Differentiation of the Wright functions with respect to parameters and other results

TL;DR

This survey explores derivatives of Wright functions with respect to parameters, revealing their series representations, connections to Bessel and Mittag-Leffler functions, and their Laplace transform properties.

Contribution

It provides a comprehensive analysis of Wright functions' derivatives, including series expansions, closed-form results, and their relation to Bessel and Mittag-Leffler functions.

Findings

Derivatives lead to infinite series with digamma and gamma functions as coefficients.

Functional forms resemble those of Mittag-Leffler functions.

Laplace transforms of Wright functions facilitate deriving explicit Mittag-Leffler functions.

Abstract

In this survey we discuss derivatives of the Wright functions (of the first and the second kind) with respect to parameters. Differentiation of these functions leads to infinite power series with coefficient being quotients of the digamma (psi) and gamma functions. Only in few cases it is possible to obtain the sums of these series in a closed form. Functional form of the power series resembles those derived for the Mittag-Leffler functions. If the Wright functions are treated as the generalized Bessel functions, differentiation operations can be expressed in terms of the Bessel functions and their derivatives with respect to the order. It is demonstrated that in many cases it is possible to derive the explicit form of the Mittag-Leffler functions by performing simple operations with the Laplace transforms of the Wright functions. The Laplace transform pairs of the both kinds of the Wright functions are discussed for particular values of the parameters. Some transform pairs serve to obtain functional limits by applying the shifted Dirac delta function.