Basic Hypergeometric Series

Essays on the theory of elliptic hypergeometric functions

“…For |q| < 1 its particular solution, analytical at the point z = 0, is determined by a simple iteration. This yields the standard qgamma function γ q (z) = 1/(z; q) ∞ (which can be considered also as a q-exponential function [3]). This expression differs from the Jackson q-gamma function 13) satisfying the equation Γ (J)…”

“…Other proofs of formula (3.8) are given in [3,45,46,47,48,49]. For p → 0 and fixed parameters, equality (3.8) is reduced to the Jackson sum for the terminating very-well-poised balanced q-hypergeometric series 8 ϕ 7 [1].…”

“…The Gauss hypergeomeric function 2 F 1 (a, b; c; x) is a canonical example of functions of such a type. According to the approach by Pochhammer and Horn [2], the generalized plain hypergeometric series can be defined as the sums n c n for which the ratio c n+1 /c n is a rational function of n. In 1847 Heine has introduced a q-analogue of the 2 F 1 -series 2 ϕ 1 (a, b; c; q; x) [3]. The general series of such type n c n are characterized by the property that for them the ratio c n+1 /c n is a rational function of q n , where q is some complex parameter.…”

Uncertainty principles

“…Throughout this paper, we assume q ∈]0, 1[, and refer to the general reference [6] for the definitions, notations and properties of the q-shifted factorials and the qhypergeometric functions.…”

Multiple Hypergeometric Series: Appell Series and Beyond

“…For basic (or q-series) analogues of Appell functions, see G. Gasper and M. Rahman's text [13,Ch. 10].…”