On Schwarzian Triangle Functions, Automorphic Forms and a Generalization of Ramanujan.s Triple Differential Equations

Let G be the group of the fractional linear transformations generated by T(.)=.+.,S(.)=.cos.n+sin.n..sin.n+cos.n; where .=2cos.m+cos.nsin.n; m, n is a pair of integers with either n . 2,m . 3 or n . 3,m . 2; . lies in the upper half plane H. A fundamental set of functions f0, fi and f. automorphic with respect to G will be constructed from the conformal mapping of the fundamental domain of G. We derive an analogue of Ramanujan.s triple differential equations associated with the group G and establish the connection of f0, fi and f. with a family of hypergeometric functions.