## Abstract

In the first part of this investigation, we considered the parameter differentiation of the Whittaker function (Formula presented.). In this second part, first derivatives with respect to the parameters of the Whittaker function (Formula presented.) 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 (Formula presented.). 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 (Formula presented.) and its derivative with respect to (Formula presented.), as well as some reduction formulas for the integral Whittaker functions (Formula presented.) and (Formula presented.), are calculated.

Original language | English |
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Article number | 382 |

Journal | Axioms |

Volume | 12 |

Issue number | 4 |

DOIs | |

State | Published - 1 Apr 2023 |

## Keywords

- Whittaker functions
- derivatives with respect to parameters
- incomplete gamma functions
- infinite integrals involving Bessel functions
- integral Whittaker functions
- sums of infinite series of psi and gamma

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Mathematical Physics
- Logic
- Geometry and Topology

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