Approximations and complex multiplication according to Ramanujan

On Certain Hypergeometric Summation Theorems Motivated by the Works of Ramanujan, Chudnovsky and Borwein

“…In three further papers , (Borwein, J. M, 1988) and (Borwein, J. M, 1993), they established several additional formulas of this type. D. V. and G. V. Chudnovsky (1988) not only also proved formulas of this sort, but they, moreover, found representations for other transcendental constants, some involving gamma functions, by hypergeometric series of the same kin.…”

Ramanujan’s Contributions to Eisenstein Series, Especially in His Lost Notebook

“…The numbers a n and b n arise in series representations for 1/π proved by D. V. and G. V. Chudnovsky [21] and J. M. and P. B. Borwein [17]. We now have sufficient notation to state our first theorem.…”

“…Let N n denote the number of digits of π which agree with the decimal expansion of A n . Although not mentioned by Ramanujan on page 211 in [44], the ideas needed to prove the results on this page lead to a very general infinite series expansion for 1/π in the spirit of those given by Ramanujan [38], and later by the Chudnovskys [21] and Borweins [17]. To state this expansion, we need further definitions.…”

Eisenstein series and Ramanujan-type series for 1/π

“…These authors subsequently discovered many further series for 1/π [15], [16], [17], [18], [19], where [17] is coauthored with D. H. Bailey. D. V. Chudnovsky and G. V. Chudnovsky [28] independently proved several of Ramanujan's series representations for 1/π and established new ones as well. Further particular series representations for 1/π as well as some general formulas have subsequently been derived by Berndt and H. H. Chan [10], Berndt, Chan, and W.-C. Liaw [11], H. H. Chan, S. H. Chan, and Z. Liu [20], H. H. Chan and Liaw [21], H. H. Chan and K. P. Loo [23], H. H. Chan, Liaw, and V. Tan [22], and H. H. Chan and H. Verrill [24].…”