In his 1984 AMS memoir, Andrews introduced the family of k-colored generalized Frobenius partition functions. For any positive integer k, let $c{\varphi }_k(n)$ denote the number of k-colored generalized Frobenius partitions of n. Among many other things, Andrews proved that for any $n\ge 0$, $c{\varphi }_2(5n+3)\equiv 0(\mathrm{mod}5)$. 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 $\mathrm{\text{C}}{\mathrm{\Phi }}_k(q)$ with $2\le k\le 17$ by employing the theory of modular forms, where $\mathrm{\text{C}}{\mathrm{\Phi }}_k(q)$ denotes the generating function of $c{\varphi }_k(n)$. We notice that many coefficients in the expressions of $\mathrm{\text{C}}{\mathrm{\Phi }}_k(q)$ are not integers. In this paper, we first observe that $\mathrm{\text{C}}{\mathrm{\Phi }}_k(q)$ 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 $\mathrm{\text{C}}{\mathrm{\Phi }}_k(q)$ with integral coefficients. As an immediate consequence, we prove some infinite families of congruences satisfied by $c{\varphi }_k(n)$, where k is allowed to grow arbitrary large.

在他 1984 年的 AMS 回忆录中，Andrews 介绍了k色广义 Frobenius配分函数系列。对于任何正整数k，令$C{\phi }_k（n）$表示n的k色广义 Frobenius 分区的数量。在许多其他事情中，安德鲁斯证明了对于任何$n\ge 0$,$C{\phi }_2（5n+3）==0（\mathrm{模组}5）$。从那时起，许多学者随后考虑了各种k色广义 Frobenius 配分函数（通常具有少量颜色）的同余性质。

2019年，Chan、Wang和Yang系统地研究了$\mathrm{\text{C}}{\mathrm{\Phi }}_k（q）$和$2\le k\le \mathrm{17\; 号}$通过采用模形式理论，其中$\mathrm{\text{C}}{\mathrm{\Phi }}_k（q）$表示生成函数$C{\phi }_k（n）$。我们注意到表达式中的许多系数$\mathrm{\text{C}}{\mathrm{\Phi }}_k（q）$不是整数。在本文中，我们首先观察到$\mathrm{\text{C}}{\mathrm{\Phi }}_k（q）$与一族双变量函数的常数项相关，然后在这些双变量函数的系数上建立一般的对称和递推关系。基于这种关系，我们接下来导出许多双变量恒等式。通过提取和计算这些双变量恒等式的常数项，我们建立了表达式$\mathrm{\text{C}}{\mathrm{\Phi }}_k（q）$具有积分系数。直接的结果是，我们证明了一些无限的同余族，满足$C{\phi }_k（n）$，其中k可以任意增大。