On Hecke groups,Schwarzian triangle functions and a class of hyper-elliptic functions

Let m be a positive integer (ge )3 and (lambda =2cos frac{pi }{m}). The Hecke group (mathfrak {G}(lambda )) is generated by the fractional linear transformations (tau + lambda ) and (-frac{1}{tau }) for (tau ) in the upper half plane (mathbb H) of the complex plane (mathbb C). We consider a set of functions (mathfrak {f}_0, mathfrak {f}_i) and (mathfrak {f}_{infty }) automorphic with respect to (mathfrak {G}(lambda )), constructed from the conformal mapping of the fundamental domain of (mathfrak {G}(lambda )) to the upper half plane (mathbb H), and establish their connection with the Legendre functions and a class of hyper-elliptic functions. Many well-known classical identities associated with the cases of (lambda =1) and 2 are preserved. As an application, we will establish a set of identities expressing the reciprocal of (pi ) in terms of the hypergeometric series.