M-Lauricella hypergeometric functions: integral representations and solutions of fractional differential equations

Abstract

In this paper, using the modified beta function involving the generalized M-series in its kernel, we described new extensions for the Lauricella hypergeometric functions $F_{A}^{(r)}$, $F_{B}^{(r)}$, $F_{C}^{(r)}$ and $F_{D}^{(r)}$. Furthermore, we obtained various integral representations for the newly defined extended Lauricella hypergeometric functions. Then, we obtained solution of fractional differential equations involving new extensions of Lauricella hypergeometric functions, as examples.

Keywords

• Fractional derivatives and integrals
• beta function
• confluent hypergeometric function
• Lauricelle functions
• fractional differential equations
• Laplace transform

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