Infinite Series and Logarithmic Integrals Associated to Differentiation with Respect to Parameters of the Whittaker Wκ,μx Function II

Abstract

In the first part of this investigation, we considered the parameter differentiation of the Whittaker function Mκ,μx. In this second part, first derivatives with respect to the parameters of the Whittaker function Wκ,μx are calculated. Using the confluent hypergeometric function, these derivatives can be expressed as infinite sums of quotients of the digamma and gamma functions. Furthermore, 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 Wκ,μx. 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 wiκ,μx and its derivative with respect to κ, as well as some reduction formulas for the integral Whittaker functions Wiκ,μx and wiκ,μx, are calculated.