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 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 Full-Text :

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