The method of constant terms and k-colored generalized Frobenius partitions

Abstract

In his 1984 AMS memoir, Andrews introduced the family of k-colored generalized Frobenius partition functions. For any positive integer k, let $c{\varphi }_k\left(n\right)$ denote the number of k-colored generalized Frobenius partitions of n. Among many other things, Andrews proved that for any $n\ge 0$, $c{\varphi }_2(5n+3)\equiv 0\left(\mathrm{mod}5\right)$. Since then, many scholars subsequently considered congruence properties of various k-colored generalized Frobenius partition functions, typically with a small number of colors.

In 2019, Chan, Wang and Yang systematically studied arithmetic properties of $\text{C}{\Phi }_k\left(q\right)$ with $2\le k\le 17$ by employing the theory of modular forms, where $\text{C}{\Phi }_k\left(q\right)$ denotes the generating function of $c{\varphi }_k\left(n\right)$. We notice that many coefficients in the expressions of $\text{C}{\Phi }_k\left(q\right)$ are not integers. In this paper, we first observe that $\text{C}{\Phi }_k\left(q\right)$ is related to the constant term of a family of bivariable functions, then establish a general symmetric and recurrence relation on the coefficients of these bivariable functions. Based on this relation, we next derive many bivariable identities. By extracting and computing the constant terms of these bivariable identities, we establish the expressions of $\text{C}{\Phi }_k\left(q\right)$ with integral coefficients. As an immediate consequence, we prove some infinite families of congruences satisfied by $c{\varphi }_k\left(n\right)$, where k is allowed to grow arbitrary large.