A SIMPLE PROOF OF QUOTIENTS OF THETA SERIES AS RATIONAL FUNCTIONS OF J

For two even unimodular positive definite integral quadratic forms A[X], B[X] in n-variables, J. K. Koo [1, Theorem 1] showed that ${\theta}_A(\tau)/{\theta}_B(\tau)$ is a rational function of J, satisfying a certain condition. Where ${\theta}_A(\tau)$ and ${\theta}_B(\tau)$ are theta series related to A[X] and B[X], respectively, and J is the classical modular invariant. In this paper we give a simple proof of Theorem 1 of [1].