Recep Sahin | Searcharxiv

arXiv2021

Derivatives of Srivastava's hypergeometric functions with respect to their parameters

This paper studies derivatives with respect to the parameters of Srivastava triple hypergeometric functions HA, HB and HC. Using basic properties of the Gamma function and Pochhammer symbols, we obtain explicit formulas for first and higher order derivatives. These derivatives are expressed in terms of Pathan quadruple hypergeometric function F. We also derive Euler type differential operator identities, contiguous relations for unit shifts in the parameters, and recurrence relations satisfied by these derivatives. In addition, we show that derivatives of arbitrary order satisfy systems of linear partial differential equations in the underlying variables. The results extend known differentiation formulas for classical and multivariable hypergeometric functions and provide tools for potential applications in mathematical physics and engineering.

math.CA↗

arXiv2021

Derivatives of Humbert confluent hypergeometric functions with respect to their parameters

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.

math.CA↗