×

Sporadic sequences, modular forms and new series for \(1/\pi\). (English) Zbl 1336.11031

Summary: Two new sequences, which are analogues of six sporadic examples of D. Zagier, are presented. The connection with modular forms is established and some new series for \(1/\pi\) are deduced. The experimental procedure that led to the discovery of these results is recounted. Proofs of the main identities will be given, and some congruence properties that appear to be satisfied by the sequences will be stated as conjectures.

MSC:

11F11 Holomorphic modular forms of integral weight
11F27 Theta series; Weil representation; theta correspondences
11Y60 Evaluation of number-theoretic constants
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Almkvist, G., van Enckevort, C., van Straten, D., Zudilin, W.: Tables of Calabi–Yau equations. http://wain.mi.ras.ru/publications.html (webpage accessed July 14, 2011) · Zbl 1223.33007
[2] Almkvist, G., van Straten, D., Zudilin, W.: Generalizations of Clausen’s formula and algebraic transformations of Calabi–Yau differential equations. Proc. Edinb. Math. Soc. 54, 273–295 (2011) · Zbl 1223.33007
[3] Barrucand, P.: Sur la somme des puissances des coefficients multinomiaux et les puissances successives d’une fonction de Bessel. C. R. Acad. Sci. Paris 258, 5318–5320 (1964) · Zbl 0126.28602
[4] Baruah, N.D., Berndt, B.C., Chan, H.H.: Ramanujan’s series for 1/{\(\pi\)}: a survey. Am. Math. Mon. 116, 567–587 (2009) · Zbl 1229.11162
[5] Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer, New York (1991) · Zbl 0733.11001
[6] Chan, H.H., Chan, S.H., Liu, Z.-G.: Domb’s numbers and Ramanujan–Sato type series for 1/{\(\pi\)}. Adv. Math. 186, 396–410 (2004) · Zbl 1122.11087
[7] Chan, H.H., Cooper, S.: Powers of theta functions. Pac. J. Math. 235, 1–14 (2008) · Zbl 1231.11048
[8] Chan, H.H., Cooper, S.: Rational analogues of Ramanujan’s series for 1/{\(\pi\)}, preprint · Zbl 1268.11165
[9] Chan, H.H., Cooper, S., Sica, F.: Congruences satisfied by Apéry-like numbers. Int. J. Number Theory 6, 89–97 (2010) · Zbl 1303.11009
[10] Chan, H.H., Loo, K.P.: Ramanujan’s cubic continued fraction revisited. Acta Arith. 126, 305–313 (2007) · Zbl 1123.11042
[11] Chan, H.H., Ong, Y.L.: On Eisenstein series and $\(\backslash\)sum_{m,n=-\(\backslash\)infty}\^{\(\backslash\)infty }q\^{m\^{2}+mn+2n\^{2}}$ . Proc. Am. Math. Soc. 127, 1735–1744 (1999) · Zbl 0922.11039
[12] Chan, H.H., Tanigawa, Y., Yang, Y., Zudilin, W.: New analogues of Clausen’s identities arising from the theory of modular forms. Adv. Math. 228, 1294–1314 (2011) · Zbl 1234.33009
[13] Chan, H.H., Verrill, H.: The Apéry numbers, the Almkvist–Zudilin numbers and new series for 1/{\(\pi\)}. Math. Res. Lett. 16, 405–420 (2009) · Zbl 1193.11038
[14] Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc. 11, 308–339 (1979) · Zbl 0424.20010
[15] Cooper, S.: Series and iterations for 1/{\(\pi\)}. Acta Arith. 141, 33–58 (2010) · Zbl 1226.11133
[16] Cooper, S.: Series for 1/{\(\pi\)} from level 10, J. Ramanujan Math. Soc., to appear · Zbl 1282.11032
[17] Cooper, S., Toh, P.C.: Quintic and septic Eisenstein series. Ramanujan J. 19, 163–181 (2009) · Zbl 1259.11046
[18] Elaydi, S.: An Introduction to Difference Equations, 3rd edn. Springer, New York (2005) · Zbl 1071.39001
[19] Elsner, C.: On recurrence formulae for sums involving binomial coefficients. Fibonacci Q. 43, 31–45 (2005) · Zbl 1136.40301
[20] Franel, J.: Solution to question 42. L’Intermédiaire des Math. 1, 45–47 (1894)
[21] Franel, J.: Solution to question 170. L’Intermédiaire des Math. 2, 33–35 (1895)
[22] Maier, R.S.: On rationally parametrized modular equations. J. Ramanujan Math. Soc. 24, 1–73 (2009) · Zbl 1214.11049
[23] Ramanujan, S.: Modular equations and approximations to {\(\pi\)}. Q. J. Math. 45, 350–372 (1914) · JFM 45.1249.01
[24] Ramanujan, S.: Notebooks (2 vols.). Tata, Bombay (1957) · Zbl 0138.24201
[25] Rogers, M.D.: New 5 F 4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/{\(\pi\)}. Ramanujan J. 18, 327–340 (2009) · Zbl 1226.11113
[26] Sato, T.: Apéry numbers and Ramanujan’s series for 1/{\(\pi\)}. Abstract of a talk presented at the Annual meeting of the Mathematical Society of Japan, 28–31 March (2002)
[27] Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999) · Zbl 0928.05001
[28] van der Poorten, A.: A proof that Euler missed ...Apéry’s proof of the irrationality of {\(\zeta\)}(3). An informal report. Math. Intell. 1(4), 195–203 (1978/1979) · Zbl 0409.10028
[29] Verrill, H.A.: Some congruences related to modular forms, preprint. http://www.math.lsu.edu/\(\sim\)verrill/ (webpage accessed: Sept. 15, 2010) · Zbl 1209.11047
[30] Zagier, D.: Integral solutions of Apéry-like recurrence equations. In: Groups and Symmetries. CRM Proc. Lecture Notes, vol. 47, pp. 349–366. Am. Math. Soc., Providence (2009) · Zbl 1244.11042
[31] Zudilin, W.: Ramanujan-type formulae for 1/{\(\pi\)}: a second wind? In: Modular Forms and String Duality. Fields Inst. Commun., vol. 54, pp. 179–188. Am. Math. Soc., Providence (2008) · Zbl 1159.11053
[32] Zudilin, W.: Private communication, March 23 (2011)
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.