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Ramanujan’s cubic transformation and generalized modular equation

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Abstract

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 µ1/3*(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.

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References

  1. Abramowitz M, Stegun I A. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. New York: Dover Publications, 1965

    MATH  Google Scholar 

  2. Anderson G D, Barnard R W, Richards K C, et al. Inequalities for zero-balanced hypergeometric functions. Trans Amer Math Soc, 1995, 347: 1713–1723

    Article  MathSciNet  MATH  Google Scholar 

  3. Anderson G D, Qiu S L, Vamanamurthy M K, et al. Generalized elliptic integrals and modular equations. Pacific J Math, 2000, 192: 1–37

    Article  MathSciNet  Google Scholar 

  4. Anderson G D, Vamanamurthy M K, Vuorinen M. Funcitonal inequalities for hypergeometric functions and complete elliptic integrals. SIAM J Math Anal, 1992, 23: 512–524

    Article  MathSciNet  MATH  Google Scholar 

  5. Anderson G D, Vamanamurthy M K, Vuorinen M. Conformal Invariants, Inequalities, and Quasiconformal Maps. New York: John Wiley & Sons, 1997

    MATH  Google Scholar 

  6. Askey R. Ramanujan and hypergeometric and basic hypergeometric series. In: Ramanujan International Symposium on Analysis. New Delhi: Macmillan of India, 1989, 1–83

    Google Scholar 

  7. Baricz Á. Turán type inequalities for generalized complete elliptic integrals. Math Z, 2007, 256: 895–911

    Article  MathSciNet  MATH  Google Scholar 

  8. Baricz Á. Landen inequalities for special functions. Proc Amer Math Soc, 2014, 142: 3059–3066

    Article  MathSciNet  MATH  Google Scholar 

  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: 403–419

    Article  MathSciNet  MATH  Google Scholar 

  10. Baruah N D, Berndt B C. Partition identities and Ramanujan’s modular equations. J Combin Theory Ser A, 2007, 114: 1024–1045

    Article  MathSciNet  MATH  Google Scholar 

  11. Berndt B C. Ramanujan’s Notebooks, Part I. New York: Springer-Verlag, 1985

    Book  MATH  Google Scholar 

  12. Berndt B C. Ramanujan’s Notebooks, Part II. New York: Springer-Verlag, 1989

    Book  MATH  Google Scholar 

  13. Berndt B C. Ramanujan’s Notebooks, Part III. New York: Springer-Verlag, 1991

    Book  MATH  Google Scholar 

  14. Berndt B C. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag, 1994

    Book  MATH  Google Scholar 

  15. Berndt B C, Bhargava S, Garvan F G. Ramanujan’s theories of elliptic functions to alternative bases. Trans Amer Math Soc, 1995, 347: 4163–4244

    MathSciNet  MATH  Google Scholar 

  16. Beukers F, Heckman G. Monodromy for the hypergeometric function n F n−1. Invent Math, 1989, 95: 325–354

    Article  MathSciNet  MATH  Google Scholar 

  17. Borwein J M, Borwein P M. Explicit Ramanujan-type approximations to pi of high order. Proc Indian Acad Sci Math Sci, 1987, 97: 53–59

    Article  MathSciNet  MATH  Google Scholar 

  18. Borwein J M, Borwein P B. Pi and the AGM. New York: John Wiley & Sons, 1987

    MATH  Google Scholar 

  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, 27–31

    Chapter  Google Scholar 

  20. Borwein J M, Borwein P B. A cubic counterpart of Jacobi’s identity and the AGM. Trans Amer Math Soc, 1991, 323: 691–701

    MathSciNet  MATH  Google Scholar 

  21. Heikkala V, Vamanamurthy M K, Vuorinen M. Generalized elliptic integrals. Comput Methods Funct Theory, 2009, 9: 75–109

    Article  MathSciNet  MATH  Google Scholar 

  22. Olver F W J, Lozier D W, Boisvert R F, et al. NIST Handbook of Mathematical Functions. Cambridge: Cambridge University Press, 2010

    MATH  Google Scholar 

  23. Ponnusamy S, Vuorinen M. Asymptotic expansions and inequalities for hypergeometric functions. Mathematika, 1997, 44: 278–301

    Article  MathSciNet  MATH  Google Scholar 

  24. Qiu S L. Grötzsch ring and Ramanujan’s modular equations (in Chinese). Acta Math Sinica, 2000, 43: 283–290

    MATH  Google Scholar 

  25. Qiu S L, Vuorinen M. Infinite products and the normalized quotients of hypergeometric functions. SIAM J Math Anal, 1999, 30: 1057–1075

    Article  MathSciNet  MATH  Google Scholar 

  26. Qiu S L, Vuorinen M. Duplication inequalities for the ratios of hypergeometric functions. Forum Math, 2000, 12: 109–133

    MathSciNet  MATH  Google Scholar 

  27. Rainville E D. Special Functions. New York: MacMillan, 1960

    MATH  Google Scholar 

  28. Ramanujan S. Notebooks (2 volumes). Bombay: Tata Institute of Fundamental Research, 1957

    MATH  Google Scholar 

  29. Ramanujan S. Collected Papers. New York: Chelsea, 1962

    Google Scholar 

  30. Ramanujan S. The Lost Notebook and Other Unpublished Papers. New Delhi: Narosa, 1988

    MATH  Google Scholar 

  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: 71–76

    MathSciNet  MATH  Google Scholar 

  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: 125–134

    Article  MathSciNet  Google Scholar 

  33. Simić S, Vuorinen M. Landen inequalities for zero-balanced hypergeometric functions. Abstr Appl Anal, 2012, Article ID 932061, 11 pages

    Google Scholar 

  34. Venkatachaliengar K. Development of Elliptic Functions According to Ramanujan. Madurai: Madurai Kamaraj University, 1988

    MATH  Google Scholar 

  35. Vuorinen M. Singular values, Ramanujan modular equations, and Landen transformations. Studia Math, 1996, 121: 221–230

    MathSciNet  MATH  Google Scholar 

  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: 1275–1283

    Article  MathSciNet  MATH  Google Scholar 

  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

    Google Scholar 

  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: 221–229

    Article  MathSciNet  MATH  Google Scholar 

  39. Whittaker E T, Watson G N. A Course of Modern Analysis, 4th ed. London: Cambridge University Press, 1996

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China

    MiaoKun Wang & YuMing Chu

  2. School of Mathematics and Computation Science, Hunan City University, Yiyang, 413000, China

    YingQing Song

Authors
  1. MiaoKun Wang
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  2. YuMing Chu
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  3. YingQing Song
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Correspondence to MiaoKun Wang.

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Wang, M., Chu, Y. & Song, Y. Ramanujan’s cubic transformation and generalized modular equation. Sci. China Math. 58, 2387–2404 (2015). https://doi.org/10.1007/s11425-015-5023-3

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