Journal of the
Korean Mathematical Society JKMS

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [\emph{Integral Transforms Spec. Funct.} 23 (2012), 659--683] and the second Appell function [\emph{Appl. Math. Comput.} 219 (2013), 8332--8337] by means of the incomplete Pochhammer symbols $ \left(\lambda; \kappa \right)_{\nu} $ and $ \left[\lambda; \kappa \right]_{\nu}$, we introduce here the family of the incomplete generalized $\tau$-hypergeometric functions $_{2}\gamma_{1}^{\tau}(z) $ and $ _{2}\Gamma_{1}^{\tau}(z) $. The main object of this paper is to study these extensions and investigate their several properties including, for example, their integral representations, derivative formulas, Euler-Beta transform and associated with certain fractional calculus operators. Further, we introduce and investigate the family of incomplete second $\tau$-Appell hypergeometric functions $\Gamma_{2}^{\tau_{1},\tau_{2}}$ and $\gamma_{2}^{\tau_{1},\tau_{2}}$ of two variables. Relevant connections of certain special cases of the main results presented here with some known identities are also pointed out.

Keywords: gamma functions, incomplete gamma functions, Pochhammer symbol, incomplete Pochhammer symbols, incomplete generalized hypergeometric functions, generalized $\tau$-hypergeometric functions, incomplete generalized $\tau$-hypergeometric functions, Euler-Beta

MSC numbers: Primary 33B20, 33C20; Secondary 33B15, 33C05