Classical automorphic forms and hypergeometric functions

Abstract

We exhibit a graded algebra of hypergeometric functions and show how to canonically identify it with the graded algebra of modular forms for the full modular group SL 2( Z). We also show how Dedekind's eta function is related to the square root of a hypergeometric function and give yet another simple proof of its functional equation. The methods permit the simple translation of integrals of modular forms (e.g., Mellin transforms and special values of associated Dirichlet series) into integrals of hypergeometric functions where the theory of these classical special functions can be brought to bear. As an example, we express the Mellin transform of an Eisenstein series (which involves Riemann's zeta function) in terms of hypergeometric functions.