## Abstract

In this paper, first derivatives of the Whittaker function (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) and integral Whittaker functions (Formula presented.) and (Formula presented.) are calculated.

Original language | English |
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Article number | 381 |

Journal | Axioms |

Volume | 12 |

Issue number | 4 |

DOIs | |

State | Published - 1 Apr 2023 |

## Keywords

- Whittaker functions
- derivatives with respect to parameters
- finite and infinite logarithmic integrals and Bessel functions
- incomplete gamma 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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