Elementary proofs of congruences modulo 5 for overpartitions with restricted odd differences

RESUMO

In 2015, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function $\overline{t}(n)$ to be the number of overpartitions of weight $n$ where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. In their work, they proved that $\overline{t}(n)$ satisfies an elegant congruence modulo 3. Since then, a number of authors have studied arithmetic properties satisfied by $\overline{t}(n)$. In particular, in 2023, Hanson and Smith utilized the theory of modular forms to prove the following two congruences modulo 5: For all $n\geq 0$, \begin{align*} \overline{t}(80n+40)\equiv \overline{t}(80n+60)\equiv 0 \pmod{5}. \end{align*} Our goal in this work is to provide a truly elementary proof of this pair of congruences.