Commun. Korean Math. Soc. 2022; 37(4): 1055-1072
Online first article May 13, 2022 Printed October 31, 2022
https://doi.org/10.4134/CKMS.c210344
Copyright © The Korean Mathematical Society.
Purnima Chopra, Mamta Gupta, Kanak Modi
Marudhar Engineering College; Amity University; Amity University
Our aim is to establish certain image formulas of the $ (p,\nu)$--extended Gauss' hypergeometric function $F_{\,p,\nu}(a,b;c;z)$ by using Saigo's hypergeometric fractional calculus (integral and differential) operators. Corresponding assertions for the classical Riemann-Liouville(R-L) and Erd\'elyi-Kober(E-K) fractional integral and differential operators are deduced. All the results are represented in terms of the Hadamard product of the $ (p,\nu)$--extended Gauss's hypergeometric function $F_{\,p,\nu}(a,b;c;z)$ and Fox-Wright function $_{r}\Psi_{s}(z)$. We also established Jacobi and its particular assertions for the Gegenbauer and Legendre transforms of the $ (p,\nu)$--extended Gauss' hypergeometric function $F_{\,p,\nu}(a,b;c;z)$.
Keywords: $(p,\nu)$-extended Gauss hypergeometric function $F_{\,p,\nu}(a,b,c,z)$, extended beta function, fractional calculus operators
MSC numbers: Primary 26A33, 33B20, 33C20; Secondary 26A09, 33B15, 33C05
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