Certain Generalizations of Quadratic Transformations of Hypergeometric and Generalized Hypergeometric Functions
There have been numerous investigations on the hypergeometric series 2F1 and the generalized hypergeometric series pFq such as differential equations, integral representations, analytic continuations, asymptotic expansions, reduction cases, extensions of one and several variables, continued fraction...
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| Veröffentlicht in: | Symmetry (Basel) Jg. 14; H. 5; S. 1073 |
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| Abstract | There have been numerous investigations on the hypergeometric series 2F1 and the generalized hypergeometric series pFq such as differential equations, integral representations, analytic continuations, asymptotic expansions, reduction cases, extensions of one and several variables, continued fractions, Riemann’s equation, group of the hypergeometric equation, summation, and transformation formulae. Among the various approaches to these functions, the transformation formulae for the hypergeometric series 2F1 and the generalized hypergeometric series pFq are significant, both in terms of applications and theory. The purpose of this paper is to establish a number of transformation formulae for pFq, whose particular cases would include Gauss’s and Kummer’s quadratic transformation formulae for 2F1, as well as their two extensions for 3F2, by making advantageous use of a recently introduced sequence and some techniques commonly used in dealing with pFq theory. The pFq function, which is the most significant function investigated in this study, exhibits natural symmetry. |
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| AbstractList | There have been numerous investigations on the hypergeometric series 2F1 and the generalized hypergeometric series pFq such as differential equations, integral representations, analytic continuations, asymptotic expansions, reduction cases, extensions of one and several variables, continued fractions, Riemann’s equation, group of the hypergeometric equation, summation, and transformation formulae. Among the various approaches to these functions, the transformation formulae for the hypergeometric series 2F1 and the generalized hypergeometric series pFq are significant, both in terms of applications and theory. The purpose of this paper is to establish a number of transformation formulae for pFq, whose particular cases would include Gauss’s and Kummer’s quadratic transformation formulae for 2F1, as well as their two extensions for 3F2, by making advantageous use of a recently introduced sequence and some techniques commonly used in dealing with pFq theory. The pFq function, which is the most significant function investigated in this study, exhibits natural symmetry. |
| Author | Qureshi, Mohd Idris Choi, Junesang Shah, Tafaz Rahman |
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| Cites_doi | 10.1002/mana.19720530114 10.1016/B978-0-12-385218-2.00002-5 10.1186/s13660-018-1842-4 10.1007/978-3-662-11761-3 10.3390/fractalfract5040150 |
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| Copyright | 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| References | Rakha (ref_6) 2011; 1 Srivastava (ref_15) 1969; 9 ref_11 Rezk (ref_17) 2020; 2020 ref_10 Rakha (ref_12) 2009; 14 ref_1 Srivastava (ref_13) 1972; 53 ref_2 Lehnigk (ref_16) 2018; 2018 ref_9 ref_8 Srivastava (ref_14) 1969; 31 ref_5 ref_4 ref_7 Gauss (ref_3) 1866; Volume III |
| References_xml | – ident: ref_9 – ident: ref_8 – ident: ref_4 – ident: ref_5 – ident: ref_2 – volume: 53 start-page: 151 year: 1972 ident: ref_13 article-title: A note on the convergence of Kampé de Fériet’s double hypergeometric series publication-title: Math. Nachr. doi: 10.1002/mana.19720530114 – volume: 1 start-page: 171 year: 2011 ident: ref_6 article-title: On an extension of a quadratic transformation formula due to Gauss publication-title: Int. J. Math. Model. Comput. – ident: ref_11 – volume: 2020 start-page: 6427378 year: 2020 ident: ref_17 article-title: Inequalities of Hardy type via superquadratic functions with general kernels and measures for several variables on time scales publication-title: J. Funct. Spaces – ident: ref_1 doi: 10.1016/B978-0-12-385218-2.00002-5 – volume: 9 start-page: 199 year: 1969 ident: ref_15 article-title: On Eulerian integrals associated with Kampé de Fériet’s function publication-title: Publ. Inst. Math. – volume: 2018 start-page: 253 year: 2018 ident: ref_16 article-title: Inequalities involving hypergeometric and related functions publication-title: J. Inequ. Appl. doi: 10.1186/s13660-018-1842-4 – volume: 31 start-page: 449 year: 1969 ident: ref_14 article-title: Certain generalized Neumann expansions associated with the Kampé De Fériet function publication-title: Nederl. Akad. Wetensch. Proc. Ser. A 72 = Indag. Math. – ident: ref_10 doi: 10.1007/978-3-662-11761-3 – volume: 14 start-page: 207 year: 2009 ident: ref_12 article-title: On an extension of a quadratic transformation formula due to Kummer publication-title: Math. Commun. – volume: Volume III start-page: 207 year: 1866 ident: ref_3 article-title: Disquisitiones Generales Circa Seriem Infinitam publication-title: Ph.D. Thesis – ident: ref_7 doi: 10.3390/fractalfract5040150 |
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| SubjectTerms | Asymptotic series Differential equations Hypergeometric functions Mathematical analysis Transformations (mathematics) |
| Title | Certain Generalizations of Quadratic Transformations of Hypergeometric and Generalized Hypergeometric Functions |
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