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

Abstract

Let $c\unicode[STIX]{x1D719}_{k}(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. Recently, new Ramanujan-type congruences associated with $c\unicode[STIX]{x1D719}_{4}(n)$ 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 $c\unicode[STIX]{x1D719}_{4}(14n+6)\equiv 0\;\text{mod}\;7$ and Seller’s congruence $c\unicode[STIX]{x1D719}_{4}(10n+6)\equiv 0\;\text{mod}\;5$.

Congruences Modulo Powers of 3 for 3- and 9-Colored Generalized Frobenius Partitions

“…Since the terms of the form q 3n+1 do not appear in the series expansion of E 5 3 , we deduce that cφ 3 (9n + 5) ≡ 0 (mod 3 5 ). In the proof of Lemma 2.6 we have seen that x 2,1 = 2 • 3 5 .…”

“…In 2016, Gu, Wang and Xia [9] found many congruences modulo powers of 3 for cφ 6 (n). For example, for any integer n ≥ 0, we have that cφ 6 (27n + 16) ≡ 0 (mod 3 5 ), (1.18) cφ 6 (243n + 142) ≡ 0 (mod 3 6 ).…”

“…where we denote For more results on k-colored generalized Frobenius partitions, see [2,3], [5]- [22], [24]- [26].…”

“…Observe that the cases k = 1 and 2 of (1.10) and (1.11) give the following congruences: Moreover, there are some congruences which are not included in (1.10) and (1.11). For example, it appears that for any n ≥ 0, cφ 3 (9n + 5) ≡ 0 (mod 3 5 ), (1.30) cφ 3 (27n + 26) ≡ 0 (mod 3 9 ).…”