THE INCOMPLETE GENERALIZED τ-HYPERGEOMETRIC AND SECOND τ-APPELL FUNCTIONS

Title & Authors
THE INCOMPLETE GENERALIZED τ-HYPERGEOMETRIC AND SECOND τ-APPELL FUNCTIONS
Parmar, Rakesh Kumar; Saxena, Ram Kishore;

Abstract
Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [Integral Transforms Spec. Funct. 23 (2012), 659-683] and the second Appell function [Appl. Math. Comput. 219 (2013), 8332-8337] by means of the incomplete Pochhammer symbols $\small{({\lambda};{\kappa})_{\nu}}$ and $\small{[{\lambda};{\kappa}}$$\small{]}$$\small{_{\nu}}$, we introduce here the family of the incomplete generalized $\small{{\tau}}$-hypergeometric functions $\small{2{\gamma}_1^{\tau}(z)}$ and $\small{2{\Gamma}_1^{\tau}(z)}$. 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 $\small{{\tau}}$-Appell hypergeometric functions $\small{{\Gamma}_2^{{\tau}_1,{\tau}_2}}$ and $\small{{\gamma}_2^{{\tau}_1,{\tau}_2}}$ 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 $\small{{\tau}}$-hypergeometric functions;incomplete generalized $\small{{\tau}}$-hypergeometric functions;Euler-Beta transform;fractional calculus;incomplete second Appell functions;incomplete second $\small{{\tau}}$-Appell function;
Language
English
Cited by
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