In 1984, Andrews introduced the family of k-colored generalized Frobenius partition functions. For any \(k\ge 1\) , let \(c\phi _k(n)\) denote the number of k-colored generalized Frobenius partitions of n, and let \(\text {C}\Phi _k(q)\) represent its generating function. Baruah and Sarmah derived integral expressions for \(\text {C}\Phi _k(q)\) with \(k\in \{4,5,6\}\) by applying the method of integer matrix exact covering systems developed by Cao. Subsequently, Chan, Wang and Yang systematically investigated the generating function of \(c\phi _k(n)\) by utilizing the theory of modular forms. In this paper, we propose a unified framework to derive expressions for \(\text {C}\Phi _k(q)\) with integral coefficients, extending several known results about \(\text {C}\Phi _k(q)\) .
