Differentiation of integral Mittag-Leffler and integral Wright functions with respect to parameters

Abstract

Derivatives with respect to the parameters of the integral Mittag-Leffler function and the integral Wright function, recently introduced by us, are calculated. These derivatives can be expressed in the form of infinite sums of quotients of the digamma and gamma functions. In some particular cases, these infinite sums are calculated in closed-form with the help of MATHEMATICA. However, parameter differentiation reduction formulas are explicitly derived in order to check some of the results given by MATHEMATICA, as well as to provide many other new results. In addition, we present these infinite sums graphically for particular values of the parameters. Finally, new results for parameter derivatives of the Mittag-Leffler and Wright functions are reported in the Appendices.