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

Abstract

Let m be a positive integer $\ge$3 and $\lambda =2\cos \frac{\pi }{m}$. 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.