# Part 1. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{M}_{\kappa ,\mu }\left( x\right) $ function

## TL;DR

This paper derives formulas for the derivatives of the Whittaker function with respect to its parameters, expressing them as infinite sums and integrals, and applies these results to obtain reduction formulas and evaluate related integrals.

## Contribution

It provides new closed-form expressions and reduction formulas for parameter derivatives of the Whittaker and confluent hypergeometric functions, including integral representations.

## Key findings

- Derived parameter derivatives of the Whittaker function in closed-form.
- Expressed derivatives as infinite sums involving digamma and gamma functions.
- Obtained reduction formulas for Whittaker functions and related integrals.

## Abstract

First derivatives of the Whittaker function $\mathrm{M}_{\kappa ,\mu }\left(x\right) $ with respect to the parameters are calculated. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Also, it is possible to obtain these parameter derivatives in terms of finite and infinite integrals with integrands containing elementary functions (products of algebraic, exponential and logarithmic functions) from the integral representation of $\mathrm{M}_{\kappa ,\mu }\left( x\right) $. These infinite sums and integrals can be expressed in closed-form for particular values of the parameters. For this purpose, we have obtained the parameter derivative of the incomplete gamma function in closed-form. As an application, reduction formulas for parameter derivatives of the confluent hypergeometric function has been derived, as well as some finite and infinite integrals containing products of algebraic, exponential, logarithmic and Bessel functions. Finally, some reduction formulas for the Whittaker functions $\mathrm{M}_{\kappa ,\mu }\left( x\right) $ and integral Whittaker functions $\mathrm{Mi}_{\kappa ,\mu }\left( x\right) $ and $\mathrm{mi}_{\kappa ,\mu }\left( x\right) $ are calculated.