Part 2. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{W}_{κ,μ}\left( x\right) $ function

Alexander Apelblat, Juan Luis González-Santander

Published 2023-02-24Version 1

First derivatives with respect to the parameters of the Whittaker function $\mathrm{W}_{\kappa ,\mu }\left( x\right) $ 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 infinite integrals with integrands containing elementary functions (products of algebraic, exponential and logarithmic functions) from the integral representation of $\mathrm{W}_{\kappa ,\mu }\left( x\right) $. These infinite sums and integrals can be expressed in closed-form for particular values of the parameters. Finally, an integral representation of the integral Whittaker function $\mathrm{wi}_{\kappa ,\mu }\left( x\right) $ and its derivative with respect to $\kappa $, as well as some reduction formulas for the integral Whittaker functions $\mathrm{Wi}_{\kappa ,\mu }\left( x\right) $ and $\mathrm{wi}_{\kappa ,\mu }\left( x\right) $ are calculated.