Discrete Analogues of the Erdélyi Type Integrals for Hypergeometric Functions

Abstract

Gasper followed the fractional calculus proof of an Erdélyi integral to derive its discrete analogue in the form of a hypergeometric expansion. To give an alternative proof, we derive it by following a procedure analogous to a triple series manipulation-based proof of the Erdélyi integral, due to “Joshi and Vyas”. Motivated from this alternative way of proof, we establish the discrete analogues corresponding to many of the Erdélyi type integrals due to “Joshi and Vyas” and “Luo and Raina” in the form of new hypergeometric expansion formulas. Moreover, the applications of investigated discrete analogues in deriving some expansion formulas involving orthogonal polynomials of the Askey-scheme and a new generalization of Whipple’s transformation for a balanced 4F3 in the form of an ${}_{m+4}{F}_{m+3}$ transformation, are also discussed.