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Derivatives of Humbert confluent hypergeometric functions with respect to their parameters | ||
| Computational Methods for Differential Equations | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 19 تیر 1405 اصل مقاله (1.13 M) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22034/cmde.2026.70778.3535 | ||
| نویسندگان | ||
| Recep Şahin1؛ Ayman Shehata2؛ Shimaa I. Moustafa2؛ Oğuz Yağcı* 1 | ||
| 1Department of Mathematics, Faculty of Arts and Sciences, Kırıkkale University, Kırıkkale 71450, Turkey. | ||
| 2Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt. | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Generalized hypergeometric functions؛ Srivastava's hypergeometric function؛ Humbert confluent hypergeometric functions | ||
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