Hypergeometric modular equations. (English) Zbl 1428.33014

Cooper, Shaun; Zudilin, Wadim

J. Aust. Math. Soc. 107, No. 3, 338-366 (2019).

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.

MSC:

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

Keywords:

Ramanujan-type formula; hypergeometric modular equations

References:

[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