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

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.