Congruences Modulo 5 and 7 for 4-Colored Generalized Frobenius Partitions

Let cx1D719₊ (n) denote the number of k -colored generalized Frobenius partitions of n. Recently, new Ramanujan-type congruences associated with cx1D719₄ (n) were discovered. In this article, we discuss two approaches in proving such congruences using the theory of modular forms. Our methods allow us to prove congruences such as cx1D719₄ (14n+6) 0\;mod\;7 and Seller’s congruence cx1D719₄ (10n+6) 0\;mod\;5.