We prove a master theorem for hypergeometric functions of Karlsson–Minton type, stating that a very general multilateral U(n) Karlsson–Minton type hypergeometric series may be reduced to a finite sum. This identity contains the Karlsson–Minton summation formula and many of its known generalizations as special cases, and it also implies several “Bailey-type” identities for U(n) hypergeometric se...

We study the action of the Hecke operators Un on the set of hypergeometric functions, as well as on formal power series. We show that the spectrum of these operators on the set of hypergeometric functions is the set {na : n ∈ N and a ∈ Z}, and that the polylogarithms play a dominant role in the study of the eigenfunctions of the Hecke operators Un on the set of hypergeometric functions. As a co...

In this paper we compute the Galois groups of basic hypergeometric equations. In this paper q is a complex number such that 0 < |q| < 1. 1 Basic hypergeometric series and equations The theory of hypergeometric functions and equations dates back at least as far as Gauss. It has long been and is still an integral part of the mathematical literature. In particular, the Galois theory of (generalize...

Some integral properties of zonal spherical functions, hypergeometric functions and invariant polynomials are studied for real normed division algebras.

Euler’s two-term transformation formula for the Gauss hypergeometric function 2F1 is extended to hypergeometric functions of higher order. The generalized two-term transformation follows from the theory of Fuchsian differential equations with accessory parameters, although it also has a combinatorial proof. Unusually, it constrains the hypergeometric function parameters algebraically and not li...

Throughout the paper, we denote this equation by E(a, b, c). Since Kummer we know that in general there are 24 hypergeometric series which express solutions of E(a, b, c). The subject of this paper is Gauss hypergeometric functions when some or all of the numbers a, b, c, c − a, c − b, a − b, c − a − b are integers. We refer to these functions and to corresponding hypergeometric equations as de...

Though connections between a well established theory of analytic univalent functions and hypergeometric functions have been investigated by several researchers, yet analogous connections between planer harmonic mappings and hypergeometric functions have not been explored. The purpose of this paper is to uncover some of the inequalities associating hypergeometric functions with planer harmonic m...

The paper classifies algebraic transformations of Gauss hypergeometric functions and pull-back transformations between hypergeometric differential equations. This classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of the Gauss hypergeometric function.

We formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hy-pergeometric series. Single and multivariable elliptic hypergeometric series are considered in detail. A characterization theorem for a single variable totally elliptic hypergeometric series is proved.

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...