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hypergeometric Functions: From One Scalar Variable to Several Matrix Arguments, in Statistics and Beyond

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Open Journal of Statistics 2016年第5期6卷 951-994页

作者： T. Pham-Gia Dinh Ngoc Thanh Department of Mathematics and Statistics Université de Moncton Moncton Canada The Applied Multivariate Research Group Université de Moncton Moncton Canada Department of Mathematics and Computer Science University of Science Vietnam National University Ho Chi Minh City Vietnam

hypergeometric functions have been increasingly present in several disciplines including Statistics, but there is much confusion on their proper uses, as well as on their existence and domain of definition. In this article, we try to clarify several points and give a general overview of the topic, going from the univariate case to the matrix case, in one and then in several arguments. We also survey some results in fields close to Statistics, where hypergeometric functions are actively used, studied and developed.

关键词： hypergeometric Zonal Polynomial Fractional Calculus Lie Group Cohomology Computation

The Origin and Mathematical Characteristics of the Super-Universal Associated-Legendre Polynomials

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Communications in Theoretical Physics 2014年第9期61卷 331-337页

作者：陈昌远尤源陆法林孙东升董世海 School of Physics and Electronics Yancheng Teachers University Escuela Superior de Físicay Matemáticas Instituto Politécnico Nacional Edificio 9 Unidad Profesional Adolfo López Mateos Mexico D.F. 07738 Mexico Department of Physics and Astronomy Louisiana State University Baton Rouge LA 70803-4001 USA

We study the mathematical characteristics of the super-universal associated-Legendre polynomials arising from a kind of double ring-shaped potentials and obtain their polar angular wave functions with certain parity. We find that there exists the even or odd parity for the polar angular wave functions when the parameter η is taken to be positive integer, while there exist both even and odd parities when η is taken as positive non-integer real values. The relations among the super-universal associated-Legendre polynomials, the hypergeometric polynomials, and the Jacobi polynomials are also established.

关键词： double ring-shaped potential super-universal associated Legendre polynomials parity hypergeometric functions Jacobi polynomials

A Subclass of Harmonic Functions Associated with Wright’s hypergeometric Functions

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Applied Mathematics 2010年第2期1卷 87-93页

作者： Gangadharan Murugusundaramoorthy Kalliyapan Vijaya 不详

We introduce a new class of complex valued harmonic functions associated with Wright hypergeometric functions which are orientation preserving and univalent in the open unit disc. Further we define, Wright generalized operator on harmonic function and investigate the coefficient bounds, distortion inequalities and extreme points for this generalized class of functions.

关键词： Harmonic Univalent Starlike Functions Harmonic Convex Functions Wright hypergeometric Functions Raina-Dziok Operator Distortion Bounds Extreme Points

MONOTONICITY,CONVEXITY AND INEQUALITIES INVOLVING THE GENERALIZED ELLIPTIC INTEGRALS

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Acta Mathematica Scientia 2019年第5期39卷 1440-1450页

作者：王淼坤张稳褚玉明 Department of Mathematics Huzhou University Huzhou 313000 China Friedman Brain Institute Icahn School of Medicine at Mount Sinai New York NY 10029 USA College of Science Hunan City University Yiyang 313000 China

We establish the monotonicity and convexity properties for several special functions involving the generalized elliptic integrals, and present some new analytic inequalities.

关键词： Gaussian hypergeometric function generalized elliptic integrals convexity inequality

Bergman kernel on generalized exceptional Hua domain

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Science China Mathematics 2002年第3期45卷 321-334页

作者：殷慰萍赵振刚 Department of Mathematics Capital Normal University Beijing 100037 China

We have computed the Bergman kernel functions explicitly for two types of generalized exceptional Hua domains, and also studied the asymptotic behavior of the Bergman kernel function of exceptional Hua domain near bou... 详细信息

关键词： Bergman kernel function, exceptional domain, hypergeometric function, Hua domain.

Generalized Form of Hermite Matrix Polynomials via the hypergeometric Matrix Function

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Advances in Linear Algebra & Matrix Theory 2014年第2期4卷 134-141页

作者： Raed S. Batahan Department of Mathematics Faculty of Science Hadhramout University Hadhranout Yemen

The object of this paper is to present a new generalization of the Hermite matrix polynomials by means of the hypergeometric matrix function. An integral representation, differential recurrence relation and some other properties of these generalized forms are established here. Moreover, some new properties of the Hermite and Chebyshev matrix polynomials are obtained. In particular, the two-variable and two-index Chebyshev matrix polynomials of two matrices are presented.

关键词： Hermite and Chebyshev Matrix Polynomials Three Terms Recurrence Relation hypergeometric Matrix Function and Gamma Matrix Function

Wright Type hypergeometric Function and Its Properties

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Advances in Pure Mathematics 2013年第3期3卷 335-342页

作者： Snehal B. Rao Jyotindra C. Prajapati Ajay K. Shukla Department of Applied Mathematics and Humenities S.V. National Institute of Technology Surat India Department of Applied Mathematics The M.S. University of Baroda Vadodara India Department of Mathematical Sciences Faculty of Applied Sciences Charotar University of Science and Technology Anand India

Let s and z be complex variables, Γ(s) be the Gamma function, and for any complex v be the generalized Pochhammer symbol. Wright Type hypergeometric Function is defined (Virchenko et al. [1]), as: where which is a direct generalization of classical Gauss hypergeometric Function 2F1(a,b;c;z). The principal aim of this paper is to study the various properties of this Wright type hypergeometric function 2R1(a,b;c;τ;z);which includes differentiation and integration, representation in terms of pFq and in terms of Mellin-Barnes type integral. Euler (Beta) transforms, Laplace transform, Mellin transform, Whittaker transform have also been obtained;along with its relationship with Fox H-function and Wright hypergeometric function.

关键词： Euler Transform Fox H-Function Wright Type hypergeometric Function Laplace Transform Mellin Transform Whittaker Transform Wright hypergeometric Function

Abel’s lemma on summation by parts and partial q-series transformations

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Science China Mathematics 2009年第4期52卷 720-748页

作者： CHU WenChang WANG ChenYing Department of Applied Mathematics Dalian University of TechnologyDalian 116024China

The partial sums of basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several transformation and summation formulae for well-poised, quadratic, cubic and quartic q... 详细信息

关键词： Abel’s lemma on summation by parts basic hypergeometric series well-poised series quadratic series cubic series quartic series reciprocal relation

Torsional wave frequency in heterogeneous earth crust lying over dry sandy semi-infinite substratum

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Applied Mathematics and Mechanics(English Edition) 2019年第10期40卷 1399-1412页

作者： S.K.VISHWAKARMA R.KAUR T.R.PANIGRAHI Department of Mathematics Birla Institute of Technology and SciencePilani Hyderabad Campus

The present study is carried out to investigate the transference of torsional surface waves in a heterogeneous anisotropic crust lying over a dry sandy half-space. The rigidities and densities as well as the initial stress are assumed varying as a function of depth in both the media. These variations are the product of the polynomial function of depth in degree n (n ∈ R) and the exponential function of depth. Following the theory of elastic waves, the mathematical model is established. Separation of variables is used to obtain the displacement in the layer and the half-space. Intrinsic boundary conditions are imposed to derive the dispersion equation. The inhomogeneity parameters associated with the rigidity, the density, and the initial stress of the medium are found to have substantial influence on the phase velocity of the torsional surface wave. The graphical presentations are drawn to exhibit the findings. The results thus obtained are significant for the investigation and characterization of torsional surface wave in the heterogeneous anisotropic layer.

关键词： hypergeometric function Laguerre function torsional wave inhomogeneity gravity initial stress sandy medium