Infinite Series and Logarithmic Integrals Associated to Differentiation with Respect to Parameters of the Whittaker Mκ,μ(x) Function I

Abstract

In this paper, first derivatives of the Whittaker function Mκ,μx are calculated with respect to the parameters. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Moreover, from the integral representation of Mκ,μx 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). 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 are derived, along with finite and infinite integrals containing products of algebraic, exponential, logarithmic, and Bessel functions. Finally, reduction formulas for the Whittaker functions Mκ,μx and integral Whittaker functions Miκ,μx and miκ,μx are calculated.