A Transformation Law of an Eta Product and the Invariance of a Class of Entire Modular Functions Under $\Gamma _0 (n)$

Abstract

We present a new proof, using Residue Calculus, of the transformation law of a general eta product under $\Gamma_0(n)$ where $n$ is any integer, then we deduce the invariance of a special case of this product under this group and we prove the transformation law of another special case. Our proof is inspired by Siegel's proof [7] of the transformation law of the Dedekind eta function and by Rademacher's generalization [5].