Related papers: Degenerate Miller-Paris transformations
Miller-Paris transformations are extensions of Euler's transformations for the Gauss hypergeometric functions to generalized hypergeometric functions of higher-order having integral parameter differences (IPD). In our recent work we…
In this paper we give several independent extensions of the Karlsson-Minton summation formula for the generalized hypergeometric function with integral parameters differences. In particular, we examine the "prohibited" values for the…
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…
In our recent work we proposed a generalization of the beta integral method for derivation of the hypergeometric identities which can by analogy be termed "the G function integral method". In this paper we apply this technique to the cubic…
In this paper we continue investigation of the hypergeometric function ${}_4F_3(1)$ as the function of its seven parameters. We deduce several reduction formulas for this function under additional conditions that one of the top parameters…
We give multidimensional generalizations of several transformation formulae for basic hypergeometric series of a specific type. Most of the upper parameters of the series differ multiplicatively from corresponding lower parameters by a…
Euler's transformation formula for the Gauss hypergeometric function 2F1 is extended to hypergeometric functions of higher order. Unusually, the generalized transformation constrains the hypergeometric function parameters algebraically but…
As a generalization of Riemann-Liouville integral, we introduce integral transformations of convergent power series which can be applied to hypergeometric functions with several variables.
A master formula of transformation formulas for bilinear sums of basic hypergeometric series is proposed. It is obtained from the author's previous results on a transformation formula for Milne's multivariate generalization of basic…
In this paper, we give an algorithm to generate connection formulas of generalized hypergeometric functions ${}_p F_{p-1}$ for degenerated values of parameters. We also show that these connection formulas give a fast method for numerical…
The differential-reduction algorithm, which allows one to express generalized hypergeometric functions with parameters of arbitrary values in terms of such functions with parameters whose values differ from the original ones by integers, is…
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and…
We consider the ratio of two Gauss hypergeometric functions, in which the parameters of the numerator function differ from the respective parameters of the denominator function by integers. We derive explicit integral representations for…
In this note we state (with minor corrections) and give an alternative proof of a very general hypergeometric transformation formula due to Slater. As an application, we obtain a new hypergeometric transformation formula for a ${}_5F_4(-1)$…
In this paper we study decay of correlations and limit theorems for generalized baker's transformations. Our examples are piecewise non-uniformly hyperbolic maps on the unit square that posses two spatially separated lines of indifferent…
In this article three expansion formulas for a generalized hypergeometric function $_4F_3$ are derived, when its upper parameters differ by integers. Though the results are special cases of a general continuation formula for $_pF_q$, they…
The Gauss hypergeometric functions 2F1 with arbitrary values of parameters are reduced to two functions with fixed values of parameters, which differ from the original ones by integers. It is shown that in the case of integer and/or…
We compute bilinear integrals involving Macdonald and Gegenbauer functions. These integrals are convergent only for a limited range of parameters. However, when one uses generalized integrals they can be computed essentially without…
We develop a theoretical study of non-terminating hypergeometric summations with one free parameter. Composing various methods in complex and asymptotic analysis, geometry and arithmetic of certain transcendental curves and rational…
We provide new formulae for the degenerations of the bilateral basic hypergeometric function ${}_1\psi_1 ( a; b; q, z )$ with using the $q$-Borel-Laplace transformation. These are thought of as the first step to construct connection…