## Domb’s numbers and Ramanujan-Sato type series for $$1/\pi$$.(English)Zbl 1122.11087

Summary: We construct a general series for $$\frac{1}{\pi}$$. We indicate that Ramanujan’s $$\frac{1}{\pi}$$-series are all special cases of this general series and we end the paper with a new class of $$\frac{1}{\pi}$$-series. Our work is motivated by series recently discovered by Takeshi Sato.

### MSC:

 11Y60 Evaluation of number-theoretic constants 11F11 Holomorphic modular forms of integral weight 33C20 Generalized hypergeometric series, $${}_pF_q$$ 33C05 Classical hypergeometric functions, $${}_2F_1$$
Full Text:

### References:

 [1] Berndt, B.C., Ramanujan’s notebooks, part III, (1991), Springer New York · Zbl 0733.11001 [2] Berndt, B.C.; Bhargava, S.; Garvan, F.G., Ramanujan’s theories of elliptic functions to alternative bases, Trans. amer. math. soc., 347, 4163-4244, (1995) · Zbl 0843.33012 [3] Berndt, B.C.; Chan, H.H., Eisenstein series and approximations to π, Illinois J. math., 45, 1, 75-90, (2001) · Zbl 0998.33003 [4] Berndt, B.C.; Chan, H.H.; Huang, S.S., Incomplete elliptic integrals in Ramanujan’s lost notebook, Contemp. math., 254, 79-126, (2000) · Zbl 0971.33012 [5] Berndt, B.C.; Chan, H.H.; Liaw, W.C., On Ramanujan’s quartic theory of elliptic functions, J. number theory, 88, 129-156, (2001) · Zbl 1005.33009 [6] Borwein, J.M.; Borwein, P.B., Pi and the AGM, (1987), Wiley New York · Zbl 0699.10044 [7] Chan, H.H., On Ramanujan’s cubic transformation formula for $$2F1(13,23;1;z)$$, Math. proc. Cambridge philos. soc., 124, 193-204, (1998) [8] Chan, H.H.; Liaw, W.C.; Tan, V., Ramanujan’s class invariant λn and a new class of series for 1/π, J. London math. soc., 64, 2, 93-106, (2001) · Zbl 1110.11300 [9] Domb, C., On the theory of cooperative phenomena in crystals, Adv. phys., 9, 149-361, (1960) [10] M. Kontsevich, D. Zagier, Periods, Mathematics Unlimited—2001 and beyond, Springer, Berlin, 2001, pp. 771-808. [11] Petkovsek, M.; Wilf, H.S.; Zeilberger, D., A=B, (1996), AK Peters USA [12] Ramanujan, S., Modular equations and approximations to π, Quart. J. math. (Oxford), 45, 350-372, (1914) · JFM 45.1249.01 [13] T. Sato, 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. [14] N.J.A. Sloane, http://www.research-att.com/njas/sequences/index.html. [15] Venkatachaliengar, K., Development of elliptic functions according to Ramanujan, (1988), Madurai Kamaraj University Madurai · Zbl 0913.33012 [16] Y.F. Yang, On differential equations satisfied by modular forms, preprint. · Zbl 1108.11040
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.