N-colored generalized Frobenius partitions: Generalized Kolitsch identities

Let $N\geq 1$ be squarefree with $(N,6)=1$. Let $c\phi_N(n)$ denote the number of $N$-colored generalized Frobenius partition of $n$ introduced by Andrews in 1984. We prove $$c\phi_N(n)= \sum_{d \mid N} N/d \cdot P\left( \frac{ N}{d^2}n - \frac{N^2-d^2}{24d^2} \right) + b(n)$$ where $C(z) := (q;q)^N_\infty\sum_{n=1}^{\infty} b(n) q^n$ 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 $c\phi_N(n)$ in terms of the classical partition function.