On a Class of New q-Hypergeometric Expansions as Discrete Analogues of the Erdélyi Type q-Integrals

Abstract

Recently, Vyas et al. have developed an alternative way of proof for the Gasper’s discrete analogue of an Erdélyi integral and inspired from this new type of derivation they resolved the problem of finding the discrete extensions of all the Erdélyi type integrals in the form of several new hypergeometric expansions for certain $$_{q + 1} F_{q}$$ . Motivated from the above-mentioned work, here in this paper, our objective is to resolve the problem of finding the discrete extensions of the Erdélyi type q-integrals in the form of several new q-hypergeometric expansions for certain $$_{r + 1} \Phi_{r}$$ . The motivation behind this work is the fact that the q-series and basic q-polynomials, specifically the q-gamma and basic q- hypergeometric functions and basic q-hypergeometric polynomials, are applicable particularly in several diverse areas of science and engineering, viz. Statistics, number theory, combinatorial analysis, nonlinear electric circuit theory, combinatorial generating functions, quantum mechanics, mechanical engineering, lie theory, theory of heat conduction, particle physics and cosmology.