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Fast and stable rational approximation of generalized hypergeometric functions

NUMERICAL ALGORITHMS 2025年第2期98卷 587-624页

作者： Slevinsky, Richard Mikael Univ Manitoba Dept Math Winnipeg MB Canada

Rational approximations of generalized hypergeometric functions pFq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_pF_q$$\end{document} of type (n+k,k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+k,k)$$\end{document} are constructed by the Drummond and factorial Levin-type sequence transformations. We derive recurrence relations for these rational approximations that require O[max{p,q}(n+k)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}[\max \{p,q\}(n+k)]$$\end{document} flops. These recurrence relations come in two forms: for the successive numerators and denominators;and, for an auxiliary rational sequence and the rational approximations themselves. Numerical evidence suggests that these recurrence relations are much more stable than the original formul AE for the Drummond and factorial Levin-type sequence transformations. Theoretical results on the placement of the poles of both transformations confirm the superiority of factorial Levin-type transformation over the Drummond transformation.

关键词： generalized hypergeometric functions Drummond sequence transformation Factorial Levin-type sequence transformation

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Pearson equations for discrete orthogonal polynomials: I. generalized hypergeometric functions and Toda equations

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STUDIES IN APPLIED MATHEMATICS 2022年第3期148卷 1141-1179页

作者： Manas, Manuel Fernandez-Irisarri, Itsaso Gonzalez-Hernandez, Omar F. Univ Complutense Madrid Dept Fis Teor Plaza Ciencias 1 Madrid 28040 Spain Inst Ciencias Matemat ICMAT Campus Cantoblanco UAM Madrid 28049 Spain

The Cholesky factorization of the moment matrix is applied to discrete orthogonal polynomials on the homogeneous lattice. In particular, semiclassical discrete orthogonal polynomials, which are built in terms of a discrete Pearson equation, are studied. The Laguerre-Freud structure semiinfinite matrix that models the shifts by +/- 1 in the independent variable of the set of orthogonal polynomials is introduced. In the semiclassical case it is proven that this Laguerre-Freud matrix is banded. From the well-known fact that moments of the semiclassical weights are logarithmic derivatives of generalized hypergeometric functions, it is shown how the contiguous relations for these hypergeometric functions translate as symmetries for the corresponding moment matrix. It is found that the 3D Nijhoff-Capel discrete Toda lattice describes the corresponding contiguous shifts for the squared norms of the orthogonal polynomials. The continuous 1D Toda equation for these semiclassical discrete orthogonal polynomials is discussed and the compatibility equations are derived. It is also shown that the Kadomtesev-Petviashvilii equation is connected to an adequate deformed semiclassical discrete weight, but in this case, the deformation does not satisfy a Pearson equation.

关键词： 3D Nijhoff-Capel discrete Toda equation Cholesky factorization contigous relations discrete orthogonal polynomials generalized hypergeometric functions Pearson equations Toda hierarchy

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Inductance Calculations for Circular Coils With Rectangular Cross Section and Parallel Axes Using Inverse Mellin Transform and generalized hypergeometric functions

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IEEE TRANSACTIONS ON POWER ELECTRONICS 2017年第2期32卷 1367-1374页

作者： Luo, Yao Wang, Xiaowei Zhou, Xinglong Wuhan Univ Sch Elect Engn Wuhan 430072 Peoples R China South Cent Univ Nationalities Coll Elect & Informat Wuhan 430074 Peoples R China

The inductance integrals of air-core circular coils with rectangular cross section and parallel axes are difficult to tackle with analytical method due to the inclusion of Struve functions. Using the inverse Mellin transform, these Bessel-Struve integrals will be continued analytically to the complex plane, and then by virtue of contour deformation and residue theorem, they can be expanded to the series containing the generalized hypergeometric functions which are easy to execute in common mathematical software packages. The series solutions are preferable in case the difficulties arise in the evaluations of Bessel-Struve integrals, though they do not cover the whole region of coil geometric parameters. In addition, the obtained series were compared numerically with the corresponding integrals, complete consistency was shown clearly. By means of the proposed method, the computation efficiency can be improved by several orders of magnitude compared to that of the corresponding integral methods.

关键词： generalized hypergeometric functions inductance calculations inverse mellin transform residue theorem

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HYPERDIRE-hypergeometric functions DIfferential REduction: Mathematica-based packages for the differential reduction of generalized hypergeometric functions: Lauricella function F_C of three variables

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COMPUTER PHYSICS COMMUNICATIONS 2016年 206卷 78-83页

作者： Bytev, Vladimir V. Kniehl, Bernd A. Univ Hamburg Inst Theoret Phys 2 D-22761 Hamburg Germany Joint Inst Nucl Res Dubna 141980 Moscow Region Russia

We present a further extension of the HYPERDIRE project, which is devoted to the creation of a set of Mathematica-based program packages for manipulations with Horn-type hypergeometric functions on the basis of differential equations. Specifically, we present the implementation of the differential reduction for the Lauricella function F-C of three variables. Program summary Program title: HYPERDIRE Catalogue identifier: AEPP_v4_0 Program summary URL: http://***/summaries/AEPP_v4_*** Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 3 No. of lines in distributed program, including test data, etc.: 243461 No. of bytes in distributed program, including test data, etc.: 61610782 Distribution format: *** Programming language: Mathematica. Computer: All computers running Mathematica. Classification: 4.4. Does the new version supersede the previous version?: No, it significantly extends the previous version. Nature of problem: Reduction of hypergeometric function F-C of three variables to a set of basis functions. Solution method: Differential reduction. Reasons for new version: The extension package allows the user to handle the Lauricella function F-C of three variables. Summary of revisions: The previous version goes unchanged. Running time: Depends on the complexity of the problem. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Feynman integrals generalized hypergeometric functions Differential reduction

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Radially geometric stable distributions and processes

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ADVANCES IN APPLIED PROBABILITY 2025年

作者： Torricelli, Lorenzo Univ Bologna Bologna Italy Dept Stat Sci P Fortunati Via Belle Arti 41 I-40126 Bologna Italy

Motivated by the investigation of probability distributions with finite variance but heavy tails, we study infinitely divisible laws whose L & eacute;vy measure is characterized by a radial component of geometric (tempered) stable type. We closely investigate the univariate case: characteristic exponents and cumulants are calculated, as well as spectral densities;absolute continuity relations are shown, and short- and long-time scaling limits of the associated L & eacute;vy processes analyzed. Finally, we derive some properties of the involved probability density functions.

关键词： Geometric stable distributions tempered stable distributions L & eacute vy processes Mittag-Leffler function generalized hypergeometric functions 60 33C

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Meijer's G-Function and Euler's Differential Equation Revisited

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COMPUTATIONAL METHODS AND FUNCTION THEORY 2024年 1-23页

作者： Gesztesy, Fritz Hunziker, Markus Baylor Univ Dept Math Sid Richardson Bldg 1410 S 4th St Waco TX 76706 USA

We consider the generalized eigenvalue problem for the classical Euler differential equation and demonstrate its intimate connection with Meijer's G-functions. In the course of deriving the solution of the generalized Euler eigenvalue equation we review some of the basics of generalized hypergeometric functions and Meijer's G-functions and some of its special cases where the underlying Mellin-type integrand exhibits higher-order poles.

关键词： Euler's differential equation Meijer's G-function generalized hypergeometric functions Strongly singular coefficients

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EXPANSION OF hypergeometric functions IN TERMS OF POLYLOGARITHMS WITH A NONTRIVIAL CHANGE OF VARIABLES

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THEORETICAL AND MATHEMATICAL PHYSICS 2024年第3期219卷 871-896页

作者： Bezuglov, M. A. Onishchenko, A. I. Joint Inst Nucl Res Bogoliubov Lab Theoret Phys Dubna Russia Budker Inst Nucl Phys Novosibirsk Russia Lomonosov Moscow State Univ Skobeltsyn Inst Nucl Phys Moscow Russia

hypergeometric functions of one and many variables play an important role in various branches of modern physics and mathematics. We often encounter hypergeometric functions with indices linearly dependent on a small parameter with respect to which we need to perform Laurent expansions. Moreover, it is desirable that such expansions be expressed in terms of well-known functions that can be evaluated with arbitrary precision. To solve this problem, we use the method of differential equations and the reduction of corresponding differential systems to a canonical basis. In this paper, we are interested in the generalized hypergeometric functions of one variable and in the Appell and Lauricella functions and their expansions in terms of the Goncharov polylogarithms. Particular attention is paid to the case of rational indices of the considered hypergeometric functions when the reduction to the canonical basis involves a nontrivial variable change. The paper comes with a Mathematica package Diogenes, which provides an algorithmic implementation of the required steps.

关键词： generalized hypergeometric functions Appell and Lauricella functions

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Asymptotic behavior of Clebsch-Gordan coefficients

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JOURNAL OF MATHEMATICAL CHEMISTRY 2024年第10期62卷 2761-2775页

作者： Akdemir, S. Ozay, S. Oztekin, E. Sinop Univ Fac Educ Dept Sci Educ Sinop Turkiye Ondokuz Mayis Univ Fac Sci Dept Phys Samsun Turkiye

In this study, the Clebsch-Gordan coefficients are expressed using the generalized hypergeometric functions of unit argument and binomial coefficients, and their behavior as a function of quantum numbers is examined. The Clebsch-Gordan coefficients are also numerically computed for integer and non-integer quantum numbers. Subsequently, the convergence between the Clebsch-Gordan and asymptotic Clebsch-Gordan coefficients is investigated based on the orbital angular momentum quantum numbers.

关键词： Clebsch-Gordan coefficients Vector addition coefficients Wigner coefficients Classical limits of the Clebsch-Gordan coefficients generalized hypergeometric functions

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Highly Accurate Method for a Singularly Perturbed Coupled System of Convection-Diffusion Equations with Robin Boundary Conditions

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JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS 2024年第1期31卷 17页

作者： Ahmed, H. M. Helwan Univ Fac Technol & Educ Dept Math Helwan Egypt

This paper's major goal is to provide a numerical approach for estimating solutions to a coupled system of convection-diffusion equations with Robin boundary conditions (RBCs). We devised a novel method that used four homogeneous RBCs to generate basis functions using generalized shifted Legendre polynomials (GSLPs) that satisfy these RBCs. We provide new operational matrices for the derivatives of the developed polynomials. The collocation approach and these operational matrices are utilized to find approximate solutions for the system under consideration. The given system subject to RBCs is turned into a set of algebraic equations that can be solved using any suitable numerical approach utilizing this technique. Theoretical convergence and error estimates are investigated. In conclusion, we provide three illustrative examples to demonstrate the practical implementation of the theoretical study we have just presented, highlighting the validity, usefulness, and applicability of the developed approach. The computed numerical results are compared to those obtained by other approaches. The methodology used in this study demonstrates a high level of concordance between approximate and exact solutions, as shown in the presented tables and figures.

关键词： Legendre polynomials generalized hypergeometric functions Collocation method Boundary value problems Robin boundary conditions

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Highly Accurate Method for Boundary Value Problems with Robin Boundary Conditions

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JOURNAL OF NONLINEAR MATHEMATICAL PHYSICS 2023年第3期30卷 1239-1263页

作者： Ahmed, Hany. M. Helwan Univ Fac Technol & Educ Dept Math Helwan Egypt

The main aim of the current paper is to construct a numerical algorithm for the numerical solutions of second-order linear and nonlinear differential equations subject to Robin boundary conditions. A basis function in terms of the shifted Chebyshev polynomials of the first kind that satisfy the homogeneous Robin boundary conditions is constructed. It has established operational matrices for derivatives of the constructed polynomials. The obtained solutions are spectral and are consequences of the application of collocation method. This method converts the problem governed by their boundary conditions into systems of linear or nonlinear algebraic equations, which can be solved by any convenient numerical solver. The theoretical convergence and error estimates are discussed. Finally, we support the presented theoretical study by presenting seven examples to ensure the accuracy, efficiency, and applicability of the constructed algorithm. The obtained numerical results are compared with the exact solutions and results from other methods. The method produces highly accurate agreement between the approximate and exact solutions, which are displayed in tables and figures.

关键词： Chebyshev polynomials of the first kind generalized hypergeometric functions Collocation method Boundary value problems Robin boundary conditions

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