The INCOMPLETE GENERALIZED Τ-Hypergeometric AND SECOND Τ-Appell FUNCTIONS

Abstract

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 (λ; κ) ν and [λ; κ] ν , we introduce here the family of the incomplete generalized τ -hypergeometric functions 2 γ τ 1 (z) and 2 Γ τ 1 (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 τ -Appell hypergeometric functions Γ τ 1 ,τ 2 2 and γ τ 1 ,τ 2 2 of two variables. Relevant connections of certain special cases of the main results presented here with some known identities are also pointed out.

Incomplete Extended Hurwitz-Lerch Zeta Functions and Associated Properties

“…Motivated essentially by the demonstrated potential for applications of these incomplete hypergeometric functions p γ q and p Γ q in many diverse areas of mathematical, physical, engineering and statistical sciences (see, for details, [2,6,7,8,16,30,31,32,33,34] and the references cited therein), here, we aim here at systematically investigating the family of the incomplete extended Hurwitz-Lerch Zeta function. We, first introduce incomplete Fox-Wright Psi function.…”

Certain integral transforms concerning the product of family of polynomials and generalized incomplete functions

“…Incomplete special functions have recently been applicable to a broad spectrum of problems relating to reaction, combustion, reactiondiffusion, electronics and communication, fractional differential and integral equations, other fields of theoretical mechanics, statistical probability theory, etc. Consequently, several research papers [4,9,15,16,17,19,24,25,26] on incomplete special functions have recently been published by various researchers along with related other higher transcendental special functions.…”

$H_{B}^{\tau _{1},\tau _{2},\tau _{3}}$ Srivastava Hypergeometric Function

“…The interested reader may be referred to several recent papers on the subject [10,16,17,27,30,31]. The special case when τ 1, τ 2 = 1 in (1.7) give the familiar representations of Appell's hypergeometric function [4,5,14,15,17,18,20,22,23]. However, recently there has been an increasing interest in and widely extended use of differential equations and systems of fractional order (that is, of arbitrary order), as better models of phenomena of various physics, engineering, automatization, biology and biomedicine, chemistry, earth science, economics, nature and so on.…”

“…The interested reader may be referred to several recent papers on the subject [10,16,17,27,30,31]. The special case when τ 1, τ 2 = 1 in (1.7) give the familiar representations of Appell's hypergeometric function [4,5,14,15,17,18,20,22,23].…”