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Rational analogues of Ramanujan's series for 1/π†

Published online by Cambridge University Press:  17 May 2012

HENG HUAT CHAN  and
SHAUN COOPER
HENG HUAT CHAN
Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore117543. e-mail: [email protected]
SHAUN COOPER
Affiliation:
Institute of Information and Mathematical Sciences, Massey University-Albany, Private Bag 102904, North Shore Mail Centre, Auckland, New Zealand. e-mail: [email protected]
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Abstract

A general theorem is stated that unifies 93 rational Ramanujan-type series for 1/π, 40 of which are believed to be new. Moreover, each series is shown to have a companion identity, thereby giving another 93 series, the majority of which are new.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 2 , September 2012 , pp. 361 - 383
Copyright
Copyright © Cambridge Philosophical Society 2012

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Footnotes

†

This work is supported by National University of Singapore Academic Research Fund R-146-000-103-112.

References

REFERENCES

[1]Almkvist, G., van Straten, D. and Zudilin, W.Generalizations of Clausen's formula and algebraic transformations of Calabi–Yau differential equations. Proc. Edinburgh Math. Soc. 54 (2011), 273–295.CrossRefGoogle Scholar
[2]Andrews, G. E., Askey, R. A. and Roy, R.Special Functions (Cambridge University Press, 1999).CrossRefGoogle Scholar
[3]Baruah, N. D., Berndt, B. C. and Chan, H. H.Ramanujan's series for 1/π: a survey. Amer. Math. Monthly 116 (2009), 567–587.CrossRefGoogle Scholar
[4]Bauer, G.Von den Coefficienten der Reihen von Kugelfunctionen einer Variablen. J. Reine Angew. Math. 56 (1859), 101–121.Google Scholar
[5]Berndt, B. C.Ramanujan's Notebooks, Part III (Springer-Verlag, New York, 1991).CrossRefGoogle Scholar
[6]Berndt, B. C.Ramanujan's Notebooks, Part V (Springer-Verlag, New York, 1998).CrossRefGoogle Scholar
[7]Berndt, B. C.Number theory in the spirit of Ramanujan (Amer. Math. Soc., Providence, RI, 2006).CrossRefGoogle Scholar
[8]Berndt, B. C., Bhargava, S. and Garvan, F. G.Ramanujan's theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (1995), 4163–4244.Google Scholar
[9]Berndt, B. C. and Chan, H. H.Eisenstein series and approximations to π. Illinois J. Math. 45 (2001), 75–90.CrossRefGoogle Scholar
[10]Berndt, B. C., Chan, H. H., Huang, S.-S., Kang, S.-Y., Sohn, J. and Son, S. H.The Rogers–Ramanujan continued fraction. J. Comput. Appl. Math. 105 (1999), 9–24.CrossRefGoogle Scholar
[11]Berndt, B. C., Chan, H. H. and Liaw, W.–C.On Ramanujan's quartic theory of elliptic functions. J. Number Theory 88 (2001), 129–156.CrossRefGoogle Scholar
[12]Berndt, B. C., Chan, S. H., Liu, Z.–G. and Yesilyurt, H. A.A new identity for (q;q) with an application to Ramanujan's partition congruence modulo 11. Quart. J. Math. 55 (2004), 13–30.CrossRefGoogle Scholar
[13]Borwein, J. M. and Borwein, P. B.More Ramanujan-type series for 1/π. Ramanujan revisited (Urbana-Champaign, Ill., 1987), 359–374 (Academic Press, Boston, MA, 1988).Google Scholar
[14]Borwein, J. M. and Borwein, P. B.Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity (Wiley, New York, 1987).Google Scholar
[15]Borwein, J. M. and Borwein, P. B.Class number three Ramanujan type series for 1/π. J. Comput. Appl. Math. 46 (1993), 281–290.CrossRefGoogle Scholar
[16]Borwein, J. M. and Garvan, F. G.Approximations to π via the Dedekind eta function, In: Organic Mathematics (Burnaby, BC, 1995), 89–115, CMS Conf. Proc., 20 (Amer. Math. Soc., Providence, RI, 1997).Google Scholar
[17]Borwein, J. M., Borwein, P. B. and Bailey, D. H.Ramanujan, modular equations, and approximations to pi, or How to compute one billion digits of pi. Amer. Math. Monthly 96 (1989), 201–219.CrossRefGoogle Scholar
[18]Beukers, F.Irrationality of π2, periods of an elliptic curve and Γ1(5), in Diophantine approximations and transcendental numbers (Luminy, 1982), 47–66. Progr. Math. 31 (Birkhäuser, Boston, 1983).Google Scholar
[19]Chan, H. H., Chan, S. H. and Liu, Z.–G.Domb's numbers and Ramanujan–Sato type series for 1/π. Adv. Math. 186 (2004), 396–410.CrossRefGoogle Scholar
[20]Chan, H. H. and Lang, M. L.Ramanujan's modular equations and Atkin–Lehner involutions. Israel J. Math. 103 (1998), 1–16.CrossRefGoogle Scholar
[21]Chan, H. H., Liaw, W.–C. and Tan, V.Ramanujan's class invariant λn and a new class of series for 1/π. J. London Math. Soc. (2) 64 (2001), 93–106.CrossRefGoogle Scholar
[22]Chan, H. H. and Loo, K. P.Ramanujan's cubic continued fraction revisited. Acta Arith. 126 (2007), 305–313.CrossRefGoogle Scholar
[23]Chan, H. H., Tanigawa, Y., Yang, Y. and Zudilin, W.New analogues of Clausen's identities arising from the theory of modular forms. Adv. Math. 228 (2011), 1294–1314.CrossRefGoogle Scholar
[24]Chan, H. H. and Verrill, H.The Apéry numbers, the Almkvist–Zudilin numbers and new series for 1/π. Math. Res. Lett. 16 (2009), 405–420.CrossRefGoogle Scholar
[25]Chan, S. H.Generalized Lambert series identities. Proc. London Math. Soc. (3) 91 (2005), 598–622.CrossRefGoogle Scholar
[26]Chudnovsky, D. V. and Chudnovsky, G. V.Approximations and complex multiplication according to Ramanujan, Ramanujan revisited (Urbana-Champaign, Ill., 1987), 375–472 (Academic Press, Boston, MA, 1988).Google Scholar
[27]Chudnovsky, D. V. and Chudnovsky, G. V.The computation of classical constants Proc. Nat. Acad. Sci. U.S.A. 86 (1989), 8178–8182.CrossRefGoogle ScholarPubMed
[28]Cooper, S.A simple proof of an expansion of an eta-quotient as a Lambert series. Bull. Austral. Math. Soc. 71 (2005), 353–358.CrossRefGoogle Scholar
[29]Cooper, S.Inversion formulas for elliptic functions. Proc. London Math. Soc. 99 (2009), 461–483.CrossRefGoogle Scholar
[30]Cooper, S.Series and iterations for 1/π. Acta Arith. 141 (2010), 33–58.CrossRefGoogle Scholar
[31]Cooper, S.Level 10 Analogues of Ramanujan's Series for 1/π. J. Ramanujan Math. Soc. 27 (2012), 75–92.Google Scholar
[32]Cooper, S. Sporadic sequences, modular forms and new series for 1/π. Ramanujan J., to appear.Google Scholar
[33]Dobbie, J. M.A simple proof of some partition formulae of Ramanujan's. Quart. J. Math. Oxford, Ser. (2) 6 (1955), 193–196.CrossRefGoogle Scholar
[34]Duke, W.Continued fractions and modular functions. Bull. Amer. Math. Soc. 42 (2005), 137–162.CrossRefGoogle Scholar
[35]Elaydi, S.An Introduction to Difference Equations, Third edition (Springer, New York, 2005).Google Scholar
[36]Farkas, H. M. and Kra, I.Theta constants, Riemann surfaces and the modular group. An introduction with applications to uniformization theorems, partition identities and combinatorial number theory. Graduate Studies in Mathematics 37 (Amer. Math. Soc., Providence, RI, 2001).Google Scholar
[37]Hirschhorn, M. D.A simple proof of an identity of Ramanujan. J. Austral. Math. Soc. Ser. A 34 (1983), 31–35.CrossRefGoogle Scholar
[38]Hirschhorn, M. D.An identity of Ramanujan, and applications, q-series from a contemporary perspective (South Hadley, MA, 1998), 229–234. Contemp. Math. 254 (Amer. Math. Soc., Providence, RI, 2000).Google Scholar
[39]Ramanujan, S.Modular equations and approximations to π. Quart. J. Math. 45 (1914), 350–372.Google Scholar
[40]Ramanujan, S.Notebooks (2 volumes) (Tata Institute of Fundamental Research, Bombay, 1957).Google Scholar
[41]Rogers, M. D.New 5F 4 hypergeometric transformations, three-variable Mahler measures, and formulas for 1/π. Ramanujan J. 18 (2009), 327–340.CrossRefGoogle Scholar
[42]Sato, T.Apéry numbers and Ramanujan's series for 1/π. Abstract of a talk presented at the Annual meeting of the Mathematical Society of Japan, 28–31 (March 2002).Google Scholar
[43]Verrill, H. A. Some congruences related to modular forms. Preprint, http://www.math.lsu.edu/~verrill/ (webpage accessed: Jan. 25, 2012).Google Scholar
[44]Watson, G. N.Theorems stated by Ramanujan. VII: theorems on continued fractions. J. London Math. Soc. 4 (1929), 39–48.CrossRefGoogle Scholar
[45]Zagier, D.Integral solutions of Apéry-like recurrence equations. Groups and symmetries, 349–366, CRM Proc. Lecture Notes 47 (Amer. Math. Soc., Providence, RI, 2009).Google Scholar
[46]Zudilin, W.Ramanujan-type formulae for 1/π: a second wind? In: Modular forms and string duality, 179–188. Fields Inst. Commun. 54 (Amer. Math. Soc., Providence, RI, 2008).Google Scholar