A further look at the overpartition function modulo \(2^4\) and \(2^5\)

In this paper, we describe a systematic way of obtaining the exact generating functions for \(\overline{p}(2n)\) $\overline{p}(2n)$ , \(\overline{p}(4n)\) $\overline{p}(4n)$ (first proved by Fortin et al.), \(\overline{p}(8n)\) $\overline{p}(8n)$ , \(\overline{p}(16n)\) $\overline{p}(16n)$ , etc. where \(\overline{p}(n)\) $\overline{p}(n)$ denotes the number of overpartitions of n. We further establish several new infinite families of congruences modulo \(2^4\) $2^4$ and \(2^5\) $2^5$ for \(\overline{p}(n)\) $\overline{p}(n)$ . For example, we prove that for all \(n, \alpha , \beta \ge 0\) $n,\alpha ,\beta \ge 0$ and primes \(p\ge 5\) $p\ge 5$ , \(\begin{aligned} \overline{p}\left( 3^{4\alpha +1}p^{2\beta +1}\left( 24pn+24j+7p\right) \right)&\equiv 0\pmod {2^5} \end{aligned}\) $$\begin{array}{cc}\hfill \overline{p}\left(3^{4\alpha +1},p^{2\beta +1},\left(24,p,n,+,24,j,+,7,p\right)\right)& \equiv 0(mod{2}^5)\hfill \end{array}$$ and \(\begin{aligned} \overline{p}\left( 3^{2\alpha +1}(24n+23)\right)&\equiv 0\pmod {2^5}, \end{aligned}\) $$\begin{array}{cc}\hfill \overline{p}\left(3^{2\alpha +1},(24n+23)\right)& \equiv 0(mod{2}^5),\hfill \end{array}$$ where \(\bigl (\frac{-6}{p}\bigr )=-1\) $(\frac{-6}{p})=-1$ and \(1\le j\le p-1\) $1\le j\le p-1$ . The last congruence was proved by Xiong (Int J Number Theory 12:1195–1208, 2016) for modulo \(2^4\) $2^4$ .