RAMANUJAN'S CLASS INVARIANT λ_n AND A NEW CLASS OF SERIES FOR 1/π

Abstract

AOn page 212 of his lost notebook, Ramanujan defined a new class invariant λ n and constructed a table of values for λ n . The paper constructs a new class of series for 1\π associated with λ n . The new method also yields a new proof of the Borweins' general series for 1\π belonging to Ramanujan's ' theory of q # '.

Eisenstein series and Ramanujan-type series for 1/π

“…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]. J. Guillera [29]- [33] discovered some beautiful series for 1/π as well as for 1/π 2 .…”

Ramanujan's Eisenstein series and new hypergeometric-like series for 1/π2

“…Chudnovsky and Chudnovsky [27] independently proved several of Ramanujan's series representations for 1/ and established new ones as well. Further particular series representations for 1/ as well as some general formulas were subsequently derived by Berndt and Chan [11], Berndt et al [12], Chan et al [21], Chan and Liaw [22], Chan et al [23], and Chan and Verrill [24]. In a recently communicated paper, the authors [3] employed Ramanujan's ideas expressed in Section 13 of his fundamental paper [37,38, p. 36] and used them in conjunction with 12 identities for Eisenstein series recorded without proofs by Ramanujan in Section 10 of [37,38, pp.…”

Error estimates of Ramanujan-type series

“…Gosper used this formula to compute 17 million digits of π. Since then, series of Ramanujan's type were widely studied by other authors, and we refer here to [8] 1500…”