Infinite Families of Congruences Modulo 81 for Overpartitions with Restricted Odd Differences

Abstract

Let $\overline{t}(n)$ denote the number of overpartitions of weight $n$ in which (i) the difference between the 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 this paper, by employing fundamental generating function dissection techniques and various $q$-series identities, we prove several new infinite families of congruences modulo 81 for $\overline{t}(n)$. For example, for $\alpha \geq 0$ and $n \geq 0$,
\[
\overline{t}\left(4^{2\alpha+1}(8n+5)\right) \equiv 0 \pmod{81}.
\]
Consequently, we extend a result of Hirschhorn and Sellers to an infinite family, namely, for $\alpha \geq 0$ and $n \geq 0$,
\[
\overline{t}\left(4^{\alpha}(8n+5)\right) \equiv 0 \pmod{9}.
\]