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

TL;DR

This paper provides an elementary proof confirming Saikia's conjecture on congruences for t-colored overpartitions for all odd integers t, using classical generating functions, dissections, and induction.

Contribution

It offers a simple, elementary proof of Saikia's conjecture extending its validity from prime to all odd integers t.

Findings

Saikia's conjecture holds for all odd t.

Elementary proof using classical methods.

Confirmed specific congruences for overpartition functions.

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 $t\geq 1$ 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 $n\geq 0$ 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).