Derivatives of Humbert confluent hypergeometric functions with respect to their parameters

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Arts and Sciences, Kırıkkale University, Kırıkkale 71450, Turkey.

2 Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt.

Abstract

Humbert confluent hypergeometric functions of two variables arise in many problems of mathematical physics and applied analysis, yet their behaviour with respect to parameters has not been systematically studied. In this paper we investigate derivatives with respect to numerator and denominator parameters for the seven classical Humbert functions
\(\Phi_{1}\), \(\Phi_{2}\), \(\Phi_{3}\), \(\Psi_{1}\), \(\Psi_{2}\), \(\Xi_{1}\) and \(\Xi_{2}\).
Using their double--series representations together with elementary properties of the Gamma and digamma functions, we derive explicit formulas for first--order parameter derivatives and express them in compact form in terms of Srivastava's triple hypergeometric function \(F^{(3)}\). By differentiating the underlying partial differential equations, we further obtain simple operator recurrences for derivatives of arbitrary order, which yield closed differentiation and reduction formulas in terms of contiguous Humbert functions. Finally, we indicate how these results lead to Taylor-type parameter expansions and illustrate their use with basic numerical examples and plots.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 10 July 2026

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History

  • Receive Date: 17 December 2025
  • Revise Date: 29 June 2026
  • Accept Date: 07 July 2026

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