Three General Double-Series Identities and Associated Reduction Formulas and Fractional Calculus

Abstract

In this article, we introduce three general double-series identities using Whipple transformations for terminating generalized hypergeometric 4F3 and 5F4 functions. Then, by employing the left-sided Riemann–Liouville fractional integral on these identities, we show the ability to derive additional identities of the same nature successively. These identities are used to derive transformation formulas between the Srivastava–Daoust double hypergeometric function (S–D function) and Kampé de Fériet’s double hypergeometric function (KDF function) with equal arguments. We also demonstrate reduction formulas from the S–D function or KDF function to the generalized hypergeometric function pFq. Additionally, we provide general summation formulas for the pFq and S–D function (or KDF function) with specific arguments. We further highlight the connections between the results presented here and existing identities.