A New Extension of the Pochhammer Symbol and Its Application to Hypergeometric Functions

Our main goal in this present paper is to define first a new extension of the Pochhammer symbol and the gamma functions which involving the Mittag-Leffler function in their kernels. By using this extended Pochhammer symbol, we then introduce and investigate the corresponding extension of the generalized hypergeometric functions and of some of its special cases. Also, we establish the basic properties and results for the extended τ-Gauss hypergeometric function, which includes integral and derivative formulas involving the Mellin transform and fractional calculus approaches. Some new and known results as consequences of our proposed extension of the τ-Gauss hypergeometric function are also established.