<i>N</i>-colored generalized Frobenius partitions: generalized Kolitsch identities

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