Derivatives of Humbert confluent hypergeometric functions with respect to their parameters

2021/04/11 by Ayman Shehata, Shehata, Ayman, Sahin, Recep +2
Mathematics · #05A30 #33C15 #33C20 #33C65 #33D70 #Advanced Mathematical Identities #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2104.05051

openalex publication_date 2021/04/11 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

Humbert confluent hypergeometric functions of two variables arise in many problems of mathematical physics and applied analysis, yet their behavior 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 Φ1, Φ2, Φ₃, Psi₁, Psi₂, Ξ₁ and Ξ₂. 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 triple hypergeometric function F3. 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.

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