Arithmetic Relations for Overpartitions

Abstract

In recent work, Corteel and Lovejoy extensively studied overpartitions as a means of better understanding and interpreting various $q$-series identities. Our goal in this article is quite different. We wish to prove a number of arithmetic relations satisfied by the overpartition function. Employing elementary generating function dissection techniques, we will prove identities such as

$$\sum _{n\ge 0}\overline{p}(8n+7)q^n=64\frac{(q^2{)}_{\mathrm{\infty }}^{22}}{(q{)}_{\mathrm{\infty }}^{23}}$$

and congruences such as

$$\overline{p}(9n+6)\equiv 0(\mathrm{mod}8)$$

where $\overline{p}(n)$ denotes the number of overpartitions of $n$.