Ramanujan's cubic transformation and generalized modular equation

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• 作者：MiaoKun Wang ; YuMing Chu ; YingQing Song
• 关键词：Gaussian hypergeometric function ; Ramanujan’s cubic transformation ; generalized modular equation ; infinite product ; modular function
• 刊名：SCIENCE CHINA Mathematics
• 出版年：2015
• 出版时间：November 2015
• 年：2015
• 卷：58
• 期：11
• 页码：2387-2404
• 全文大小：286 KB
• 参考文献：1.Abramowitz M, Stegun I A. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. New York: Dover Publications, 1965MATH
  2.Anderson G D, Barnard R W, Richards K C, et al. Inequalities for zero-balanced hypergeometric functions. Trans Amer Math Soc, 1995, 347: 1713–1723MathSciNet CrossRef MATH
  3.Anderson G D, Qiu S L, Vamanamurthy M K, et al. Generalized elliptic integrals and modular equations. Pacific J Math, 2000, 192: 1–37MathSciNet CrossRef
  4.Anderson G D, Vamanamurthy M K, Vuorinen M. Funcitonal inequalities for hypergeometric functions and complete elliptic integrals. SIAM J Math Anal, 1992, 23: 512–524MathSciNet CrossRef MATH
  5.Anderson G D, Vamanamurthy M K, Vuorinen M. Conformal Invariants, Inequalities, and Quasiconformal Maps. New York: John Wiley & Sons, 1997MATH
  6.Askey R. Ramanujan and hypergeometric and basic hypergeometric series. In: Ramanujan International Symposium on Analysis. New Delhi: Macmillan of India, 1989, 1–83
  7.Baricz Á. Turán type inequalities for generalized complete elliptic integrals. Math Z, 2007, 256: 895–911MathSciNet CrossRef MATH
  8.Baricz Á. Landen inequalities for special functions. Proc Amer Math Soc, 2014, 142: 3059–3066MathSciNet CrossRef MATH
  9.Barnard R W, Pearce K, Richards K C. A monotonicity property involving 3 F 2 and comparisons of the classical approximations of elliptical arc length. SIAM J Math Anal, 2000, 32: 403–419MathSciNet CrossRef MATH
  10.Baruah N D, Berndt B C. Partition identities and Ramanujan’s modular equations. J Combin Theory Ser A, 2007, 114: 1024–1045MathSciNet CrossRef MATH
  11.Berndt B C. Ramanujan’s Notebooks, Part I. New York: Springer-Verlag, 1985CrossRef MATH
  12.Berndt B C. Ramanujan’s Notebooks, Part II. New York: Springer-Verlag, 1989CrossRef MATH
  13.Berndt B C. Ramanujan’s Notebooks, Part III. New York: Springer-Verlag, 1991CrossRef MATH
  14.Berndt B C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994CrossRef MATH
  15.Berndt B C, Bhargava S, Garvan F G. Ramanujan’s theories of elliptic functions to alternative bases. Trans Amer Math Soc, 1995, 347: 4163–4244MathSciNet MATH
  16.Beukers F, Heckman G. Monodromy for the hypergeometric function n F n−1. Invent Math, 1989, 95: 325–354MathSciNet CrossRef MATH
  17.Borwein J M, Borwein P M. Explicit Ramanujan-type approximations to pi of high order. Proc Indian Acad Sci Math Sci, 1987, 97: 53–59MathSciNet CrossRef MATH
  18.Borwein J M, Borwein P B. Pi and the AGM. New York: John Wiley & Sons, 1987MATH
  19.Borwein J M, Borwein P B. A Remarkable cubic mean iteration. In: Computational Methods and Function Theory. Lecture Notes in Mathematics, vol. 1435. New York: Springer-Verlag, 1990, 27–31CrossRef
  20.Borwein J M, Borwein P B. A cubic counterpart of Jacobi’s identity and the AGM. Trans Amer Math Soc, 1991, 323: 691–701MathSciNet MATH
  21.Heikkala V, Vamanamurthy M K, Vuorinen M. Generalized elliptic integrals. Comput Methods Funct Theory, 2009, 9: 75–109MathSciNet CrossRef MATH
  22.Olver F W J, Lozier D W, Boisvert R F, et al. NIST Handbook of Mathematical Functions. Cambridge: Cambridge University Press, 2010MATH
  23.Ponnusamy S, Vuorinen M. Asymptotic expansions and inequalities for hypergeometric functions. Mathematika, 1997, 44: 278–301MathSciNet CrossRef MATH
  24.Qiu S L. Grötzsch ring and Ramanujan’s modular equations (in Chinese). Acta Math Sinica, 2000, 43: 283–290MATH
  25.Qiu S L, Vuorinen M. Infinite products and the normalized quotients of hypergeometric functions. SIAM J Math Anal, 1999, 30: 1057–1075MathSciNet CrossRef MATH
  26.Qiu S L, Vuorinen M. Duplication inequalities for the ratios of hypergeometric functions. Forum Math, 2000, 12: 109–133MathSciNet MATH
  27.Rainville E D. Special Functions. New York: MacMillan, 1960MATH
  28.Ramanujan S. Notebooks (2 volumes). Bombay: Tata Institute of Fundamental Research, 1957MATH
  29.Ramanujan S. Collected Papers. New York: Chelsea, 1962
  30.Ramanujan S. The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa, 1988MATH
  31.Saigo M, Srivastava H M. The behavior of the zero-balanced hypergeometric series p F p−1 near the boundary of its convergence region. Proc Amer Math Soc, 1990, 110: 71–76MathSciNet MATH
  32.Shen L C. On an identity of Ramanujan based on the hypergeometric series 2 F 1(1/3, 2/3; 1/2; x). J Number Theory, 1998, 69: 125–134MathSciNet CrossRef
  33.Simić S, Vuorinen M. Landen inequalities for zero-balanced hypergeometric functions. Abstr Appl Anal, 2012, Article ID 932061, 11 pages
  34.Venkatachaliengar K. Development of Elliptic Functions According to Ramanujan. Madurai: Madurai Kamaraj University, 1988MATH
  35.Vuorinen M. Singular values, Ramanujan modular equations, and Landen transformations. Studia Math, 1996, 121: 221–230MathSciNet MATH
  36.Wang G D, Zhang X H, Chu Y M. Inequalities for the generalized elliptic integrals and modular functions. J Math Anal Appl, 2007, 331: 1275–1283MathSciNet CrossRef MATH
  37.Wang M K, Chu Y M, Jiang Y P. Ramanujan’s cubic transformation inequalities for zero-balanced hypergeometric functions. Rocky Mountain J Math, arXiv:1210.6126, 2012
  38.Wang M K, Qiu S L, Chu Y M, et al. Generalized Hersch-Pfluger distortion function and complete elliptic integrals. J Math Anal Appl, 2012, 385: 221–229MathSciNet CrossRef MATH
  39.Whittaker E T, Watson G N. A Course of Modern Analysis, 4th ed. London: Cambridge University Press, 1996CrossRef MATH
  
• 作者单位：MiaoKun Wang (1)
  YuMing Chu (1)
  YingQing Song (2)
  
  1. College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China
  2. School of Mathematics and Computation Science, Hunan City University, Yiyang, 413000, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Chinese Library of Science
  Applications of Mathematics
  
• 出版者：Science China Press, co-published with Springer
• ISSN：1869-1862

文摘

We study the quotient of hypergeometric functions $$\mu _a^* (r) = \frac{\pi } {{2\sin (\pi a)}}\frac{{F(a,1 - a;1;1 - r^3 )}} {{F(a,1 - a;1;r^3 )}},r \in (0,1) $$ in the theory of Ramanujan’s generalized modular equation for a ∈ (0, 1/2], find an infinite product formula for µ_1/3*(r) by use of the properties of µ _a * and Ramanujan’s cubic transformation. Besides, a new cubic transformation formula of hypergeometric function is given, which complements the Ramanujan’s cubic transformation. Keywords Gaussian hypergeometric function Ramanujan’s cubic transformation generalized modular equation infinite product modular function MSC(2010) 33C05 11F03 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (39) References1.Abramowitz M, Stegun I A. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. New York: Dover Publications, 1965MATH2.Anderson G D, Barnard R W, Richards K C, et al. Inequalities for zero-balanced hypergeometric functions. Trans Amer Math Soc, 1995, 347: 1713–1723MathSciNetCrossRefMATH3.Anderson G D, Qiu S L, Vamanamurthy M K, et al. Generalized elliptic integrals and modular equations. Pacific J Math, 2000, 192: 1–37MathSciNetCrossRef4.Anderson G D, Vamanamurthy M K, Vuorinen M. Funcitonal inequalities for hypergeometric functions and complete elliptic integrals. SIAM J Math Anal, 1992, 23: 512–524MathSciNetCrossRefMATH5.Anderson G D, Vamanamurthy M K, Vuorinen M. Conformal Invariants, Inequalities, and Quasiconformal Maps. New York: John Wiley & Sons, 1997MATH6.Askey R. Ramanujan and hypergeometric and basic hypergeometric series. In: Ramanujan International Symposium on Analysis. New Delhi: Macmillan of India, 1989, 1–837.Baricz Á. Turán type inequalities for generalized complete elliptic integrals. Math Z, 2007, 256: 895–911MathSciNetCrossRefMATH8.Baricz Á. Landen inequalities for special functions. Proc Amer Math Soc, 2014, 142: 3059–3066MathSciNetCrossRefMATH9.Barnard R W, Pearce K, Richards K C. A monotonicity property involving ₃ F ₂ and comparisons of the classical approximations of elliptical arc length. SIAM J Math Anal, 2000, 32: 403–419MathSciNetCrossRefMATH10.Baruah N D, Berndt B C. Partition identities and Ramanujan’s modular equations. J Combin Theory Ser A, 2007, 114: 1024–1045MathSciNetCrossRefMATH11.Berndt B C. Ramanujan’s Notebooks, Part I. New York: Springer-Verlag, 1985CrossRefMATH12.Berndt B C. Ramanujan’s Notebooks, Part II. New York: Springer-Verlag, 1989CrossRefMATH13.Berndt B C. Ramanujan’s Notebooks, Part III. New York: Springer-Verlag, 1991CrossRefMATH14.Berndt B C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994CrossRefMATH15.Berndt B C, Bhargava S, Garvan F G. Ramanujan’s theories of elliptic functions to alternative bases. Trans Amer Math Soc, 1995, 347: 4163–4244MathSciNetMATH16.Beukers F, Heckman G. Monodromy for the hypergeometric function _n F _n−1. Invent Math, 1989, 95: 325–354MathSciNetCrossRefMATH17.Borwein J M, Borwein P M. Explicit Ramanujan-type approximations to pi of high order. Proc Indian Acad Sci Math Sci, 1987, 97: 53–59MathSciNetCrossRefMATH18.Borwein J M, Borwein P B. Pi and the AGM. New York: John Wiley & Sons, 1987MATH19.Borwein J M, Borwein P B. A Remarkable cubic mean iteration. In: Computational Methods and Function Theory. Lecture Notes in Mathematics, vol. 1435. New York: Springer-Verlag, 1990, 27–31CrossRef20.Borwein J M, Borwein P B. A cubic counterpart of Jacobi’s identity and the AGM. Trans Amer Math Soc, 1991, 323: 691–701MathSciNetMATH21.Heikkala V, Vamanamurthy M K, Vuorinen M. Generalized elliptic integrals. Comput Methods Funct Theory, 2009, 9: 75–109MathSciNetCrossRefMATH22.Olver F W J, Lozier D W, Boisvert R F, et al. NIST Handbook of Mathematical Functions. Cambridge: Cambridge University Press, 2010MATH23.Ponnusamy S, Vuorinen M. Asymptotic expansions and inequalities for hypergeometric functions. Mathematika, 1997, 44: 278–301MathSciNetCrossRefMATH24.Qiu S L. Grötzsch ring and Ramanujan’s modular equations (in Chinese). Acta Math Sinica, 2000, 43: 283–290MATH25.Qiu S L, Vuorinen M. Infinite products and the normalized quotients of hypergeometric functions. SIAM J Math Anal, 1999, 30: 1057–1075MathSciNetCrossRefMATH26.Qiu S L, Vuorinen M. Duplication inequalities for the ratios of hypergeometric functions. Forum Math, 2000, 12: 109–133MathSciNetMATH27.Rainville E D. Special Functions. New York: MacMillan, 1960MATH28.Ramanujan S. Notebooks (2 volumes). Bombay: Tata Institute of Fundamental Research, 1957MATH29.Ramanujan S. Collected Papers. New York: Chelsea, 196230.Ramanujan S. The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa, 1988MATH31.Saigo M, Srivastava H M. The behavior of the zero-balanced hypergeometric series _p F _p−1 near the boundary of its convergence region. Proc Amer Math Soc, 1990, 110: 71–76MathSciNetMATH32.Shen L C. On an identity of Ramanujan based on the hypergeometric series ₂ F ₁(1/3, 2/3; 1/2; x). J Number Theory, 1998, 69: 125–134MathSciNetCrossRef33.Simić S, Vuorinen M. Landen inequalities for zero-balanced hypergeometric functions. Abstr Appl Anal, 2012, Article ID 932061, 11 pages34.Venkatachaliengar K. Development of Elliptic Functions According to Ramanujan. Madurai: Madurai Kamaraj University, 1988MATH35.Vuorinen M. Singular values, Ramanujan modular equations, and Landen transformations. Studia Math, 1996, 121: 221–230MathSciNetMATH36.Wang G D, Zhang X H, Chu Y M. Inequalities for the generalized elliptic integrals and modular functions. J Math Anal Appl, 2007, 331: 1275–1283MathSciNetCrossRefMATH37.Wang M K, Chu Y M, Jiang Y P. Ramanujan’s cubic transformation inequalities for zero-balanced hypergeometric functions. Rocky Mountain J Math, arXiv:1210.6126, 201238.Wang M K, Qiu S L, Chu Y M, et al. Generalized Hersch-Pfluger distortion function and complete elliptic integrals. J Math Anal Appl, 2012, 385: 221–229MathSciNetCrossRefMATH39.Whittaker E T, Watson G N. A Course of Modern Analysis, 4th ed. London: Cambridge University Press, 1996CrossRefMATH About this Article Title Ramanujan’s cubic transformation and generalized modular equation Journal Science China Mathematics Volume 58, Issue 11 , pp 2387-2404 Cover Date2015-11 DOI 10.1007/s11425-015-5023-3 Print ISSN 1674-7283 Online ISSN 1869-1862 Publisher Science China Press Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Applications of Mathematics Keywords Gaussian hypergeometric function Ramanujan’s cubic transformation generalized modular equation infinite product modular function 33C05 11F03 Industry Sectors Aerospace IT & Software Telecommunications Authors MiaoKun Wang (1) YuMing Chu (1) YingQing Song (2) Author Affiliations 1. College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China 2. School of Mathematics and Computation Science, Hunan City University, Yiyang, 413000, China Continue reading... 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