The Rogers-Ramanujan continued fraction and a quintic iteration for 1/𝜋

Chan, Heng Huat; Cooper, Shaun; Liaw, Wen-Chin

doi:10.1090/S0002-9939-07-09031-4

The Rogers-Ramanujan continued fraction and a quintic iteration for $1/\pi$
HTML articles powered by AMS MathViewer

by Heng Huat Chan, Shaun Cooper and Wen-Chin Liaw PDF

Proc. Amer. Math. Soc. 135 (2007), 3417-3424 Request permission

Abstract:

Properties of the Rogers-Ramanujan continued fraction are used to obtain a formula for calculating $1/\pi$ with quintic convergence.

References

George E. Andrews and Bruce C. Berndt, Ramanujan’s lost notebook. Part I, Springer, New York, 2005. MR 2135178
Bruce C. Berndt, Ramanujan’s notebooks. Part III, Springer-Verlag, New York, 1991. MR 1117903, DOI 10.1007/978-1-4612-0965-2
Bruce C. Berndt, Ramanujan’s notebooks. Part V, Springer-Verlag, New York, 1998. MR 1486573, DOI 10.1007/978-1-4612-1624-7
Bruce C. Berndt, Number theory in the spirit of Ramanujan, Student Mathematical Library, vol. 34, American Mathematical Society, Providence, RI, 2006. MR 2246314, DOI 10.1090/stml/034
Bruce C. Berndt and Robert A. Rankin, Ramanujan, History of Mathematics, vol. 9, American Mathematical Society, Providence, RI; London Mathematical Society, London, 1995. Letters and commentary. MR 1353909
Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1987. A study in analytic number theory and computational complexity; A Wiley-Interscience Publication. MR 877728
J. M. Borwein and P. B. Borwein, Approximating $\pi$ with Ramanujan’s modular equations, Rocky Mountain J. Math. 19 (1989), no. 1, 93–102. Constructive Function Theory—86 Conference (Edmonton, AB, 1986). MR 1016163, DOI 10.1216/RMJ-1989-19-1-93
J. M. Borwein, P. B. Borwein, and D. H. Bailey, Ramanujan, modular equations, and approximations to pi, or How to compute one billion digits of pi, Amer. Math. Monthly 96 (1989), no. 3, 201–219. MR 991866, DOI 10.2307/2325206
J. M. Borwein and F. G. Garvan, Approximations to $\pi$ via the Dedekind eta function, Organic mathematics (Burnaby, BC, 1995) CMS Conf. Proc., vol. 20, Amer. Math. Soc., Providence, RI, 1997, pp. 89–115. MR 1483915
Heng Huat Chan, Ramanujan’s elliptic functions to alternative bases and approximations to $\pi$, Number theory for the millennium, I (Urbana, IL, 2000) A K Peters, Natick, MA, 2002, pp. 197–213. MR 1956226
H. H. Chan and K. P. Loo, Ramanujan’s cubic continued fraction revisited, Acta Arith., 126 (2007), no. 4, 305–313.
J. M. Dobbie, A simple proof of some partition formulae of Ramanujan’s, Quart. J. Math. Oxford Ser. (2) 6 (1955), 193–196. MR 72896, DOI 10.1093/qmath/6.1.193
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 2, 137–162. MR 2133308, DOI 10.1090/S0273-0979-05-01047-5
M. D. Hirschhorn, A simple proof of an identity of Ramanujan, J. Austral. Math. Soc. Ser. A 34 (1983), no. 1, 31–35. MR 683175
Michael D. Hirschhorn, An identity of Ramanujan, and applications, $q$-series from a contemporary perspective (South Hadley, MA, 1998) Contemp. Math., vol. 254, Amer. Math. Soc., Providence, RI, 2000, pp. 229–234. MR 1768930, DOI 10.1090/conm/254/03956
Srinivasa Ramanujan, Notebooks. Vols. 1, 2, Tata Institute of Fundamental Research, Bombay, 1957. MR 0099904
Srinivasa Ramanujan, The lost notebook and other unpublished papers, Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988. With an introduction by George E. Andrews. MR 947735