Certain image formulas of $(p,\nu)$--extended Gauss' hypergeometric function and related Jacobi Transforms
Commun. Korean Math. Soc.
Published online May 13, 2022
Purnima Chopra, Mamta Gupta, and Kanak Modi
Marudhar Engineering College, Bikaner, India; Amity School of Applied Sciences, Amity University, Jaipur, India; Amity School of Applied Sciences, Amity University, Jaipur, India
Abstract : 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 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 : 26A33, 33B20, 33C05, 33C20
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