Hypergeometric series with noncommutative parameters and argument, in the special case involving square matrices, have recently been studied by a number of researchers including (in alphabetical order) Durán, Duval, Grünbaum, Iliev, Ovsienko, Pacharoni, Tirao, and others. See [3, 6, 8, 9, 10, 11, 12, 16] for some selected papers. The subject of hypergeometric series involving matrices is closel...

The transformation which assigns to a linear operator L the recurrence satis ed by coe cient se quences of the polynomial series in its kernel is shown to be an isomorphism of the corresponding operator algebras We use this fact to help factoring q di erence and recurrence operators and to nd nice power series solutions of linear di erential equations In particular we characterize generalized h...

It is suggested that the (p, q)-hypergeometric series studied by Burban and Klimyk (in Integral Transforms and Special Functions 2 (1994) 15-36) can be considered as a special case of a more general (P, Q)-hypergeometric series. 1. Introduction I propose a general (P, Q)-hypergeometric series. In this, I am inspired mainly by the paper of Burban and Klimyk [1] titled " P, Q-differentiation, P, ...

In this minicourse I would like to present computer algebra algorithms for the work with orthogonal polynomials and special functions. This includes • the computation of power series representations of hypergeometric type functions, given by “expressions”, like arcsin(x)/x , • the computation of holonomic differential equations for functions, given by expressions, • the computation of holonomic...

The considered problem is uniform convergence of sequences of hypergeometric series. We give necessary and sufficient conditions for uniformly dominated convergence of infinite sums of proper bivariate hypergeometric terms. These conditions can be checked algorithmically. Hence the results can be applied in Zeilberger type algorithms for nonterminating hypergeometric series.

We consider a q-Painlevé IV equation which is the A (1) 4 -surface type in the Sakai’s classification. We find three distinct types of classical solutions with determinantal structures whose elements are basic hypergeometric functions. Two of them are expressed by 2φ1 basic hypergeometric series and the other is given by 2ψ2 bilateral basic hypergeometric series.

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...

The main goal of our survey is to give common characteristics of auxiliary hypergeometric functions (and their generalisations), functions which occur in number-theoretical problems. Originally designed as a tool for solving these problems, the hypergeometric series have become a connecting link between different areas of number theory and mathematics in general. Bibliography: 183 titles.