N-colored generalized Frobenius partitions: Generalized Kolitsch identities

TL;DR

This paper derives a new formula for counting N-colored generalized Frobenius partitions, extending previous results and providing an asymptotic estimate based on classical partition functions.

Contribution

It generalizes Kolitsch's identities to squarefree N coprime with 6, linking partition counts to cusp forms and classical partition functions.

Findings

Derived explicit formula for cφ_N(n) involving divisor sums and partition functions.

Extended previous prime N results to squarefree N coprime with 6.

Provided asymptotic behavior of cφ_N(n) in terms of classical partitions.

Abstract

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.