Applications of Tannery's theorem to Bailey's ψ66 summation
Ramanujan's ψ11 summation is obtained from Bailey's ψ66 summation by taking a limit and then specializing one of the variables. The process is shown to yield a three-term transformation for the general ψ22 series as an intermediate stage. Tannery's theorem is applied carefully, leadin...
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| Vydané v: | Journal of mathematical analysis and applications Ročník 541; číslo 1 |
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01.01.2025
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| Abstract | Ramanujan's ψ11 summation is obtained from Bailey's ψ66 summation by taking a limit and then specializing one of the variables. The process is shown to yield a three-term transformation for the general ψ22 series as an intermediate stage. Tannery's theorem is applied carefully, leading to the identification of two counterintuitive instances of series for which interchanging the limit and the summation yields incorrect results. Analogous limiting processes involving ordinary hypergeometric series are considered. |
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| AbstractList | Ramanujan's ψ11 summation is obtained from Bailey's ψ66 summation by taking a limit and then specializing one of the variables. The process is shown to yield a three-term transformation for the general ψ22 series as an intermediate stage. Tannery's theorem is applied carefully, leading to the identification of two counterintuitive instances of series for which interchanging the limit and the summation yields incorrect results. Analogous limiting processes involving ordinary hypergeometric series are considered. |
| ArticleNumber | 128659 |
| Author | Bradley-Thrush, J.G. |
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| DOI | 10.1016/j.jmaa.2024.128659 |
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| Keywords | Basic hypergeometric series Tannery's theorem Special functions |
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| References | Andrews (br0010) 1970; 25 Bailey (br0060) 1936; 7 Koornwinder (br0180) 1994; vol. 311 Bailey (br0070) 1950; 1 Warnaar (br0300) 2024 Bradley-Thrush (br0080) 2023 Atkin, Swinnerton-Dyer (br0050) 1954; 4 Weierstrass (br0290) 1882; vol. 1 Watson (br0280) 1929; 4 Chu (br0100) 2011; 25 Jackson (br0170) 1921; 50 Carlson (br0090) 1969 Andrews, Berndt (br0030) 2013 Andrews, Askey, Roy (br0020) 1999; vol. 71 Askey (br0040) 1984; 90 Halphen (br0140) 1886 Slater (br0250) 1969; 37 Dougall (br0110) 1907; 25 Guo, Schlosser (br0130) 2007; 334 Tannery (br0260) 1904 Gasper, Rahman (br0120) 2004; vol. 96 Ramanujan (br0200) 1988 Heine (br0160) 1847; 34 Rogers (br0210) 1895; 26 Slater (br0230) 1952; 3 Slater (br0240) 1966 Ramanujan (br0190) 1957 Upadhyay (br0270) 1968; 75 Hardy (br0150) 1940 Schlosser (br0220) 2005; vol. 13 |
| References_xml | – volume: vol. 96 year: 2004 ident: br0120 article-title: Basic Hypergeometric Series publication-title: Encyclopedia of Mathematics and Its Applications – year: 1966 ident: br0240 article-title: Generalized Hypergeometric Functions – volume: 25 start-page: 121 year: 2011 end-page: 139 ident: br0100 article-title: -extensions of Dougall's bilateral publication-title: Ramanujan J. – volume: 34 start-page: 285 year: 1847 end-page: 328 ident: br0160 article-title: Untersuchungen über die Reihe publication-title: J. Reine Angew. Math. – volume: 90 start-page: 575 year: 1984 end-page: 579 ident: br0040 article-title: The very well poised publication-title: Proc. Am. Math. Soc. – volume: 7 start-page: 105 year: 1936 end-page: 115 ident: br0060 article-title: Series of hypergeometric type which are infinite in both directions publication-title: Q. J. Math. Oxf. Ser. – year: 1886 ident: br0140 article-title: Traité des fonctions elliptiques et de leurs applications, vol. 1 – volume: 50 start-page: 101 year: 1921 end-page: 112 ident: br0170 article-title: Summation of publication-title: Messenger Math. – volume: 37 start-page: 310 year: 1969 ident: br0250 publication-title: J. Math. Rev. – volume: 4 start-page: 4 year: 1929 end-page: 9 ident: br0280 article-title: A new proof of the Rogers-Ramanujan identities publication-title: J. Lond. Math. Soc. (1) – year: 2023 ident: br0080 article-title: Symmetries in the theory of basic hypergeometric series – volume: vol. 13 start-page: 383 year: 2005 end-page: 400 ident: br0220 article-title: Abel–Rothe type generalizations of Jacobi's triple product identity publication-title: Theory and Applications of Special Functions – year: 1957 ident: br0190 article-title: Notebooks of Srinivasa Ramanujan, vol. 2 – volume: vol. 71 year: 1999 ident: br0020 article-title: Special Functions publication-title: Encyclopedia of Mathematics and Its Applications – year: 2013 ident: br0030 article-title: Ramanujan's Lost Notebook. Part IV – volume: 334 start-page: 393 year: 2007 end-page: 403 ident: br0130 article-title: Curious extensions of Ramanujan's publication-title: J. Math. Anal. Appl. – volume: vol. 311 start-page: 46 year: 1994 end-page: 128 ident: br0180 article-title: Compact quantum groups and publication-title: Representations of Lie Groups and Quantum Groups – year: 1904 ident: br0260 article-title: Introduction à la théorie des fonctions d'une variable, vol. 1 – volume: vol. 1 start-page: 505 year: 1882 end-page: 508 ident: br0290 article-title: Zur Theorie der Jacobi'schen Functionen von mehreren Veränderlichen publication-title: Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften – volume: 25 start-page: 114 year: 1907 end-page: 132 ident: br0110 article-title: On Vandermonde's theorem, and some more general expansions publication-title: Proc. Edinb. Math. Soc. – year: 1988 ident: br0200 article-title: The Lost Notebook and Other Unpublished Papers – volume: 4 start-page: 84 year: 1954 end-page: 106 ident: br0050 article-title: Some properties of partitions publication-title: Proc. Lond. Math. Soc. (3) – volume: 26 start-page: 15 year: 1895 end-page: 32 ident: br0210 article-title: Third memoir on the expansion of certain infinite products publication-title: Proc. Lond. Math. Soc. – volume: 1 start-page: 194 year: 1950 end-page: 198 ident: br0070 article-title: On the basic bilateral hypergeometric series publication-title: Q. J. Math. Oxf. Ser. (2) – volume: 3 start-page: 73 year: 1952 end-page: 80 ident: br0230 article-title: General transformations of bilateral series publication-title: Q. J. Math. Oxf. Ser. (2) – year: 1940 ident: br0150 article-title: Ramanujan: Twelve lectures on subjects suggested by his life and work – year: 2024 ident: br0300 article-title: The publication-title: The Ramanujan Encyclopedia – start-page: 391 year: 1969 ident: br0090 publication-title: Zentralblatt für Mathematik und ihre Grenzgebiete, vol. 157 – volume: 75 start-page: 171 year: 1968 end-page: 172 ident: br0270 article-title: Certain transformations and summation formulae for basic bilateral hypergeometric series publication-title: Am. Math. Mon. – volume: 25 start-page: 554 year: 1970 end-page: 558 ident: br0010 article-title: On a transformation of bilateral series with applications publication-title: Proc. Am. Math. Soc. |
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| Snippet | Ramanujan's ψ11 summation is obtained from Bailey's ψ66 summation by taking a limit and then specializing one of the variables. The process is shown to yield a... |
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| SubjectTerms | Basic hypergeometric series Special functions Tannery's theorem |
| Title | Applications of Tannery's theorem to Bailey's ψ66 summation |
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