Internal congruences modulo powers of 2 for $k$-colored overpartitions into odd parts

Abstract

Let ${\overline{po}}_k(n)$ denote the number of $k$-colored overpartitions of $n$ into odd parts. For every odd number of colors, we establish infinite families of internal congruences modulo high powers of $2$. The proof proceeds by expressing the relevant difference generating functions as a fixed theta quotient multiplied by polynomials in an auxiliary function. Uniform lower bounds for the $2$-adic valuations of the polynomial coefficients then yield the stated internal congruences. These results generalize an internal congruence of Saikia, Sarma and Sellers (J. Ramanujan Math. Soc.~2025) on overpartition triples into odd parts.

arXiv comment16 pages