Explicit closed forms are derived for the probability density function of the stable distribution Sα(β, γ, μ), α ∈ (1, 2]. Consequent asymptotic expansions are given. The expressions involve the Srivastava-Daoust generalized Kampé de Fériet hypergeometric S-function, the Fox-Wright generalized hypergeometric Ψ-function, and the Gauss hypergeometric function 2F1. 2000 Mathematics subject classif...

In this paper, we derive conditions on the parameters a, b, c so that the function zF (a, b; c; z) is starlike in D, where F (a, b; c; z) denotes the classical hypergeometric function. We give some consequences of our results including some mapping properties of convolution operator.

In a recent paper [A-O], the author and K. Ono study the “Gaussian” hypergeometric series 4F3(1)p over the finite field Fp. They describe relationships between values of these series, Fourier coefficients of modular forms, and the arithmetic of a certain algebraic variety. These relationships, together with tools from p-adic analysis and some unexpected combinatorial identities, lead to the pro...

Let p be an odd prime. The purpose of this paper is to refine methods of Ahlgren and Ono [2] and Kilbourn [13] in order to prove a general mod p congruence for the Gaussian hypergeometric series n+1Fn(λ) where n is an odd positive integer. As a result, we extend three recent supercongruences. The first is a result of Ono and Ahlgren [2] on a supercongruence for Apéry numbers which was conjectur...

In this paper, we propose a matrix-variate generalization of the Gauss hypergeometric distribution and study several of its properties. We also derive probability density functions of the product of two independent random matrices when one of them is Gauss hypergeometric. These densities are expressed in terms of Appell’s first hypergeometric function F1 and Humbert’s confluent hypergeometric f...

Based on Spiridonov's analysis of elliptic generalizations of the Gauss hypergeomet-ric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E 7 (ellip-tic, hyperbolic) and of type E 6...

Recently, J. A. Tirao [Proc. Nat. Acad. Sci. 100 (14) (2003), 8138–8141] considered a matrix-valued analogue of the 2F1 Gauß hypergeometric function and showed that it is the unique solution of a matrix-valued hypergeometric equation analytic at z = 0 with value I, the identity matrix, at z = 0. We give an independent proof of Tirao’s result, extended to the more general setting of hypergeometr...

the purpose of this paper is to derive various useful subordination properties and characteristics for certain subclass of multivalent meromorphic functions, which are defined here by the multiplier transformation. also, we obtained inclusion relationship for this subclass.

Various methods to obtain the analytic continuation near z = 1 of the hypergeometric series p+1Fp(z) are reviewed together with some of the results. One approach is to establish a recurrence relation with respect to p and then, after its repeated use, to resort to the well-known properties of the Gaussian hypergeometric series. Another approach is based on the properties of the underlying gener...