On the generalized eta- and theta-transformation formulas as the Hecke modular relation

Abstract We prove that both the eta transformation formula and the transformation formula for Dieter’s Lambert series are consequences of the Hecke functional equation for the generating zeta-function. Then using the product expression for the theta-function, we prove the theta-transformation formula under a general modular substitution is a consequence of those two transformation formulas, and consequently, theta transformation formula itself is a consequence of the Hecke modular relation.