N-colored generalized Frobenius partitions: generalized Kolitsch identities

Abstract

AbstractLet $N\geq 1$ be squarefree with $(N,6)=1$ . Let $c\phi _N(n)$ denote the number of N-colored generalized Frobenius partitions of n introduced by Andrews in 1984, and $P(n)$ denote the number of partitions of 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 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 $P(n)$ .