N-colored generalized Frobenius partitions: Generalized Kolitsch identities

Abstract

Let $N\ge 1$ be squarefree with $(N,6)=1$. Let $c{\varphi }_N(n)$ denote the number of $N$-colored generalized Frobenius partition of $n$ introduced by Andrews in 1984. We prove $$c{\varphi }_N(n)=\sum _{d\mid N}N/d\cdot P\left(\frac{N}{d^2}n-\frac{N^2-d^2}{24d^2}\right)+b(n)$$ where $C(z):=(q;q{)}_{\infty }^N{\sum }_{n=1}^{\infty }b(n)q^n$ is a cusp form in $S_{(N-1)/2}({\Gamma }_0(N),{\chi }_N)$. This extends and strengthens earlier results of Kolitsch and Chan-Wang-Yan treating the case when $N$ is a prime. As an immediate application, we obtain an asymptotic formula for $c{\varphi }_N(n)$ in terms of the classical partition function.