We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey’s very-well-poised 6ψ6 summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via...

This paper investigates a new family of special functions referred to as hypergeometric zeta functions. Derived from the integral representation of the classical Riemann zeta function, hypergeometric zeta functions exhibit many properties analogous to their classical counterpart, including the intimate connection to Bernoulli numbers. These new properties are treated in detail and are used to d...

Abstract The paper aims to generalize Clausen’s identity to the square of any Gauss hypergeometric function. Accordingly, solutions of the related 3rd order linear differential equation are found in terms of certain bivariate series that can reduce to 3F2 series similar to those in Clausen’s identity. The general contiguous variation of Clausen’s identity is found. The related Chaundy’s identit...

The HYPERDIRE is a project devoted to creation of a set of Mathematica based programs for differential reduction of hypergeometric functions. The current version includes two parts: one, pfq, is relevant to manipulation with hypergeometric functions p+1Fp, and second, AppellF1F4, to manipulations with Appell’s hypergeometric functions of two variables F1, F2, F3, F4. As application, a few multi...

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...

Hypergeometric sequences are such that the quotient of two successive terms is a xed rational function of the index. We give a generalization of M. Petkov sek's algorithm to nd all hypergeometric sequence solutions of linear recurrences, and we describe a program to nd all hypergeometric functions that solve a linear diierential equation. Solutions hyperg eom etriques des equations dii erentiel...

Wilf and Zeilberger conjectured in 1992 that a hypergeometric term is proper-hypergeometric if and only if it is holonomic. We prove a version of this conjecture in the case of two discrete variables.

We give an introduction to the Gauss hypergeometric function, the hypergeometric equation and their properties in an elementary way. Moreover we explicitly and uniformly describe the connection coefficients, the reducibility of the equation and the monodromy group of the solutions.

Wilf and Zeilberger conjectured in 1992 that a hypergeometric term is proper-hypergeometric if and only if it is holonomic. We prove a version of this conjecture in the case of two discrete variables.

We present a criterion for the existence of telescopers for mixed hypergeometric terms, which is based on multiplicative and additive decompositions. The criterion enables us to determine the termination of Zeilberger’s algorithms for mixed hypergeometric inputs.