Cubic modular equations and new Ramanujan-type series for 1∕π

New hypergeometric-like series for $1/\pi^{2}$ arising from Ramanujan’s theory of elliptic functions to alternative base 3

“…By employing each of the remaining pair of values for x n and a n from [11] in (3.34), we can readily arrive at twenty further new series for 1/π 2 .…”

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 .…”

Schläfli modular equations for generalized Weber functions

“…We define the following functions: m A B ≡ 4 (mod 6) (uv) 6 + 11 6 /(uv) 6 (v/u) 2 + (u/v) 2 ≡ 2 (mod 6) (uv) 2 + 11 2 /(uv) 2 (v/u) 6 + (u/v) 6 ≡ 0 (mod 6) (uv) 6 + 11 6 /(uv) 6 (v/u) 6 + (u/v) 6 ≡ 9 (mod 12) (uv) 6 + 11 6 /(uv) 6 …”

“…Modular equations of this type have also been found by Watson [16]. Modular equations also remain a topic of active interest; see for example the work of Chan and Liaw [5], [6]. For more information on the literature associated with modular equations and class invariants, the reader can very profitably consult Berndt's book [2].…”