On some generalizations of hypergeometric function and tau-Gauss hypergeometric function

Abstract

We introduce the extended Pochhammer symbol and gamma function by additional extensions using the $\,_{r}R_{s}$ function \cite{TO}. We also provide extension of generalized hypergeometric function using this extended Pochhammer symbol and derive some of its particular cases. We present extension of the $\tau$-Gauss hypergeometric function and also gives us a few more interesting and well-known results. Integral and derivative formulae for this extended $\tau$-Gauss hypergeometric function are also provided using fractional calculus techniques.