Congruence properties of 8- and 9-colored generalized Frobenius partitions modulo 5
DOI:
https://doi.org/10.55016/ojs/cdm.v20i2.78838Abstract
In his 1984 AMS Memoir, Andrews introduced the family of functions $c\phi_k(n)$, which denotes the number of $k$-colored generalized Frobenius partitions of $n$. In this paper, by employing some $q$-series identities and elementary generating function manipulations, we prove a characterization of $c\phi_8(5n+2)$ modulo $5$. Moreover, we derive a characterization of $c\phi_9(5n+1)$ modulo $5$. These two characterizations can lead to the corresponding infinite sets of Ramanujan-type congruences modulo $5$ satisfied by $c\phi_8(n)$ and $c\phi_9(n)$.
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