An Elementary Proof of a Conjecture of Saikia on Congruences for $t$--Colored Overpartitions

Abstract

The starting point for this work is the family of functions ${\overline{p}}_{-t}(n)$ which counts the number of $t$--colored overpartitions of $n.$ In recent years, several infinite families of congruences satisfied by ${\overline{p}}_{-t}(n)$ for specific values of $t\ge 1$ have been proven. In particular, in his 2023 work, Saikia proved a number of congruence properties modulo powers of 2 for ${\overline{p}}_{-t}(n)$ for $t=5,7,11,13$. He also included the following conjecture in that paper: \newline \ %\newline \noindent Conjecture: For all $n\ge 0$ and primes $t$, we have \begin{eqnarray*} \overline{p}_{-t}(8n+1) &\equiv & 0 \pmod{2}, \\ \overline{p}_{-t}(8n+2) &\equiv & 0 \pmod{4}, \\ \overline{p}_{-t}(8n+3) &\equiv & 0 \pmod{8}, \\ \overline{p}_{-t}(8n+4) &\equiv & 0 \pmod{2}, \\ \overline{p}_{-t}(8n+5) &\equiv & 0 \pmod{8}, \\ \overline{p}_{-t}(8n+6) &\equiv & 0 \pmod{8}, \\ \overline{p}_{-t}(8n+7) &\equiv & 0 \pmod{32}. \end{eqnarray*} Using a truly elementary approach, relying on classical generating function manipulations and dissections, as well as proof by induction, we show that Saikia's conjecture holds for {\bf all} odd integers $t$ (not necessarily prime).