Derivatives of Humbert confluent hypergeometric functions with respect to their parameters

TL;DR

This paper systematically studies derivatives of Humbert confluent hypergeometric functions with respect to their parameters, deriving explicit formulas, recurrences, and expansions useful in mathematical physics and applied analysis.

Contribution

It provides the first comprehensive derivation of parameter derivatives for all seven classical Humbert functions, including explicit formulas and recurrence relations.

Findings

Explicit formulas for first order derivatives in terms of Srivastava triple hypergeometric functions.

Recurrence relations for derivatives of arbitrary order.

Applications to Taylor expansions and numerical illustrations.

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 \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 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.