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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.
#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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참고문헌 (43)
참고문헌 신청
M. Abramowitz / 1964 / Handbook of mathematical functions with formulas, graphs, and mathematical tables / U.S. Government Printing Office
A. H. Al-Shammery / 2000 / An extension of some hypergeometric functions of two variables / Rev. Acad. Canaria Cienc 12(1-2):189~196
L. C. Andrews / 1984 / Special Functions for Engineers and Applied Mathematicians / Macmillan Company
W. N. Bailey / 1935 / Generalized Hypergeometric Series / Cambridge University Press
Yu. A. Brychkov / 2014 / On some formulas for the Appell function F2[a, b, b′;c, c′; w, z] / Integral Transforms Spec. Funct 25(2):111~123
A. H. Al-Shammery / 2000 / An extension of some hypergeometric functions of two variables / Rev. Acad. Canaria Cienc 12(1-2):189~196
L. C. Andrews / 1984 / Special Functions for Engineers and Applied Mathematicians / Macmillan Company
W. N. Bailey / 1935 / Generalized Hypergeometric Series / Cambridge University Press
Yu. A. Brychkov / 2014 / On some formulas for the Appell function F2[a, b, b′;c, c′; w, z] / Integral Transforms Spec. Funct 25(2):111~123
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