Ramanujan-type Congruences for Overpartitions Modulo 5

Let $\bar{p}(n)$ denote the number of overpartitions of $n$. Hirschhorn and Sellers showed that $\bar{p}(4n+3)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}8)$ for $n\ge 0$. They also conjectured that $\bar{p}(40n+35)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}40)$ for $n\geq 0$\mathrm{.}ChenandXiaprovedthisconjecturebyusingthe$(p,k)$-parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we show that $\bar{p}(5n)\equiv (-1{)}^n\bar{p}(4\cdot 5n)(\mathrm{m}\mathrm{o}\mathrm{d}5)$ for $n \geq 0$and$\overline{p}(n)\equiv (-1)^{n}\overline{p}(4n)\pmod{8}$for$n \geq 0$byusingtherelationofthegeneratingfunctionof$\overline{p}(5n)$modulo$5$foundbyTreneerandthe$2$-adicexpansionofthegeneratingfunctionof$\overline{p}(n)$duetoMahlburg\mathrm{.}Asaconsequence,wededucethat$\overline{p}(4^k(40n+35))\equiv 0 \pmod{40}$for$n,k\geq 0$. Furthermore, applying the Hecke operator on $\varphi (q{)}^3$ and the fact that $\varphi (q{)}^3$ is a Hecke eigenform, we obtain an infinite family of congrences $\overline{p}(4^k \cdot5\ell^2n)\equiv 0 \pmod{5}$,where$k\ge 0$and$\ell$ is a prime such that $\mathrm{\ell }\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}5)$ and $\left(\frac{-n}{\mathrm{\ell }}\right)=-1$. Moreover, we show that $\bar{p}(5^{2}n)\equiv \bar{p}(5^{4}n)(\mathrm{m}\mathrm{o}\mathrm{d}5)$ for $n \ge 0$\mathrm{.}Soweareledtothecongruences$\overline{p}\big(4^k5^{2i+3}(5n\pm1)\big)\equiv 0 \pmod{5}$for$n, k, i\ge 0$\mathrm{.}Inthisway,weobtainvariousRamanujan-typecongruencesfor$\overline{p}(n)$modulo$5$suchas$\overline{p}(45(3n+1))\equiv 0 \pmod{5}$and$\overline{p}(125(5n\pm 1))\equiv 0 \pmod{5}$for$n\geq 0$.