On Ramanujan's Quartic Theory of Elliptic Functions

摘要

In his famous paper [11], [12,pp. 23-29], Ramanujan offers several elegant series for 1/π. He then remarks. "There are corresponding theories in which q is replaced by one or other of the functions" q_r := q_r(x) := exp (-πcsc(π/r)(_2F_1(1/r,(r - 1)/r;1;1 - x))/(_2F_1(1/r,(r - 1)/r;1;x))), (1.1) where r = 3,4, or 6 and where _2F_1 denotes the classical Gaussian hypergeometric function.