Hypergeometric modular equations. (English) Zbl 1428.33014

Summary: We record \(\binom{42}{2}+\binom{23}{2}+\binom{13}{2}=1192\) functional identities that, apart from being amazingly amusing in themselves, find application in the derivation of Ramanujan-type formulas for \(1/ \pi\) and in the computation of mathematical constants.


33C20 Generalized hypergeometric series, \({}_pF_q\)
11B65 Binomial coefficients; factorials; \(q\)-identities
11F11 Holomorphic modular forms of integral weight
11Y60 Evaluation of number-theoretic constants
Full Text: DOI arXiv


[1] A. M.Aldawoud, ‘Ramanujan-type series for \(1/\unicode[STIX]{x1D70B}\) with quadratic irrationals’, Master of Science Thesis, Massey University, Auckland, 2012.
[2] G.Almkvist, D.van Straten and W.Zudilin, ‘Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations’, Proc. Edinb. Math. Soc.54 (2011), 273-295. · Zbl 1223.33007
[3] G. E.Andrews, R.Askey and R.Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71 (Cambridge University Press, Cambridge, 1999).
[4] A.Aycock, ‘On proving some of Ramanujan’s formulas for \(1/\unicode[STIX]{x1D70B}\) with an elementary method’, Preprint, 2013, arXiv:1309.1140v2 [math.NT].
[5] N. D.Baruah and B. C.Berndt, ‘Eisenstein series and Ramanujan-type series for 1/𝜋’, Ramanujan J.23 (2010), 17-44. · Zbl 1204.33005
[6] B. C.Berndt, S.Bhargava and F. G.Garvan, ‘Ramanujan’s theories of elliptic functions to alternative bases’, Trans. Amer. Math. Soc.347 (1995), 4163-4244. · Zbl 0843.33012
[7] J. M.Borwein and P. B.Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Canadian Mathematical Society Series of Monographs and Advanced Texts (John Wiley, New York, 1987). · Zbl 0611.10001
[8] H. H.Chan, S. H.Chan and Z.-G.Liu, ‘Domb’s numbers and Ramanujan-Sato type series for 1/𝜋’, Adv. Math.186 (2004), 396-410. · Zbl 1122.11087
[9] H. H.Chan and S.Cooper, ‘Rational analogues of Ramanujan’s series for 1/𝜋’, Math. Proc. Cambridge Philos. Soc.153 (2012), 361-383. · Zbl 1268.11165
[10] H. H.Chan, Y.Tanigawa, Y.Yang and W.Zudilin, ‘New analogues of Clausen’s identities arising from the theory of modular forms’, Adv. Math.228 (2011), 1294-1314. · Zbl 1234.33009
[11] H. H.Chan and W.Zudilin, ‘New representations for Apéry-like sequences’, Mathematika56 (2010), 107-117. · Zbl 1275.11035
[12] D. V.Chudnovsky and G. V.Chudnovsky, ‘Approximations and complex multiplication according to Ramanujan’, in: Ramanujan Revisited (Urbana-Champaign, IL, 1987) (Academic Press, Boston, MA, 1988), 375-472. · Zbl 0647.10002
[13] S.Cooper, ‘Inversion formulas for elliptic functions’, Proc. Lond. Math. Soc.99 (2009), 461-483. · Zbl 1248.11031
[14] S.Cooper, ‘On Ramanujan’s function k (q) = r (q)r^2(q^2)’, Ramanujan J.20 (2009), 311-328. · Zbl 1239.11051
[15] S.Cooper, ‘Level 10 analogues of Ramanujan’s series for 1/𝜋’, J. Ramanujan Math. Soc.27 (2012), 59-76. · Zbl 1282.11032
[16] S.Cooper, J.Guillera, A.Straub and W.Zudilin, ‘Crouching AGM, hidden modularity’, in: Frontiers in Orthogonal Polynomials and q-Series, Contemporary Mathematics and its Applications: Monographs, Expositions and Lecture Notes, 1 (eds. M.Zuhair Nashed and X.Li) (World Scientific, Singapore, 2018), 169-187. · Zbl 1437.11058
[17] S.Cooper and D.Ye, ‘The level 12 analogue of Ramanujan’s function k’, J. Aust. Math. Soc.101 (2016), 29-53. · Zbl 1404.11040
[18] É.Goursat, ‘Sur l’équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique’, Ann. Sci. Éc. Norm. Supér. Sér. 210 (1881), 3-142. · JFM 13.0267.01
[19] J.Guillera, ‘New proofs of Borwein-type algorithms for Pi’, Integral Transforms Spec. Funct.27 (2016), 775-782. · Zbl 1402.11153
[20] J.Guillera and W.Zudilin, ‘Ramanujan-type formulae for 1/𝜋: the art of translation’, in: The Legacy of Srinivasa Ramanujan, Ramanujan Mathematical Society Lecture Notes Series, 20 (eds. B. C.Berndt and D.Prasad) (Ramanujan Mathematical Society, Mysore, 2013), 181-195. · Zbl 1371.11162
[21] R. S.Maier, ‘Algebraic hypergeometric transformations of modular origin’, Trans. Amer. Math. Soc.359 (2007), 3859-3885. · Zbl 1145.11034
[22] S.Ramanujan, Notebooks, Vol. 2 (Tata Institute of Fundamental Research, Bombay, 1957). · Zbl 0138.24201
[23] S.Ramanujan, ‘Modular equations and approximations to 𝜋’, Q. J. Math.45 (1914), 350-372; reprinted in Collected Papers, 3rd printing (American Mathematical Society/Chelsea, Providence, RI, 2000), 23-39. · JFM 45.1249.01
[24] M.Rogers, ‘New_5F_4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/𝜋’, Ramanujan J.18 (2009), 327-340. · Zbl 1226.11113
[25] R.Vidūnas, ‘Algebraic transformations of Gauss hypergeometric functions’, Funkcial. Ekvac.52 (2009), 139-180. · Zbl 1176.33005
[26] D.Zagier, ‘Integral solutions of Apéry-like recurrence equations’, in: Groups and Symmetries, CRM Proceedings and Lecture Notes, 47 (American Mathematical Society, Providence, RI, 2009), 349-366. · Zbl 1244.11042
[27] W.Zudilin, ‘Ramanujan-type formulae for 1/𝜋: a second wind?’, in: Modular Forms and String Duality (Banff, 3-8 June 2006), Fields Institute Communications, 54 (eds. N.Yui, H.Verrill and C. F.Doran) (American Mathematical Society, Providence, RI, 2008), 179-188. · Zbl 1159.11053
[28] W.Zudilin, ‘Lost in translation’, in: Advances in Combinatorics, Waterloo Workshop in Computer Algebra, W80 (Waterloo, 26-29 May 2011) (eds. I.Kotsireas and E. V.Zima) (Springer, New York, 2013), 287-293. · Zbl 1285.33008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.