N-colored generalized Frobenius partitions: generalized Kolitsch identities

Aygin, Zafer Selcuk  ; Nguyen, Khoa D.

Canadian journal of mathematics, Tome 75 (2023) no. 2, pp. 447-469

Résumé

Let $N\geq 1$ 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 $P(n)$.

Détail

DOI : 10.4153/S0008414X22000025

Mots-clés : Generalized Frobenius partitions, generating functions, theta functions, Eisenstein series