A new approach to hypergeometric transformation formulas
We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi’s canonical form of the hypergeometric differential equation. Analogy for q -hypergeometric functions is also studied.
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| Published in: | The Ramanujan journal Vol. 55; no. 2; pp. 793 - 816 |
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| Language: | English |
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01.06.2021
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| Abstract | We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi’s canonical form of the hypergeometric differential equation. Analogy for
q
-hypergeometric functions is also studied. |
|---|---|
| AbstractList | We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi’s canonical form of the hypergeometric differential equation. Analogy for q-hypergeometric functions is also studied. We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi’s canonical form of the hypergeometric differential equation. Analogy for q -hypergeometric functions is also studied. |
| Author | Otsubo, Noriyuki |
| Author_xml | – sequence: 1 givenname: Noriyuki orcidid: 0000-0002-7880-0505 surname: Otsubo fullname: Otsubo, Noriyuki email: otsubo@math.s.chiba-u.ac.jp organization: Department of Mathematics and Informatics, Chiba University |
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| Cites_doi | 10.1017/S0027763000009739 10.1619/fesi.52.203 10.1112/plms/pdp007 10.1007/BF03012437 10.1515/crll.1859.56.149 10.1090/S0002-9947-1994-1243610-6 10.24033/asens.207 10.1007/978-1-4939-0258-3_27 10.1007/978-1-4612-4530-8 10.1016/j.jnt.2006.08.002 10.1017/S0305004198002643 10.1017/CBO9780511526251 10.1090/S0002-9947-07-04128-1 10.2307/2370183 10.1619/fesi.52.139 |
| ContentType | Journal Article |
| Copyright | Springer Science+Business Media, LLC, part of Springer Nature 2020 Springer Science+Business Media, LLC, part of Springer Nature 2020. |
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| Keywords | Hypergeometric functions Basic hypergeometric functions Transformation formulas 33C05 33D15 |
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| PublicationSubtitle | An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan |
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| References | BorweinJMBorweinPBGarvanFGSome cubic modular identities of RamanujanTrans. Am. Math. Soc.199434313547124361010.1090/S0002-9947-1994-1243610-6 MatsumotoKOharaKSome transformation formulas for Lauricella’s hypergeometric functions FD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_D$$\end{document}Funkcialaj Ekvacioj200952203212254710210.1619/fesi.52.203 RamanujanSNotebooks (2 Volumes)1957BombayTata Institute of Fundamental Research0138.24201 BerndtBCRamanujan’s Notebooks, Part II1989New YorkNew YorkSpringer10.1007/978-1-4612-4530-8 GaussCFGaussCFDeterminatio seriei nostrae per aequationem differentialem secundi ordinisWerke1876GöttingenKöniglichen Gesellschaft der Wissenschaften207230 GoursatÉSur l’équation différentielle linéaire, qui admet pour intégrale la série hypergéométriqueAnn. Sci. Éc. Norm. Sup.188110314210.24033/asens.207 MaierRSOn rationally parametrized modular equationsJ. Ramanujan Math. Soc.200924117325141491214.11049 BorweinJMBorweinPBA cubic counterpart of Jacobi’s identity and the AGMTrans. Am. Math. Soc.1991323269170110104080725.33014 VidūnasRAlgebraic transformations of Gauss hypergeometric functionsFunkcialaj Ekvacioj200952139180254710010.1619/fesi.52.139 ErdélyiAHigher Transcendental Functions1953New YorkMcGrow-Hill0052.29502 AppellPKampé de FérietJFonctions Hypergéométriques et Hypersphériques1926ParisGauthier Villars52.0361.13 JacksonFHq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Difference equationsAm. J. Math.191032430531410.2307/2370183 BerndtBCBhargavaSGarvanFGRamanujan’s theories of elliptic functions to alternative basesTrans. Am. Math. Soc.1995347114163424413119030843.33012 GasperGRahmanMBasic Hypergeometric Series20042CambridgeCambridge University Press10.1017/CBO9780511526251 KatoTMatsumotoKThe common limit of a quadruple sequence and the hypergeometric function FD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_D$$\end{document} of three variablesNagoya Math. J.2009195113124255295610.1017/S0027763000009739 LauricellaGSulle Funzioni ipergeometrche a piu variabiliR. Circ. Mat. Palermo1893711115810.1007/BF03012437 ChanHHOn Ramanujan’s cubic transformation formula for 2F1(13,23;1;z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$_2F_1(\frac{1}{3},\frac{2}{3};1;z)$$\end{document}Math. Proc. Camb. Philos. Soc.1998124219320410.1017/S0305004198002643 MaierRSAlgebraic hypergeometric transformations of modular originTrans. Am. Math. Soc.2007359838593885230251610.1090/S0002-9947-07-04128-1 PooleEGCIntroduction to the Theory of Differential Equations1936OxfordOxford University Press62.1277.01 IwasakiKKimuraHShimomuraSYoshidaMFrom Gauss to Painlevé: A Modern Theory of Special Functions; Dedicated to Professor Tosihusa Kimura1992New YorkSpringer0743.34014 Kummer, E.E.: Über die hypergeometrische Reihe 1+α·β1·γx+α(α+1)β(β+1)1·2·γ(γ+1)x2+α(α+1)(α+2)β(β+1)(β+2)1·2·3·γ(γ+1)(γ+2)x3+⋯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+\frac{\alpha \cdot \beta }{1\cdot \gamma }x+\frac{\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}x^2+\frac{\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)}{1\cdot 2\cdot 3\cdot \gamma (\gamma +1)(\gamma +2)}x^3+\cdots $$\end{document}, J. Reine Angew. Math. 15(39–83), pp. 127–172. Collected Papers II, Springer, Berlin 1975, 75–166 (1836) CooperSInversion formulas for elliptic functionsProc. Lond. Math. Soc.200999461483253367210.1112/plms/pdp007 KoikeKShigaHIsogeny formulas for the Picard modular form and a three terms arithmetic geometric meanJ. Number Theory2007124123141232099410.1016/j.jnt.2006.08.002 HeineEUntersuchungen über die Reihe 1+(1-qα)(1-qβ)(1-q)(1-qγ)·x+(1-qα)(1-qα+1)(1-qβ)(1-qβ+1)(1-q)(1-q2)(1-qγ)(1-qγ+1)·x2+⋯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+\frac{(1-q^\alpha )(1-q^\beta )}{(1-q)(1-q^\gamma )}\cdot x+ \frac{(1-q^\alpha )(1-q^{\alpha +1})(1-q^\beta )(1-q^{\beta +1})}{(1-q)(1-q^2)(1-q^\gamma )(1-q^{\gamma +1})}\cdot x^2+\cdots $$\end{document}J. Reine Angew. Math.1847342853281578577 Jacobi, C.G.J.: Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe. J. Reine Angew. Math. 56, 149–165 (1859). In: C.G.J. Jacobi’s Gesammelte Werke, vol. 6, pp. 184–202. Cambridge University Press, Cambridge (2013) KarlssonPWMilovanovićGVRassiasMThGoursat’s hypergeometric transformations, revisitedAnalytic Number Theory, Approximation Theory, and Special Functions2014New YorkSpringer72173610.1007/978-1-4939-0258-3_27 R Vidūnas (286_CR26) 2009; 52 S Cooper (286_CR7) 2009; 99 K Iwasaki (286_CR13) 1992 K Matsumoto (286_CR23) 2009; 52 É Goursat (286_CR11) 1881; 10 286_CR19 E Heine (286_CR12) 1847; 34 G Lauricella (286_CR20) 1893; 7 286_CR15 JM Borwein (286_CR5) 1994; 343 K Koike (286_CR18) 2007; 124 S Ramanujan (286_CR25) 1957 JM Borwein (286_CR4) 1991; 323 (286_CR8) 1953 P Appell (286_CR1) 1926 RS Maier (286_CR22) 2009; 24 BC Berndt (286_CR2) 1989 PW Karlsson (286_CR16) 2014 EGC Poole (286_CR24) 1936 BC Berndt (286_CR3) 1995; 347 RS Maier (286_CR21) 2007; 359 CF Gauss (286_CR10) 1876 HH Chan (286_CR6) 1998; 124 G Gasper (286_CR9) 2004 FH Jackson (286_CR14) 1910; 32 T Kato (286_CR17) 2009; 195 |
| References_xml | – reference: BerndtBCRamanujan’s Notebooks, Part II1989New YorkNew YorkSpringer10.1007/978-1-4612-4530-8 – reference: CooperSInversion formulas for elliptic functionsProc. Lond. Math. Soc.200999461483253367210.1112/plms/pdp007 – reference: MaierRSOn rationally parametrized modular equationsJ. Ramanujan Math. Soc.200924117325141491214.11049 – reference: KarlssonPWMilovanovićGVRassiasMThGoursat’s hypergeometric transformations, revisitedAnalytic Number Theory, Approximation Theory, and Special Functions2014New YorkSpringer72173610.1007/978-1-4939-0258-3_27 – reference: MaierRSAlgebraic hypergeometric transformations of modular originTrans. Am. Math. Soc.2007359838593885230251610.1090/S0002-9947-07-04128-1 – reference: Kummer, E.E.: Über die hypergeometrische Reihe 1+α·β1·γx+α(α+1)β(β+1)1·2·γ(γ+1)x2+α(α+1)(α+2)β(β+1)(β+2)1·2·3·γ(γ+1)(γ+2)x3+⋯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+\frac{\alpha \cdot \beta }{1\cdot \gamma }x+\frac{\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}x^2+\frac{\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)}{1\cdot 2\cdot 3\cdot \gamma (\gamma +1)(\gamma +2)}x^3+\cdots $$\end{document}, J. Reine Angew. Math. 15(39–83), pp. 127–172. Collected Papers II, Springer, Berlin 1975, 75–166 (1836) – reference: Jacobi, C.G.J.: Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe. J. Reine Angew. Math. 56, 149–165 (1859). In: C.G.J. Jacobi’s Gesammelte Werke, vol. 6, pp. 184–202. Cambridge University Press, Cambridge (2013) – reference: AppellPKampé de FérietJFonctions Hypergéométriques et Hypersphériques1926ParisGauthier Villars52.0361.13 – reference: HeineEUntersuchungen über die Reihe 1+(1-qα)(1-qβ)(1-q)(1-qγ)·x+(1-qα)(1-qα+1)(1-qβ)(1-qβ+1)(1-q)(1-q2)(1-qγ)(1-qγ+1)·x2+⋯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+\frac{(1-q^\alpha )(1-q^\beta )}{(1-q)(1-q^\gamma )}\cdot x+ \frac{(1-q^\alpha )(1-q^{\alpha +1})(1-q^\beta )(1-q^{\beta +1})}{(1-q)(1-q^2)(1-q^\gamma )(1-q^{\gamma +1})}\cdot x^2+\cdots $$\end{document}J. Reine Angew. Math.1847342853281578577 – reference: KatoTMatsumotoKThe common limit of a quadruple sequence and the hypergeometric function FD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_D$$\end{document} of three variablesNagoya Math. J.2009195113124255295610.1017/S0027763000009739 – reference: LauricellaGSulle Funzioni ipergeometrche a piu variabiliR. Circ. Mat. Palermo1893711115810.1007/BF03012437 – reference: ErdélyiAHigher Transcendental Functions1953New YorkMcGrow-Hill0052.29502 – reference: GasperGRahmanMBasic Hypergeometric Series20042CambridgeCambridge University Press10.1017/CBO9780511526251 – reference: JacksonFHq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document}-Difference equationsAm. J. Math.191032430531410.2307/2370183 – reference: PooleEGCIntroduction to the Theory of Differential Equations1936OxfordOxford University Press62.1277.01 – reference: RamanujanSNotebooks (2 Volumes)1957BombayTata Institute of Fundamental Research0138.24201 – reference: VidūnasRAlgebraic transformations of Gauss hypergeometric functionsFunkcialaj Ekvacioj200952139180254710010.1619/fesi.52.139 – reference: IwasakiKKimuraHShimomuraSYoshidaMFrom Gauss to Painlevé: A Modern Theory of Special Functions; Dedicated to Professor Tosihusa Kimura1992New YorkSpringer0743.34014 – reference: KoikeKShigaHIsogeny formulas for the Picard modular form and a three terms arithmetic geometric meanJ. Number Theory2007124123141232099410.1016/j.jnt.2006.08.002 – reference: BorweinJMBorweinPBA cubic counterpart of Jacobi’s identity and the AGMTrans. Am. Math. Soc.1991323269170110104080725.33014 – reference: GoursatÉSur l’équation différentielle linéaire, qui admet pour intégrale la série hypergéométriqueAnn. Sci. Éc. Norm. Sup.188110314210.24033/asens.207 – reference: BerndtBCBhargavaSGarvanFGRamanujan’s theories of elliptic functions to alternative basesTrans. Am. Math. Soc.1995347114163424413119030843.33012 – reference: ChanHHOn Ramanujan’s cubic transformation formula for 2F1(13,23;1;z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$_2F_1(\frac{1}{3},\frac{2}{3};1;z)$$\end{document}Math. Proc. Camb. Philos. Soc.1998124219320410.1017/S0305004198002643 – reference: BorweinJMBorweinPBGarvanFGSome cubic modular identities of RamanujanTrans. Am. Math. Soc.199434313547124361010.1090/S0002-9947-1994-1243610-6 – reference: MatsumotoKOharaKSome transformation formulas for Lauricella’s hypergeometric functions FD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_D$$\end{document}Funkcialaj Ekvacioj200952203212254710210.1619/fesi.52.203 – reference: GaussCFGaussCFDeterminatio seriei nostrae per aequationem differentialem secundi ordinisWerke1876GöttingenKöniglichen Gesellschaft der Wissenschaften207230 – volume: 195 start-page: 113 year: 2009 ident: 286_CR17 publication-title: Nagoya Math. J. doi: 10.1017/S0027763000009739 – volume: 52 start-page: 203 year: 2009 ident: 286_CR23 publication-title: Funkcialaj Ekvacioj doi: 10.1619/fesi.52.203 – volume-title: Fonctions Hypergéométriques et Hypersphériques year: 1926 ident: 286_CR1 – volume: 24 start-page: 1 issue: 1 year: 2009 ident: 286_CR22 publication-title: J. Ramanujan Math. Soc. – volume-title: Notebooks (2 Volumes) year: 1957 ident: 286_CR25 – volume: 99 start-page: 461 year: 2009 ident: 286_CR7 publication-title: Proc. Lond. Math. Soc. doi: 10.1112/plms/pdp007 – volume: 7 start-page: 111 year: 1893 ident: 286_CR20 publication-title: R. Circ. Mat. Palermo doi: 10.1007/BF03012437 – volume-title: Higher Transcendental Functions year: 1953 ident: 286_CR8 – volume: 347 start-page: 4163 issue: 11 year: 1995 ident: 286_CR3 publication-title: Trans. Am. Math. Soc. – start-page: 207 volume-title: Werke year: 1876 ident: 286_CR10 – ident: 286_CR15 doi: 10.1515/crll.1859.56.149 – volume: 343 start-page: 35 issue: 1 year: 1994 ident: 286_CR5 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1994-1243610-6 – ident: 286_CR19 – volume: 10 start-page: 3 year: 1881 ident: 286_CR11 publication-title: Ann. Sci. Éc. Norm. Sup. doi: 10.24033/asens.207 – volume-title: From Gauss to Painlevé: A Modern Theory of Special Functions; Dedicated to Professor Tosihusa Kimura year: 1992 ident: 286_CR13 – start-page: 721 volume-title: Analytic Number Theory, Approximation Theory, and Special Functions year: 2014 ident: 286_CR16 doi: 10.1007/978-1-4939-0258-3_27 – volume-title: Ramanujan’s Notebooks, Part II year: 1989 ident: 286_CR2 doi: 10.1007/978-1-4612-4530-8 – volume: 124 start-page: 123 year: 2007 ident: 286_CR18 publication-title: J. Number Theory doi: 10.1016/j.jnt.2006.08.002 – volume-title: Introduction to the Theory of Differential Equations year: 1936 ident: 286_CR24 – volume: 124 start-page: 193 issue: 2 year: 1998 ident: 286_CR6 publication-title: Math. Proc. Camb. Philos. Soc. doi: 10.1017/S0305004198002643 – volume-title: Basic Hypergeometric Series year: 2004 ident: 286_CR9 doi: 10.1017/CBO9780511526251 – volume: 359 start-page: 3859 issue: 8 year: 2007 ident: 286_CR21 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-07-04128-1 – volume: 34 start-page: 285 year: 1847 ident: 286_CR12 publication-title: J. Reine Angew. Math. – volume: 32 start-page: 305 issue: 4 year: 1910 ident: 286_CR14 publication-title: Am. J. Math. doi: 10.2307/2370183 – volume: 323 start-page: 691 issue: 2 year: 1991 ident: 286_CR4 publication-title: Trans. Am. Math. Soc. – volume: 52 start-page: 139 year: 2009 ident: 286_CR26 publication-title: Funkcialaj Ekvacioj doi: 10.1619/fesi.52.139 |
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| SubjectTerms | Canonical forms Combinatorics Differential equations Field Theory and Polynomials Fourier Analysis Functions of a Complex Variable Hypergeometric functions Mathematics Mathematics and Statistics Number Theory Transformations (mathematics) |
| Title | A new approach to hypergeometric transformation formulas |
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