Infinite family of congruences modulo powers of $5$ for $2$ color partition

Résumé

In this work, we investigate the arithmetic properties of $p^{t}_{1,\ell}(n)$, which counts 2-color partitions of $n$ where one of the colors appears only in parts that are not multiples of $t$, and another color appears only in parts which are multiples of $\ell$, By constructing generating functions for $p^{t}_{1,\ell}$ across specific arithmetic progressions, we establish a set of Ramanujan-type infinite family of congruences modulo powers of $5$ for $p^{t}_{1,\ell}(n)$.