Derivatives of Humbert confluent hypergeometric functions with respect to their parameters
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866912760400445440 |
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| author | Shehata, Ayman Sahin, Recep Yagcı, Oguz Moustafa, Shimaa I. |
| author_facet | Shehata, Ayman Sahin, Recep Yagcı, Oguz Moustafa, Shimaa I. |
| contents | 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 Φ{1}, Φ{2}, Φ_{3}, Psi_{1}, Psi_{2}, Ξ_{1} and Ξ_{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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_05051 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Derivatives of Humbert confluent hypergeometric functions with respect to their parameters Shehata, Ayman Sahin, Recep Yagcı, Oguz Moustafa, Shimaa I. Classical Analysis and ODEs 33C15, 33C20, 33C65, 33D70, 05A30 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 Φ{1}, Φ{2}, Φ_{3}, Psi_{1}, Psi_{2}, Ξ_{1} and Ξ_{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. |
| title | Derivatives of Humbert confluent hypergeometric functions with respect to their parameters |
| topic | Classical Analysis and ODEs 33C15, 33C20, 33C65, 33D70, 05A30 |
| url | https://arxiv.org/abs/2104.05051 |