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http://dx.doi.org/10.4134/JKMS.2016.53.2.363

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

Parmar, Rakesh Kumar (Department of Mathematics Government College of Engineering and Technology)
Saxena, Ram Kishore (Department of Mathematics and Statistics Jai Narain Vyas University)
Publication Information
Journal of the Korean Mathematical Society / v.53, no.2, 2016 , pp. 363-379 More about this Journal
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 $({\lambda};{\kappa})_{\nu}$ and $[{\lambda};{\kappa}]_{\nu}$, 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 transform; fractional calculus; incomplete second Appell functions; incomplete second ${\tau}$-Appell function;
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