties of this function are investigated, some applications are given.New interesting Euler sums.https://zbmath.org/1449.110872021-01-08T12:24:00+00:00"Nimbran, Amrik Singh"https://zbmath.org/authors/?q=ai:nimbran.amrik-singh"Sofo, Anthony"https://zbmath.org/authors/?q=ai:sofo.anthonySummary: We present here some new and interesting Euler sums obtained by means of related integrals and elementary approach. We supplement Euler's general recurrence formula with two general formulas of the form
\[\sum_{n\geqslant 1} O_n^{(m)}\left(\frac{1}{(2n -1)^p} + \frac {1}{(2n)^p}\right)\quad\text{and}\quad\sum_{n\geqslant 1}\frac{O_n}{(2n-1)^p(2n+1)^q}, \]
where \(\displaystyle O_n^{(m)}= \sum_{j=1}^n \frac{1}{(2j-1)^m}\). Two formulas for \(\zeta (5)\) are also derived.A lower bound of the power exponential function.https://zbmath.org/1449.260202021-01-08T12:24:00+00:00"Nishizawa, Yusuke"https://zbmath.org/authors/?q=ai:nishizawa.yusukeSummary: In this paper, we consider the lower bound of the power exponential function \(a^{2b}+b^{2a}\) for nonnegative real numbers \(a\) and \(b\). If \(a+b=1\), then it is known that the function has the maximum value 1, but it is not known that the minimum value. In this paper, we show that \(a^{2b}+b^{2a}>6083/6144 \simeq 0.990072\) for nonnegative real numbers \(a\) and \(b\) with \(a+b=1\).Generalized multivariable Cauchy residue theorem and non-zero zeros of multivariable and multiparameters generalized Mittag-Leffler functions.https://zbmath.org/1449.330202021-01-08T12:24:00+00:00"Pathan, M. A."https://zbmath.org/authors/?q=ai:pathan.mahmood-ahmad"Kumar, Hemant"https://zbmath.org/authors/?q=ai:kumar.hemantSummary: The frequent requirements of Cauchy residue theorem in the analysis of many problems of mathematics and mathematical physics have inspired the present paper and the authors prove here the generalized multivariable Cauchy residue theorem. Then we make use of this theorem to derive Weierstrass type product formula for distribution of non-zero zeros of multivariable and multi-parameters generalized Mittag-Leffler functions.On two special functions, generalizing the Mittag-Leffler type function, their properties and applications.https://zbmath.org/1449.330192021-01-08T12:24:00+00:00"Ogorodnikov, Evgeniĭ Nikolaevich"https://zbmath.org/authors/?q=ai:ogorodnikov.evgenii-nikolaevichSummary: Two special functions, concerning Mittag-Leffler type functions, are studied. The first is the modification of generalized Mittag-Leffler function, which was introduced by A. A. Kilbas and M. Saigo; the second is the special case of the first one. The relation of these functions with some elementary and special functions and their role in solving of Abel-Volterra integral equations is indicated. The formulas of the fractional integration and differentiation in sense of Riemann-Liouville and Kober are presented. The applications to Cauchy type problems for some linear fractional differential equations with Riemann-Liouville and Kober derivatives are noticed.Chebyshev polynomials on circular arcs.https://zbmath.org/1449.300802021-01-08T12:24:00+00:00"Schiefermayr, Klaus"https://zbmath.org/authors/?q=ai:schiefermayr.klausThe Chebyshev polynomial of degree \(N\), \(N\in\mathbb{N}\), on a compact set \(K\subset\mathbb{C}\) in the complex plane is that monic polynomial \(\hat{\mathcal{P}}_N\in\hat{\mathbb{P}}_N\) which is minimal with respect to the supremum norm on \(K\) within the set of all monic polynomials, i.e. \[\hat{\mathcal{P}}_N:=\min\{\|\hat{P}_N\|_K:\hat{P}_N\in\hat{\mathbb{P}}_N\}\,, \] where \(\|\cdot\|\) denotes the supremum norm on \(K\) and \(\hat{\mathbb{P}}_N\) denotes the set of all monic polynomials of degree \(N\). In this paper, an explicit parametric representation of the complex Chebyshev polynomials \(\hat{P}_N(z)\) on a given circular arc \(A_\alpha\), defined by \[ A_\alpha:=\{z\in\mathbb{C}: |z|=1,-\alpha\leq\arg(z)\leq\alpha\},\quad 0<\alpha\leq\pi\,,\] of the unit circle (in the complex plane) in terms of real Chebyshev polynomials \(\hat{\mathcal{T}}_{N'}(x)\) on two symmetric intervals \([-1,-a]\cup[a,1]\) (on the real line) is given. For example, let \(0<\alpha<\frac{2n\pi}{2n+1}\), \(0<c<\frac{n\pi}{2n+1}\) be fixed and \(a:=\cos(\alpha/2)\). Let \(\mathcal{T}_{2n+1}\in\mathbb{P}_{2n+1}\) and \(\mathcal{U}_{2n-2}\in\mathbb{P}_{2n-2}\) be uniquely determined by \[ \mathcal{T}_{2n+1}^2(x)+(1-x^2)(x^2-a^2)(x^2-c^2)\mathcal{U}_{2n-2}^2(x)=1\,. \] Then \[\hat{P}_{2n}(z)=L_{2n}z^{n-1/2}\left(\mathcal{T}_{2n+1}(x)+i\sqrt{1-x^2}(x^2-a^2)\mathcal{U}_{2n-2}(x)\right)\,,\] is a monic polynomial of degree \(2n\) in \(z\) with real coefficients, where \(x\) and \(z\) are connected by \[ z\mapsto\frac12\left(\sqrt{z}+\frac1{\sqrt{z}}\right)=:x\,. \] Moreover, \(\hat{P}_{2n}(z)\) is the Chebyshev polynomial of degree \(2n\) on \(A_\alpha\) with minimum deviation. The case \(N=2n-1\) is also considered. It is also considered representation of Chebyshev polynomials on \([-1,-a]\cup[a,1]\) with the help of Jacobian elliptic and theta functions, which goes back to the work of Akhiezer in the 1930's.
Reviewer: Konstantin Malyutin (Kursk)Multivariable Hurwitz-Lerch Zeta function and related Apostol-Euler polynomials.https://zbmath.org/1449.110852021-01-08T12:24:00+00:00"Bin-Saad, M. G."https://zbmath.org/authors/?q=ai:binsaad.maged-g|bin-saad.maged-gumman|bin-saad.maged-gumaan"Bin-Alhag, A. Z."https://zbmath.org/authors/?q=ai:bin-alhag.ali-zSummary: The main object of this work is to introduce a new multivariable extension of the Hurwitz-Lerch Zeta function. We then systematically investigate its mathematical properties and give its explicit relationship with new defined Apostol-Euler polynomials of several variables. We also consider some important special cases.A kind of a triangular V-system construction scheme based on linear independent function groups.https://zbmath.org/1449.330152021-01-08T12:24:00+00:00"Wang, Xianghai"https://zbmath.org/authors/?q=ai:wang.xianghai"Li, Wei"https://zbmath.org/authors/?q=ai:li.wei.10|li.wei.8|li.wei.7|li.wei.9|li.wei|li.wei.5|li.wei-wayne"Lv, Fang"https://zbmath.org/authors/?q=ai:lv.fang"Song, Chuanming"https://zbmath.org/authors/?q=ai:song.chuanmingSummary: In recent years, with the development of non-continuous orthogonal function systems, a class of orthogonal complete function systems on \({L^2}[0, 1]\), U-system and V-system, have emerged, which have strong expression ability for continuous and discontinuous signals. Triangular patches are valued for their flexibility, convenience, and adaptability in complex surface modeling, and have significant advantages in 3D geometric modeling. This paper proposes a triangular domain 1st V-system construction scheme based on linear independent function groups. First, we select a set of linearly independent function groups under the 1st-level triangulation domain, then perform Gram-Schmidt orthogonalization to obtain 12 canonical orthogonal functions, and then rotate, compress and translate the generators. Other operations generate other functions of the V-system in turn. At the same time, the process of the V-system on the triangular domain in practical application is explained. The generators of the 1st V-system based on the linear independent function group constructed in this paper avoid a large number of 0, and the spectrum information of the 3D geometric model can be extracted more effectively in the application.Approximation of functions belonging to \(L[0, \infty)\) by product summability means of its Fourier-Laguerre series.https://zbmath.org/1449.420052021-01-08T12:24:00+00:00"Khatri, Kejal"https://zbmath.org/authors/?q=ai:khatri.kejal"Mishra, Vishnu Narayan"https://zbmath.org/authors/?q=ai:mishra.vishnu-narayanSummary: In this paper, we have proved the degree of approximation of functions belonging to \(L[0, \infty)\) by harmonic-Euler means of its Fourier-Laguerre series at \(x=0\). The aim of this paper is to concentrate on the approximation properties of the functions in \(L[0, \infty)\) by harmonic-Euler means of its Fourier-Laguerre series associated with the function \(f\).Solving parabolic integro-differential equations with purely nonlocal conditions by using the operational matrices of Bernstein polynomials.https://zbmath.org/1449.354252021-01-08T12:24:00+00:00"Bencheikh, Abdelkrim"https://zbmath.org/authors/?q=ai:bencheikh.abdelkrim"Chiter, Lakhdar"https://zbmath.org/authors/?q=ai:chiter.lakhdar"Li, Tongxing"https://zbmath.org/authors/?q=ai:li.tongxingSummary: Some problems from modern physics and science can be described in terms of partial differential equations with nonlocal conditions. In this paper, a numerical method which employs the orthonormal Bernstein polynomials basis is implemented to give the approximate solution of integro-differential parabolic equation with purely nonlocal integral conditions. The properties of orthonormal Bernstein polynomials, and the operational matrices for integration, differentiation and the product are introduced and are utilized to reduce the solution of the given integro-differential parabolic equation to the solution of algebraic equations. An illustrative example is given to demonstrate the validity and applicability of the new technique.On some applicable approximations of Gaussian type integrals.https://zbmath.org/1449.330232021-01-08T12:24:00+00:00"Chesneau, Christophe"https://zbmath.org/authors/?q=ai:chesneau.christophe"Navarro, Fabien"https://zbmath.org/authors/?q=ai:navarro.fabienSummary: In this paper, we introduce new applicable approximations for Gaussian type integrals. A key ingredient is the approximation of the function \(e^{-x^2}\) by the sum of three simple polynomial-exponential functions. Five special Gaussian type integrals are then considered as applications. Approximation of the so-called Voigt error function is investigated.Generalization of Bateman polynomials.https://zbmath.org/1449.330072021-01-08T12:24:00+00:00"Ali, Asad"https://zbmath.org/authors/?q=ai:ali.ali-a"Iqbal, Muhammad Zafar"https://zbmath.org/authors/?q=ai:iqbal.muhammad-zafar"Anwer, Bilal"https://zbmath.org/authors/?q=ai:anwer.bilal"Mehmood, Ather"https://zbmath.org/authors/?q=ai:mehmood.atherSummary: In this paper, we generalize the Bateman polynomials in terms of generalized hypergeometric function of the type \(_pF_p\) and we establish different forms of extended polynomials such as series expansion, generating functions and recurrence relations.Generalizations of Askey-Wilson integral with multiple variables.https://zbmath.org/1449.330182021-01-08T12:24:00+00:00"Cai, Liping"https://zbmath.org/authors/?q=ai:cai.liping"Cao, Jian"https://zbmath.org/authors/?q=ai:cao.jianSummary: In this paper, \(q\)-difference equations and related problems in special functions, whose formal solutions are \(q\)-polynomials, are discussed. Multi-variable Askey-Wilson integral and its inverse integral are extended by the method of \(q\)-difference equation. In addition, Bailey \(_6\phi_6\) summation is generalized.A variant of the Fejér-Jackson inequality.https://zbmath.org/1449.260182021-01-08T12:24:00+00:00"Alzer, Horst"https://zbmath.org/authors/?q=ai:alzer.horst"Kwong, Man Kam"https://zbmath.org/authors/?q=ai:kwong.man-kamThe following nice variant of the Fejér-Jackson inequality is proved: For all natural numbers \(n\) and real numbers \(x\in [0,\pi]\) we have \[-0.05781\ldots=\] \[-(5/48)\sqrt{130-58\sqrt{5}}\leq F_{n}(x),\] where \(F_{n}(x)=\sum_{k=1}^{n}\left( -1\right) ^{k+1}\left( \frac {\sin\left( (2k-1)x\right)}{2k-1}+\frac{\sin\left( 2kx\right)}{2k}\right) ;\) the sign of equality holds if and only if \(n=2\) and \(x=4\pi/5\).
Reviewer: Constantin Niculescu (Craiova)Generalized Apostol-type polynomial matrix and its algebraic properties.https://zbmath.org/1449.110392021-01-08T12:24:00+00:00"Quintana, Yamilet"https://zbmath.org/authors/?q=ai:quintana.yamilet"Ramírez, William"https://zbmath.org/authors/?q=ai:ramirez.william"Urieles, Alejandro"https://zbmath.org/authors/?q=ai:urieles.alejandroThe authors introduce the concept of the generalized Apostol-type polynomial matrix and of the Apostol-type matrix which involve Apostol-type polynomials and their values. Here the generalized Apostol-type polynomials in the variable \(x\) and parameters \(c,a,\lambda,\mu,\nu\) order \(\alpha\) and level \(m\) are defined by the exponential generating function \(\big(E_{1,m+1}^{(c,a; \lambda;\mu;\nu)}(z)\big)^\alpha c^{xz}\) where \(E_{1,m+1}^{(c,a;\lambda;\mu;\nu)}(z)\) is the Mittag-Leffler type function. This class of polynomials has been introduced by \textit{P. Hernández-Llanos} et al. [Result. Math. 68, No. 1--2, 203--225 (2015; Zbl 1335.11014)] and provides a unified representation of the generalized Apostol-type polynomials and the generalized Apostol-Bernoulli polynomials, Apostol-Euler polynomials and Apostol-Genocchi polynomials. In the main results of the paper the authors prove a product formula for generalized Apostol-type polynomial matrices, show their relationship with generealized Pascal matrices of the first kind and provide some factorizations of the Apostol-type matrices in terms of the Fibonacci and Lucas matrices, respectively.
Reviewer: Štefan Porubský (Praha)Concentrated force acting near the tip of an interface crack with a rigid overlay on its side.https://zbmath.org/1449.740362021-01-08T12:24:00+00:00"Vasil'eva, Yuliya Olegovna"https://zbmath.org/authors/?q=ai:vasileva.yuliya-olegovna"Sil'vestrov, Vasiliĭ Vasil'evich"https://zbmath.org/authors/?q=ai:silvestrov.vasilii-vasilevichSummary: Plane stress state near the tip of an interface crack induced by specified concentrated force is considered. One of the crack faces is partially reinforced by a rigid straight line overlay. The complex potentials, the stress intensity factors at the crack-tip are found, corresponding plots are presented.Hypergeometric type difference equations on nonuniform lattices: Rodrigues type representation for the second kind solution.https://zbmath.org/1449.330112021-01-08T12:24:00+00:00"Cheng, Jinfa"https://zbmath.org/authors/?q=ai:cheng.jinfa"Jia, Lukun"https://zbmath.org/authors/?q=ai:jia.lukunSummary: By building a second order adjoint equation, the Rodrigues type representation for the second kind solution of a second order difference equation of hypergeometric type on nonuniform lattices is given. The general solution of the equation in the form of a combination of a standard Rodrigues formula and a ``generalized'' Rodrigues formula is also established.New relation formula for generating functions.https://zbmath.org/1449.130182021-01-08T12:24:00+00:00"Chammam, Wathek"https://zbmath.org/authors/?q=ai:chammam.wathekSummary: In this paper, we develop a new relation between certain types of generating functions using formal algorithmic methods. As an application, we give a relation between the generating function and finite-type relations between polynomial sequences.Upper bounds for the approximation of some classes of bivariate functions by triangular Fourier-Hermite sums in the space \(L_{2,p}(\mathbb{R}^2)\).https://zbmath.org/1449.420442021-01-08T12:24:00+00:00"Shabozov, M. Sh."https://zbmath.org/authors/?q=ai:shabozov.mirgand-shabozovich"Dzhurakhonov, O. A."https://zbmath.org/authors/?q=ai:dzhurakhonov.olimdzhon-akmalovichThe authors study supprema of approximation of bivariate functions, generalizing research on the approximation of functions by algebraic polynomials on the real axis \(\mathbb{R}\) with Chebyshev weight \(\tilde{\rho}(x)=\exp{\{x^2\}}\). The main results are given in four theorems:
Theorem 1. Let \(m,N\in\mathbb{N}\), \(r\in\mathbb{Z}_{+}\), \(0<p\leq 2\), \(h\in (0,1]\) and let \(q\) be a nonnegative measurable summable function on \((0,h)\) which does not vanish identically. Then
\[\sup_{f\in L_{2,\rho}^{(r)}} \frac{N^rE_{N-1}(f)_{2,\rho}}{\left(\int_0^h \Omega_m^p(D^rf,t)_{2,\rho}q(t)dt\right)^{1/p}} = \frac{1}{\left\{\int_0^h (1-(1-t^2)^{N/2})^{mp}q(t)dt\right\}^{1/p}}. \]
Theorem 2. Let \(m\in\mathbb{N}\), \(r\in\mathbb{Z}_{+}\). Then, for an arbitrary \(N\in\mathbb{N}\)
\[\sup_{f\in L_{2,\rho}^{(r)}} \frac{N^r E_{N-1}(f)_{2,\rho}}{K(D^rf,\frac{1}{N^m})_{2,\rho}}=1.\]
Theorem 3. Let \(m,N\in\mathbb{N}\), \(r\in\mathbb{Z}_{+}\), \(k=0,1,2,\ldots,N\), \(0<p\leq 2\), \(0<H<1\) and \(q\geq 0\) be a measurable function summable on \((0,H)\) which is not equivalent to zero. Then \[\gamma_{N(N+1)/2+k}(H_{2,p}^r (\Omega_m,q);L_{2,\rho}) = E_{N-1}(HW_{2,p}^r(\Omega_m,q))_{2,\rho}=\] \[=N^{-r}\left\{ \int_0^H (1-(1-t^2)^{N/2})^{mpp}q(t)dt\right\}^{-1/p},\] where \(\gamma(\cdot)\) is any of the following widths: Bernstein, Gelfand, Kolmogorov, linear and projectional.
Theorem 4. Let \(\Phi\) be some majorant defining the class of functions \(W_{2,\rho}^r(K_m,\Phi)\), where \(r\in\mathbb{Z}_{+}\), \(m\in\mathbb{N}\). Then, for an arbitrary \(N\in\mathbb{N}\) and \(k=0,1,2,\ldots,N\)
\[\gamma_{N(N+2)/2+k}(W_{2,\rho}^r(K;\Phi);L_{2,\rho}) = E_{N-1}(W_{2,\rho}^r(K;\Phi))_{2,\rho}=N^{-r}\Phi(N^{-m}),\]
where \(\gamma_{nu}(\cdot)\) is any of the widths mentioned in Theorem 3. The specific definitions of the notations used would take to much space here.
The main concept is the space \(L_{2,\rho}=L_{2,\rho}(\mathbb{R}^2)\); real squared summable functions on \(\mathbb{R}^2\) with weight \(\rho(x)=\exp{\{-(x^2+y^2)\}}\). The system \(\{H_k(x)H_l(x)\}_{k,l\in\mathbb{Z}_{+}}\) of Hermite polynomials is orthogonal on the entire plane \(\mathbb{R}^2\) with weight \(\rho\) and the double Fourier-Hermite series is given by \[f(x,y)=\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\,c_{kl}(f)H_k(x)h_l(y),\] with \[c_{kl}(f)=\int\int_{\mathbb{R}^2}\,\rho(x,y)f(x,y)H_k(x)H_l(y)dx dy\] convergence in \(L_{2,\rho}(\mathbb{R}^2)\)-sense.
Reviewer: Marcel G. de Bruin (Heemstede)Some recursion formulas for \(q\)-Lauricella series.https://zbmath.org/1449.330172021-01-08T12:24:00+00:00"Verma, Ashish"https://zbmath.org/authors/?q=ai:verma.ashish"Sahai, Vivek"https://zbmath.org/authors/?q=ai:sahai.vivekSummary: We obtain certain recursion formulas for fourteen three variable \(q\)-Lauricella series. It is derived that these recursion relations can be expressed in terms of \(q\)-derivatives of the respective \(q\)-Lauricella series.Analytical properties of \((q, k)\)-Mittag Leffler function.https://zbmath.org/1449.330212021-01-08T12:24:00+00:00"Singh, V."https://zbmath.org/authors/?q=ai:singh.vishal-kumar|singh.vaibhav-kumar|singh.vijai-kumar|singh.vijaykumar|singh.v-b|singh.vinay-kumar|singh.vijeta|singh.vijay-k|singh.virendra-p|singh.vidhi|singh.vinay-pratap|singh.vipin-kumar|singh.vibha|singh.vir|singh.vasu|singh.vineet-kumar|singh.vikram|singh.vikramjeet|singh.vinai-kumar|singh.virendra-kumar|singh.vikash|singh.vijander|singh.vikrant|singh.vijay-a|singh.vikramaditya.1|singh.vijay-p|singh.vishnu-pratap|singh.vikramjit|singh.vijendra-pal|singh.vineeta|singh.vikendra|singh.virender-pal|singh.v-s|singh.viraj|singh.vanchna|singh.vandana|singh.v-n|singh.viplav-kumar|singh.vivek-kumar|singh.ved-pal|singh.vimal-pratap|singh.vanshdeep|singh.vinod-kumar|singh.v-p-n|singh.vikas-vikram|singh.veena|singh.varanasi|singh.varinder"Khan, M. A."https://zbmath.org/authors/?q=ai:khan.m-a-hakim|khan.muhammad-azam|khan.masood-amjad|khan.mohammad-azam|khan.mohd-anis|khan.maroof-a|khan.mutmaz-ahmad|khan.muhammad-ali.1|khan.mohd-ali|khan.mashir-a|khan.murtaza-ali|khan.muhammad-alamgir|khan.mushmir-ahmad|khan.muhammad-abdul-basit|khan.mashroor-ahmad|khan.md-asim-uddin|khan.majid-ali|khan.mushtaq-ahmad|khan.muhammad-aqeel-ahmad|khan.muhammad-abdul-rehman|khan.muhammad-asif|khan.md-al-amin|khan.meraj-ali|khan.mushir-ahmad|khan.muhammad-adil|khan.muhammad-ali|khan.m-a-i|khan.mohd-akram-raza|khan.mohammad-aamir|khan.mohammed-ali|khan.mohd-arsalan|khan.mohammad-asmat-ullah|khan.mohammad-aasim|khan.muhammad-ahsan|khan.muhammad-altaf|khan.m-aurangzeb|khan.mohammad-ali|khan.mohammed-a-u|khan.mumtaz-ahmad|khan.m-ashfaquzzaman|khan.md-atikur-rahman|khan.md-aquil|khan.moharram-a|khan.muazzam-ali"Khan, A. H."https://zbmath.org/authors/?q=ai:khan.abdul-hakim|khan.abdul-hamidSummary: This article refers to the study of \(q\) analogues, which play a significant role in number theory and combinatorics of annihilation. Here, we aim to present a new definition of a \((q,k)\)-Mittag Leffler function and discuss its various analytical properties. Certain special cases are derived, from the obtained results. And, we also point out their relevance with known results.A semi-Lagrangian spectral method for the Vlasov-Poisson system based on Fourier, Legendre and Hermite polynomials.https://zbmath.org/1449.652722021-01-08T12:24:00+00:00"Fatone, Lorella"https://zbmath.org/authors/?q=ai:fatone.lorella"Funaro, Daniele"https://zbmath.org/authors/?q=ai:funaro.daniele"Manzini, Gianmarco"https://zbmath.org/authors/?q=ai:manzini.gianmarcoSummary: In this work, we apply a semi-Lagrangian spectral method for the Vlasov-Poisson system, previously designed for periodic Fourier discretizations, by implementing Legendre polynomials and Hermite functions in the approximation of the distribution function with respect to the velocity variable. We discuss second-order accurate-in-time schemes, obtained by coupling spectral techniques in the space-velocity domain with a BDF time-stepping scheme. The resulting method possesses good conservation properties, which have been assessed by a series of numerical tests conducted on some standard benchmark problems including the two-stream instability and the Landau damping test cases. In the Hermite case, we also investigate the numerical behavior in dependence of a scaling parameter in the Gaussian weight. Confirming previous results from the literature, our experiments for different representative values of this parameter, indicate that a proper choice may significantly impact on accuracy, thus suggesting that suitable strategies should be developed to automatically update the parameter during the time-advancing procedure.On evaluation of Bessel functions of the first kind via Prony-like methods.https://zbmath.org/1449.330242021-01-08T12:24:00+00:00"Ji, Yu"https://zbmath.org/authors/?q=ai:ji.yu"He, Yixuan"https://zbmath.org/authors/?q=ai:he.yixuan"Wu, Guoqun"https://zbmath.org/authors/?q=ai:wu.guoqun"Wu, Min"https://zbmath.org/authors/?q=ai:wu.min.2|wu.min|wu.min.1Summary: Numerical approximations of Bessel functions are both of important theoretical significance and widely applied in mathematics, physics, and engineering. In this work, we apply two variants of Prony's method on Bessel functions of the first kind of integer order. The Prony-like methods in cosine or sine version yield approximations as sums of sinusoidal functions of Bessel functions of the first kind of integer order. In the symbolic computation software Maple, we compute the approximations for different orders and over different intervals, and compare these approximations with those obtained through the Fourier method. Experiments show that Prony-like methods perform much better than the Fourier method.Jacobi-Sobolev orthogonal polynomials and spectral methods for elliptic boundary value problems.https://zbmath.org/1449.330142021-01-08T12:24:00+00:00"Yu, Xuhong"https://zbmath.org/authors/?q=ai:yu.xuhong"Wang, Zhongqing"https://zbmath.org/authors/?q=ai:wang.zhongqing"Li, Huiyuan"https://zbmath.org/authors/?q=ai:li.huiyuanSummary: Generalized Jacobi polynomials with indexes \(\alpha\), \(\beta \in \mathbb{R}\) are introduced and some basic properties are established. As examples of applications, the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered, and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems, the Jacobi-Sobolev orthogonal basis functions are constructed, which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.Fractional-order Legendre functions for solving fractional-order differential equations.https://zbmath.org/1449.330122021-01-08T12:24:00+00:00"Kazem, S."https://zbmath.org/authors/?q=ai:kazem.saeed"Abbasbandy, S."https://zbmath.org/authors/?q=ai:abbasbandy.saeid"Kumar, Sunil"https://zbmath.org/authors/?q=ai:kumar.sunilSummary: In this article, a general formulation for the fractional-order Legendre functions (FLFs) is constructed to obtain the solution of the fractional-order differential equations. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, an efficient and reliable technique for the solution of them is important, too. For the concept of fractional derivative we will adopt Caputo's definition by using Riemann-Liouville fractional integral operator. Our main aim is to generalize the new orthogonal functions based on Legendre polynomials to the fractional calculus. Also a general formulation for FLFs fractional derivatives and product operational matrices is driven. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.A continued fraction approximation of the Gamma function related to the Gosper's formula.https://zbmath.org/1449.110772021-01-08T12:24:00+00:00"Tian, Dan"https://zbmath.org/authors/?q=ai:tian.dan"Wang, Liantang"https://zbmath.org/authors/?q=ai:wang.liantangSummary: Firstly, a continued fraction approximation of the Gamma function related to the Gosper formula is established, and the best constant and two-sided inequalities about Gamma function are obtained. Then, considering its simplest form, monotonicity, convexity and concavity are obtained.Time-stepping error bound for a stochastic parabolic Volterra equation disturbed by fractional Brownian motions.https://zbmath.org/1449.652562021-01-08T12:24:00+00:00"Qi, Ruisheng"https://zbmath.org/authors/?q=ai:qi.ruisheng"Lin, Qiu"https://zbmath.org/authors/?q=ai:lin.qiuSummary: In this paper, we consider a stochastic parabolic Volterra equation driven by the infinite dimensional fractional Brownian motion with Hurst parameter \(H \in \left[ {\frac{1}{2}, 1} \right)\). We apply the piecewise constant, discontinuous Galerkin method to discretize this equation in the temporal direction. Based on the explicit form of the scalar resolvent function and the refined estimates for the Mittag-Leffler function, we derive sharp mean-square regularity results for the mild solution. The sharp regularity results enable us to obtain the optimal error bound of the time discretization. These theoretical findings are finally accompanied by several numerical examples.Expansion and inequality involving the incomplete gamma function.https://zbmath.org/1449.330042021-01-08T12:24:00+00:00"Zhao, Wanying"https://zbmath.org/authors/?q=ai:zhao.wanyingSummary: The predecessors obtained several inequalities in terms of difference and singularity about \({I_p} (x)\). This paper first establishes an asymptotic expansion and then gets a new inequality based on this asymptotic expansion.On the functional equation \(G(x,G(y,x))=G(y,G(x,y))\) and means.https://zbmath.org/1449.330022021-01-08T12:24:00+00:00"Li, Lin"https://zbmath.org/authors/?q=ai:li.lin.2|li.lin|li.lin.1"Matkowski, Janusz"https://zbmath.org/authors/?q=ai:matkowski.januszA real valued function \(M(x,y)\) is called a mean, if \(\min(x,y)\le M(x,y)\le \max(x,y)\) for all \(x,y\) in some interval. A mean is called weighted quasi-arithmetic, if there exists a strictly monotone function \(h(x)\) and a number \(w\in (0,1)\) such that \(M(x,y) = h^{-1} (w h(x)+(1-w) h(y))\).
In the present paper the authors show that every continuous and reducible solution of the functional equation \(G(x,G(y,x)) = G(y,G(x,y))\) generates a mean resembling a weighted quasi-arithmetic mean, but no weighted quasi-arithmetic mean is a solution of this equation.
Reviewer: Khristo N. Boyadzhiev (Ada)Some applications of the \( (f,g)\)-inversion.https://zbmath.org/1449.050282021-01-08T12:24:00+00:00"Mu, Yanping"https://zbmath.org/authors/?q=ai:mu.yanping"Tong, Xiaozhou"https://zbmath.org/authors/?q=ai:tong.xiaozhouSummary: We present three kinds of applications of the \( (f, g)\)-inversion. By taking explicit functions and sequences in the \( (f, g)\)-inversion, we derive identities involving hypergeometric series and harmonic numbers. Then we give several inversion relations involving \(q\)-hypergeometric terms. Finally, we combine the \( (f, g)\)-inversion and the \(q\)-differential operators to derive some \(q\)-series identities.An unconditionally stable Laguerre based finite difference method for transient diffusion and convection-diffusion problems.https://zbmath.org/1449.651762021-01-08T12:24:00+00:00"De Sousa, Wescley T. B."https://zbmath.org/authors/?q=ai:de-sousa.wescley-t-b"Matt, Carlos F. T."https://zbmath.org/authors/?q=ai:matt.carlos-frederico-trottaSummary: This paper describes an application of weighted Laguerre polynomial functions to produce an unconditionally stable and accurate finite-difference scheme for the numerical solution of transient diffusion and convection-diffusion problems. The unconditionally stability of Laguerre-FDM (L-FDM) is guaranteed by expanding the time dependency of the unknown potential as a series of orthogonal functions in the domain \( (0,\infty)\), avoiding thus any time integration scheme. The L-FDM is a marching-on-in-degree scheme instead of traditional marching-on-in-time methods. For the two heat-transfer problems, we demonstrated the accuracy, numerical stability and computational efficiency of the proposed L-FDM by comparing its results against closed-form analytical solutions and numerical results obtained from classical finite-difference schemes as, for instance, the alternating direction implicit method.Bounds for some entropies and special functions.https://zbmath.org/1449.940492021-01-08T12:24:00+00:00"Bărar, Adina"https://zbmath.org/authors/?q=ai:barar.adina-elena"Mocanu, Gabriela Raluca"https://zbmath.org/authors/?q=ai:mocanu.gabriela-raluca"Raşa, Ioan"https://zbmath.org/authors/?q=ai:rasa.ioanSummary: We consider a family of probability distributions depending on a real parameter and including the binomial, Poisson and negative binomial distributions. The corresponding index of coincidence satisfies a Heun differential equation and is a logarithmically convex function. Combining these facts we get bounds for the index of coincidence, and consequently for Rényi and Tsallis entropies of order 2.Fractional calculus and integral transforms of the \(M\)-Wright function.https://zbmath.org/1449.330092021-01-08T12:24:00+00:00"Khan, N. U."https://zbmath.org/authors/?q=ai:khan.nabiullah-u"Kashmin, T."https://zbmath.org/authors/?q=ai:kashmin.t"Khan, S. W."https://zbmath.org/authors/?q=ai:khan.shorab-waliSummary: This paper is concerned to investigate \(M\)-Wright function, which was earlier known as transcendental function of the Wright type. \(M\)-Wright function is a special case of the Wright function given by British mathematician (E. Maitland Wright) in 1933. We have explored the cosequences of Riemann-Liouville Integral and Differential operators on \(M\)-Wright function. We have also evaluated integral transforms of the \(M\)-Wright function.An indirect finite element method for variable-coefficient space-fractional diffusion equations and its optimal-order error estimates.https://zbmath.org/1449.653312021-01-08T12:24:00+00:00"Zheng, Xiangcheng"https://zbmath.org/authors/?q=ai:zheng.xiangcheng"Ervin, V. J."https://zbmath.org/authors/?q=ai:ervin.vincent-j"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.1Summary: We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension. By the representation formula of the solutions \(u(x)\) to the proposed variable coefficient models in terms of \(v(x)\), the solutions to the constant coefficient analogues, we apply finite element methods for the constant coefficient fractional diffusion equations to solve for the approximations \(v_h(x)\) to \(v(x)\) and then obtain the approximations \(u_h(x)\) of \(u(x)\) by plugging \(v_h(x)\) into the representation of \(u(x)\). Optimal-order convergence estimates of \(u(x)-u_h(x)\) are proved in both \(L^2\) and \(H^{\alpha /2}\) norms. Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.Approximation of correlation function and power spectral density with Sonin-Laguerre orthogonal functions.https://zbmath.org/1449.420432021-01-08T12:24:00+00:00"Prokhorov, S. A."https://zbmath.org/authors/?q=ai:prokhorov.s-a"Kulikovskikh, I. M."https://zbmath.org/authors/?q=ai:kulikovskikh.i-mSummary: Approximate capabilities of Sonin-Laguerre orthogonal functions with predefined parameter of orthogonal basis are studied. Parameters of approximation expression are evaluated, that are used for construction of the models of correlation function and power spectral density in compliance with the minimal weighted quadratic error of approximation.Exponential time differencing methods for the time-space-fractional Schrödinger equation.https://zbmath.org/1449.651892021-01-08T12:24:00+00:00"Liang, Xiao"https://zbmath.org/authors/?q=ai:liang.xiao"Bhatt, Harish"https://zbmath.org/authors/?q=ai:bhatt.harish-pSummary: In this paper, exponential time differencing schemes with Padé approximation to the Mittag-Leffler function are proposed for the time-space-fractional nonlinear Schrödinger equations. Ways of increasing the efficiency of the proposed schemes are discussed. Numerical experiments are performed on the time-space-fractional nonlinear Schrödinger equations with various parameters. The accuracy, efficiency, and reliability of the proposed method are illustrated by numerical results.Notes on explicit and inversion formulas for the Chebyshev polynomials of the first two kinds.https://zbmath.org/1449.110452021-01-08T12:24:00+00:00"Qi, Feng"https://zbmath.org/authors/?q=ai:qi.feng"Niu, Da-Wei"https://zbmath.org/authors/?q=ai:niu.dawei"Lim, Dongkyu"https://zbmath.org/authors/?q=ai:lim.dongkyuSummary: In the paper, starting from the Rodrigues formulas for the Chebyshev polynomials of the first and second kinds, by virtue of the Faà di Bruno formula, with the help of two identities for the Bell polynomials of the second kind, and making use of a new inversion theorem for combinatorial coefficients, the authors derive two nice explicit formulas and their corresponding inversion formulas for the Chebyshev polynomials of the first and second kinds.A cardinal method to solve coupled nonlinear variable-order time fractional sine-Gordon equations.https://zbmath.org/1449.354372021-01-08T12:24:00+00:00"Heydari, Mohammad Hossein"https://zbmath.org/authors/?q=ai:heydari.mohammadhossein"Avazzadeh, Zakieh"https://zbmath.org/authors/?q=ai:avazzadeh.zakieh"Yang, Yin"https://zbmath.org/authors/?q=ai:yang.yin"Cattani, Carlo"https://zbmath.org/authors/?q=ai:cattani.carloSummary: In this study, a computational approach based on the shifted second-kind Chebyshev cardinal functions (CCFs) is proposed for obtaining an approximate solution of coupled variable-order time-fractional sine-Gordon equations where the variable-order fractional operators are defined in the Caputo sense. The main ideas of this approach are to expand the unknown functions in tems of the shifted second-kind CCFs and apply the collocation method such that it reduces the problem into a system of algebraic equations. To algorithmize the method, the operational matrix of variable-order fractional derivative for the shifted second-kind CCFs is derived. Meanwhile, an effective technique for simplification of nonlinear terms is offered which exploits the cardinal property of the shifted second-kind CCFs. Several numerical examples are examined to verify the practical efficiency of the proposed method. The method is privileged with the exponential rate of convergence and provides continuous solutions with respect to time and space. Moreover, it can be adapted for other types of variable-order fractional problems straightforwardly.Evaluation of a class of terminating \(_3F_2(\frac{4}{3})\)-series.https://zbmath.org/1449.330082021-01-08T12:24:00+00:00"Chen, X."https://zbmath.org/authors/?q=ai:chen.xiuyang|chen.xiaojiao|chen.xuemiao|chen.xiangtuo|chen.xueyuan|chen.xiaoying|chen.xuejun|chen.xibi|chen.xueqin|chen.xiaole|chen.xingtong|chen.xianfu|chen.xiaosong|chen.xuzong|chen.xing|chen.xinmng|chen.xintong|chen.xiyu|chen.xiaotong|chen.xumin|chen.xiaohong.3|chen.xiaochao|chen.xuechun|chen.xuqi|chen.xuejing|chen.xiangning|chen.xiaoyong|chen.xi.2|chen.xueyou|chen.xiuqin|chen.xiuping|chen.xinxin|chen.xiuidong|chen.xiaoyou|chen.xin.1|chen.xian|chen.xinghua|chen.xinxiang|chen.xuemin.2|chen.xiandong|chen.xinjiao|chen.xuerong|chen.xigeng|chen.xianyong|chen.ximeng|chen.xiangzhi|chen.xinhong|chen.xingmin|chen.xiaoming|chen.xiaolei|chen.xieyuanli|chen.xiaoqun|chen.xiaoxu|chen.xiangfu|chen.xiankang|chen.xiongda|chen.xisong|chen.xinyi|chen.xiaoqian|chen.xiangzhen|chen.xichun|chen.xiaoli|chen.xiao|chen.xiaohe|chen.xueyan|chen.xiaobiao|chen.xiaofa|chen.xuyong|chen.xianjia|chen.xudong|chen.xida|chen.xiaoshuang|chen.xianwei|chen.xinjia|chen.xiaobo|chen.xingqian|chen.xueping|chen.xueyun|chen.xueying|chen.xianzhe|chen.xihan|chen.xiuming|chen.xinzhong|chen.xuling|chen.xiaoou|chen.xuechen|chen.xueqiang|chen.xiangchuan|chen.xun|chen.xiaogen|chen.xianqiang|chen.xiaoxing|chen.xinguo|chen.xiaoyi|chen.xiyuan|chen.xiaojing|chen.xingyu|chen.xiaming|chen.xi|chen.xuerui|chen.xiaopeng|chen.xinguang|chen.xiongwen|chen.xianzhang|chen.xianming|chen.xujin|chen.xiaojuan|chen.xuzhou|chen.xiubo|chen.xue|chen.xiancheng|chen.xueling|chen.xianglian|chen.xia|chen.xu-yan|chen.xueli|chen.xianchu|chen.xiaorong|chen.xiangrui|chen.xinying|chen.xiaoru|chen.xiaoxian|chen.xiaofeng|chen.xiuqiong|chen.xiqiong|chen.xiancun|chen.xuesheng|chen.xiangwang|chen.xinjie|chen.xuzhi|chen.xihnian|chen.xiaozhu|chen.xianhua|chen.xurong|chen.xinzhao|chen.xiexiong|chen.xinlei|chen.xinquan|chen.xunchao|chen.xiqiu|chen.xiuwan|chen.xufeng|chen.xiaoqin|chen.xuemei|chen.xianyu|chen.xiuwei|chen.xiaotao|chen.xiaolin|chen.xiaoqiu|chen.xiaoyan|chen.xuanqing|chen.xiaoguang|chen.xianqing|chen.xiaojun|chen.xiaocen|chen.xuefeng|chen.xuegong|chen.xiaoxi|chen.xiwei|chen.xingshu|chen.xuwen|chen.xiaoxin|chen.xuezhang|chen.xuning|chen.xiaokun|chen.xiaoke|chen.xiangen|chen.xiaoni|chen.xiaozhi|chen.xianglan|chen.xumei|chen.xioushu|chen.xinfang|chen.xinshuang|chen.xingran|chen.xingping|chen.xinyun|chen.xuezhou|chen.xiaolong|chen.xiaosu|chen.xiangping|chen.xingfu|chen.xingguo|chen.xihao|chen.xuan|chen.xilong|chen.xiangyun|chen.xiaomei|chen.xuelin|chen.xingchi|chen.xizhen|chen.xiaohua|chen.xiumei|chen.xujun|chen.xinchu|chen.xinmeng|chen.xiangfei|chen.xinyuan|chen.xiangying|chen.xingwen|chen.xiuyuan|chen.xiaoyang|chen.xiaoji|chen.xiangguang|chen.xiuxiong|chen.xiong|chen.xiaoping|chen.xiangyu|chen.xiangsong|chen.xingfa|chen.xugunag|chen.xiaoning|chen.xiangbing|chen.xiliang|chen.xiuqing|chen.xiaoyu|chen.xi.1|chen.xinliang|chen.xianyan|chen.xiaonan|chen.xueqian|chen.xiaodong|chen.ximing|chen.xusheng|chen.xinli|chen.xibin|chen.xuming|chen.xiangyong|chen.xueguang|chen.xiuqiu|chen.xinglin|chen.xuehua|chen.xuelong|chen.xiaoman|chen.xingru|chen.xuenong|chen.xiangqing|chen.xiangyi|chen.xingyi|chen.xiaohuai|chen.xuyun|chen.xiaoheng|chen.xiangwei|chen.xiaowu|chen.xiuhua|chen.xikang|chen.xiangxian|chen.xiqing|chen.xingxing|chen.xingfan|chen.xinming|chen.xingdi|chen.xiaowu.1|chen.xiangling|chen.xiai|chen.xiaobing|chen.xiao.1|chen.xubing|chen.xianghong.1|chen.xuemin.1|chen.xuesong|chen.xianzhong|chen.xianqiao|chen.xuanyi|chen.xianmin|chen.xuejuan|chen.xuejin|chen.xiaopan|chen.xianshun|chen.xiang|chen.xugao|chen.xingguang|chen.xianjin|chen.xiaofei|chen.xiebin|chen.xuewen|chen.xingjiang|chen.xia.1|chen.xie|chen.xuhui|chen.xiangrong|chen.xiaochun|chen.xiangjian|chen.xingang|chen.xiusu|chen.xiqun|chen.xiaokai|chen.xiyang|chen.xinfu|chen.xirong|chen.xianjiang|chen.xiaowen|chen.xiaofang|chen.xinyue|chen.xioahong|chen.xueyao|chen.xueer|chen.xiaojun.1|chen.xiaoming.1|chen.xiaozhou|chen.xindu|chen.xiuyin|chen.xide|chen.xuegang|chen.xiaoyuan|chen.xiaoqing|chen.xinjian|chen.xuehui|chen.xiaogang|chen.xiazhong|chen.xiping|chen.xiqian|chen.xiangfeng|chen.xiaozheng|chen.xinxing|chen.xiangtang|chen.xubin|chen.xiaoe|chen.xuewu|chen.xiaokang|chen.xidi|chen.xianyi|chen.xuelei|chen.xiufang|chen.xinyiao|chen.xinren|chen.xiu|chen.xinjuan|chen.xinkai|chen.xuyang|chen.xinlong|chen.xiaoyun|chen.xiaobao|chen.xiuli|chen.xinglong|chen.xingyuan|chen.xiuhong|chen.xuechang|chen.xiaohui|chen.xiulong|chen.xiaoqiang|chen.xinyu|chen.xilin|chen.xizhong|chen.xiaohan|chen.xiangyang|chen.xueyong|chen.xuekun|chen.xiangjun|chen.xiaoxuan|chen.xiaojiang|chen.xingxin|chen.xuebo|chen.xiaoling|chen.xiaoxiang|chen.xinwei|chen.xiexong|chen.xixi|chen.xiongshan|chen.xi.4|chen.xiujuan|chen.xiaojie|chen.xin|chen.xuebing|chen.xiaoshi|chen.xiaoyue|chen.xuedong|chen.xianghui|chen.xianli|chen.xinbei|chen.xiaocui|chen.xianping|chen.xingding|chen.xiru|chen.xiudong|chen.xixian|chen.xiaoliang|chen.xiaoshan|chen.xiaopei|chen.xihong|chen.xiyou|chen.xingyue|chen.xu|chen.xueen|chen.xiaoting|chen.xiuhuan|chen.xiaoxia|chen.xingwang|chen.xingyong|chen.xijun|chen.xueqing|chen.xiongzhi|chen.xiying|chen.xiaozhao|chen.xuzhong|chen.xiangqun|chen.xianghong|chen.xiaoguo|chen.xiaoxiao|chen.xianze|chen.xingzhen|chen.xianfeng|chen.xiangdong|chen.xinmei|chen.xinghuan|chen.xiaotian|chen.xiangxun|chen.xucan|chen.xueqi|chen.xuanguang|chen.xiaomin|chen.xueru|chen.xinhai|chen.xi.5|chen.xueye|chen.xingrong|chen.xiaorui|chen.xianyao|chen.xiaodan|chen.xiaolan|chen.xiuzhen|chen.xuefei|chen.xinrun|chen.xiaowei"Chu, W."https://zbmath.org/authors/?q=ai:chu.weijiang|chu.wenchang|chu.wensong|chu.weijuan|chu.wanghuan|chu.weipan|chu.weiwei|chu.wenten|chu.wangli|wang.tan-chu|chu.wenqing|chu.wei|chu.wang|chu.weiqi|chu.wanglin|chu.wanmingThe authors define and study a large class of terminating \(_3F_2(\frac{4}{3})\)-series. They involve two extra integer parameters. The signs of these parameters need special treatment. At the end of the paper numerous examples are calculated for the demonstration of the results.
Reviewer: István Mező (Nanjing)Mean invariance identity.https://zbmath.org/1449.330032021-01-08T12:24:00+00:00"Matkowski, Janusz"https://zbmath.org/authors/?q=ai:matkowski.januszSummary: For a continuous and increasing function \(f\) in a real interval \(I\), and a bivariable mean \(P\) defined in \(I^2\), we prescribe a pair of bivariable means \(M\) and \(N\) such that the quasiarithmetic mean \(A_f\) generated by \(f\) is invariant with respect to the mean-type mapping \((M,N)\). This allows to find effectively the limit of the iterates of the mean-type mapping \((M,N)\). The means \(M\) and \(N\) are equal iff \(P\) is the arithmetic mean \(A\); they are symmetric iff so is \(P\). Treating \(f\) and \(P\) as the parameters, we obtain the family of all pairs of means \((M,N)\) such that the quasiarithmetic mean \(A_f\) is invariant with respect to \((M,N)\). In particular, we indicate the function \(f\) and the mean \(P\) such that the invariance identity \(A_f\circ (M,N) = A_f\) coincides with the equality \(G\circ (H,A)\), where \(G\) and \(H\) are the geometric and harmonic means, equivalent to the classical Pythagorean harmony proportion.
Some examples and an application are also presented.Nonlocal problem for a equation of mixed type of third order with generalized operators of fractional integro-differentiation of arbitrary order.https://zbmath.org/1449.353252021-01-08T12:24:00+00:00"Repin, Oleg Aleksandrovich"https://zbmath.org/authors/?q=ai:repin.oleg-aleksandrovich"Kumykova, Svetlana Kanshubievna"https://zbmath.org/authors/?q=ai:kumykova.svetlana-kanshubievnaSummary: The unique solvability of internally boundary value problem for equation of mixed type of third order with multiple characteristics is investigated. The uniqueness theorem is proved with the restrictions on certain features and different orders of fractional integro-differentiation. The existence of solution is equivalent reduced to a Fredholm integral equation of the second kind.