The incomplete generalized $\tau$-hypergeometric and second $\tau$-Appell functions
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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.
Abstract
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.
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