Arithmetic Relations for Overpartitions

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

\[
\sum\limits_{n\geq0}\overline{p}\left(8n + 7\right) q^n = 64 \frac{(q^2)_\infty^{22}}{(q)_\infty^{23}}
\]

and congruences such as

\[
\overline{p}(9n+6) \equiv 0 \pmod{8}
\]

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