Some Volterra-type integro-differential equations with a multivariate analog of generalized Mittag-Leffler function in their kernels (Q2874900)

In the paper, the properties of the following multivariable generalization of the Mittag-Leffler function NEWLINE\[NEWLINE E_{(\rho_1,\ldots,\rho_m),\lambda}^{(\gamma_1,\ldots,\gamma_m)}(z_1,\ldots,z_m) = \sum\limits_{k_1,\ldots,k_m=0}^{\infty} \frac{(\gamma_1)_{k_1}\ldots (\gamma_m)_{k_m}}{\Gamma\left[\lambda + \sum_{j=1}^{m} k_j \rho_j\right]} \frac{z_1^{k_1}\ldots z_m^{k_m}}{(k_1)!\ldots (k_m)!} NEWLINE\]NEWLINE are studied.NEWLINENEWLINENEWLINEOn the base of these properties the Cauchy type problem NEWLINE\[NEWLINE \left(I_{0+}^{(1-\nu)(\mu - 1)} h(\tau)\right) (0+) = c NEWLINE\]NEWLINE for the fractional integro-differential equation NEWLINE\[NEWLINE D_{0+}^{\mu,\nu} h(\tau) = \sigma \int\limits_{0}^{\tau} \zeta^{\lambda - 1} h(\tau - \zeta) E_{(\rho_1,\ldots,\rho_m),\lambda}^{(\gamma_1,\ldots,\gamma_m)}(\omega_1 \zeta^{\rho_1},\ldots,\omega_m \zeta^{\rho_m}) d\zeta + \kappa f(\tau) NEWLINE\]NEWLINENEWLINEis solved.NEWLINENEWLINENEWLINE{Remark.} There are a lot of misprints in the paper.