3F4 Hypergeometric Functions as a Sum of a Product of 2F3 Functions
This paper shows that certain 3F4 hypergeometric functions can be expanded in sums of pair products of 2F3 functions, which reduce in special cases to 2F3 functions expanded in sums of pair products of 1F2 functions. This expands the class of hypergeometric functions having summation theorems beyond...
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| Published in | Axioms Vol. 13; no. 3; p. 203 |
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| Format | Journal Article |
| Language | English |
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01.03.2024
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| Abstract | This paper shows that certain 3F4 hypergeometric functions can be expanded in sums of pair products of 2F3 functions, which reduce in special cases to 2F3 functions expanded in sums of pair products of 1F2 functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, 2F1 functions, and 3F2 functions into the realm of pFq functions where p<q for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation. |
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| AbstractList | This paper shows that certain 3F4 hypergeometric functions can be expanded in sums of pair products of 2F3 functions, which reduce in special cases to 2F3 functions expanded in sums of pair products of 1F2 functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, 2F1 functions, and 3F2 functions into the realm of pFq functions where p<q for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation. This paper shows that certain 3F4 hypergeometric functions can be expanded in sums of pair products of 2F3 functions, which reduce in special cases to 2F3 functions expanded in sums of pair products of 1F2 functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, 2F1 functions, and 3F2 functions into the realm of pFq functions where p<q for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation. This paper shows that certain 3F4 hypergeometric functions can be expanded in sums of pair products of 2F3 functions, which reduce in special cases to 2F3 functions expanded in sums of pair products of 1F2 functions. This expands the class of hypergeometric functions having summation theorems beyond those expressible as pair-products of generalized Whittaker functions, 2F1 functions, and 3F2 functions into the realm of pFq functions where p<q for both the summand and terms in the series. In addition to its intrinsic value, this result has a specific application in calculating the response of the atoms to laser stimulation in the Strong Field Approximation. |
| Author | Straton, Jack C. |
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| Cites_doi | 10.1093/qmath/os-13.1.69 10.1103/PhysRevLett.101.043002 10.1080/10652469.2017.1280484 10.1007/978-3-662-11761-3 10.1364/JOSAB.7.000574 10.1080/10652469.2011.595092 10.1093/qmath/os-15.1.49 10.3390/axioms13020134 10.1103/PhysRevA.22.1786 10.1016/0079-6727(92)90008-J 10.1007/978-3-540-95944-1_2 10.1007/978-3-642-90850-7 10.1007/BF01331022 10.1017/S0305004100036471 10.3390/axioms10040318 10.1017/S0305004100029765 10.1016/j.jmaa.2013.06.068 10.1007/s00025-021-01367-9 10.1080/10652469.2015.1128432 10.1088/0022-3700/6/4/011 10.1016/B978-0-12-459950-5.50013-4 |
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| Copyright | 2024 by the author. 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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| DOI | 10.3390/axioms13030203 |
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| References_xml | – volume: 13 start-page: 69 year: 1942 ident: ref_1 article-title: On basic double hypergeometric functions publication-title: Q. J. Math. doi: 10.1093/qmath/os-13.1.69 contributor: fullname: Jackson – ident: ref_28 – volume: 101 start-page: 043002 year: 2008 ident: ref_14 article-title: Limits on Tunneling Theories of Strong-Field Ionization publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.101.043002 contributor: fullname: Reiss – ident: ref_30 – ident: ref_32 – ident: ref_3 – volume: 28 start-page: 239 year: 2017 ident: ref_8 article-title: Finite summation formulas of double hypergeometric functions publication-title: Integral Transform. Spec. Funct. doi: 10.1080/10652469.2017.1280484 contributor: fullname: Wang – ident: ref_26 – ident: ref_31 doi: 10.1007/978-3-662-11761-3 – ident: ref_34 – volume: 7 start-page: 570 year: 1990 ident: ref_18 article-title: Relativistic strong-field photoionization publication-title: J. Opt. Soc. Am. B doi: 10.1364/JOSAB.7.000574 contributor: fullname: Reiss – volume: 23 start-page: 369 year: 2012 ident: ref_10 article-title: Voronoi–Nasim summation formulas and index transforms publication-title: Integral Transform. Spec. Funct. doi: 10.1080/10652469.2011.595092 contributor: fullname: Yakubovich – ident: ref_35 – volume: 15 start-page: 49 year: 1944 ident: ref_2 article-title: On basic double hypergeometric functions II publication-title: Q. J. Math. doi: 10.1093/qmath/os-15.1.49 contributor: fullname: Jackson – ident: ref_23 doi: 10.3390/axioms13020134 – volume: 22 start-page: 1786 year: 1980 ident: ref_16 article-title: Effect of an intense electromagnetic filed on a weakly bound system publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.22.1786 contributor: fullname: Reiss – ident: ref_25 – ident: ref_29 – ident: ref_33 – volume: 16 start-page: 1 year: 1992 ident: ref_17 article-title: Theoretical methods in quantum optics: S-matrix and Keldysh techniques for strong-field process publication-title: Prog. Quantum Electron. doi: 10.1016/0079-6727(92)90008-J contributor: fullname: Reiss – volume: 1 start-page: 41 year: 2010 ident: ref_15 article-title: Foundations of Strong-Field Physics publication-title: Lect. Ultrafast Intense Laser Sci. doi: 10.1007/978-3-540-95944-1_2 contributor: fullname: Reiss – ident: ref_24 doi: 10.1007/978-3-642-90850-7 – volume: 31 start-page: 79 year: 1962 ident: ref_6 article-title: Certain expansions of generalised basic hypergeometric functions publication-title: Duke Math. J. contributor: fullname: Verma – volume: 94 start-page: 250 year: 1935 ident: ref_21 article-title: Über eine Klasse von Lösungen der Diracschen Gleichung publication-title: Z. Phys. Phys. doi: 10.1007/BF01331022 contributor: fullname: Volkov – volume: 58 start-page: 239 year: 1962 ident: ref_4 article-title: Expansions of Generalized Hypergeometric Functions in Series of Products of Generalized Whittaker Functions publication-title: Math. Proc. Camb. Philos. Soc. doi: 10.1017/S0305004100036471 contributor: fullname: Ragab – ident: ref_7 doi: 10.3390/axioms10040318 – volume: 50 start-page: 628 year: 1954 ident: ref_5 article-title: Expansions of Generalized Whittaker Functions publication-title: Math. Proc. Camb. Philos. Soc. doi: 10.1017/S0305004100029765 contributor: fullname: Slater – volume: 409 start-page: 100 year: 2014 ident: ref_11 article-title: Transformation and summation formulae for Kampé de Fériet series publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2013.06.068 contributor: fullname: Liu – volume: 1 start-page: 107 year: 2019 ident: ref_12 article-title: General summation formulas for the Kampé de Fériet function publication-title: Montes Taurus J. Pure Appl. Math. contributor: fullname: Choi – volume: 76 start-page: 65 year: 2021 ident: ref_13 article-title: A Study of Extensions of Classical Summation Theorems for the Series and with Applications publication-title: Results Math doi: 10.1007/s00025-021-01367-9 contributor: fullname: Awad – ident: ref_22 – ident: ref_20 – volume: 27 start-page: 347 year: 2016 ident: ref_9 article-title: Infinite summation formulas of double hypergeometric functions publication-title: Integral Transform. Spec. Funct. doi: 10.1080/10652469.2015.1128432 contributor: fullname: Wang – volume: 6 start-page: L89 year: 1973 ident: ref_19 article-title: Multiple absorption of laser photons by atoms publication-title: J. Phys. B doi: 10.1088/0022-3700/6/4/011 contributor: fullname: Faisal – ident: ref_27 doi: 10.1016/B978-0-12-459950-5.50013-4 |
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| Snippet | This paper shows that certain 3F4 hypergeometric functions can be expanded in sums of pair products of 2F3 functions, which reduce in special cases to 2F3... This paper shows that certain 3F4 hypergeometric functions can be expanded in sums of pair products of 2F3 functions, which reduce in special cases to 2F3... This paper shows that certain 3F4 hypergeometric functions can be expanded in sums of pair products of 2F3 functions, which reduce in special cases to 2F3... |
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| SubjectTerms | 1F2 hypergeometric functions 2F3 hypergeometric functions 3F4 hypergeometric functions Approximation Hypergeometric functions Integrals laser stimulation Lasers Mathematical analysis Radiation Strong Field Approximation summation theorem Sums Theorems Whittaker functions |
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| Title | 3F4 Hypergeometric Functions as a Sum of a Product of 2F3 Functions |
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