We formulate and prove a combinatorial criterion to decide if an A-hypergeometric system of differential equations has a full set of algebraic solutions or not. This criterion generalises the so-called interlacing criterion in the case of hypergeometric functions of one variable.

UNLABELLED This paper deals with the study of a generalized function of Mittag-Leffler type. Various properties including usual differentiation and integration, Euler(Beta) transforms, Laplace transforms, Whittaker transforms, generalized hypergeometric series form with their several special cases are obtained and relationship with Wright hypergeometric function and Laguerre polynomials is also...

From numerical experiments, D. E. Knuth conjectured that 0 < Dn+4 < Dn for a combinatorial sequence (Dn) defined as the difference Dn = Rn − Ln of two definite hypergeometric sums. The conjecture implies an identity of type Ln = bRnc, involving the floor function. We prove Knuth’s conjecture by applying Zeilberger’s algorithm as well as classical hypergeometric machinery.

The Abel method on summation by parts is reformulated to present new and elementary proofs of several classical identities of terminating balanced basic hypergeometric series. The examples strengthen our conviction that as traditional analytical instrument, the revised Abel method on summation by parts is indeed a very natural choice for working with basic hypergeometric series.

In this article, we describe a new algorithm for the expansion of hypergeometric functions about half-integer parameters. The implementation of this algorithm for certain classes of hypergeometric functions in the already existing Mathematica package HypExp is described. Examples of applications in Feynman diagrams with up to four loops are given. PACS:02.10.De, 02.30.Gp

Abstract. The matrix valued analog of the Euler’s hypergeometric differential equation was introduced by Tirao in [1]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao’s equation in order to get a generalized matrix valued hypergeometri...

It is well known that many bilateral basic hypergeometric identities can be derived from unilateral identities. Using Cauchy’s method [5, 15, 20, 21] one may obtain bilateral basic hypergeometric identities from terminating unilateral identities. Starting with nonterminating unilateral basic hypergeometric series, Chen and Fu [8] developed a method to construct semi-finite forms by shifting the...

I will describe three sets of results in which computer search and computer algebra played a large role in the discovery/and or proof of results each ultimately relating to hypergeometric functions. 1. Ap ery: A generating function for (4n+3). 2. Ramanujan: Integrals, means and transformations of the hypergeometric function 2F1(1=3;2=3;1; 1 x): 3. Euler: Multivalued {function values and a produ...

In this paper, we derive a necessary and sufficient condition on the parameters of the Hypergeometric distribution for weak convergence to a Normal limit. We establish a Berry-Esseen theorem for the Hypergeometric distribution solely under this necessary and sufficient condition. We further derive a nonuniform Berry-Esseen bound where the tails of the difference between the Hypergeometric and t...