On the modular integrals and their Mellin transforms

Abstract

It is shown that if ${f}$ is an entire modular integral on $Gamma(1)$ of weight k, with multiplier system $nu$, then ${f}^m(tau^r)$ is an entire modular integral on $Gamma(1)$ of weight mk, with multiplier system $nu$. A more general formula is obtained for the Mellin transforms. Some relations among the Mellin transforms of functions ${f}(tau)$, ${f}^m(tau^r)$ and ${f}^m(tau^r/m^,,chi)$ are also deduced.