On q-Series Identities Related to Interval Orders

Abstract

We prove several power series identities involving the refined generating function of interval orders, as well as the refined generating function of the self-dual interval orders. These identities may be expressed as $\sum_{n\ge 0}(1/p;1/q)_n= \sum_{n\ge 0} pq^n(p;q)_n(q;q)_n$ and $\sum_{n\ge 0} (-1)^n(1/p;1/q)_n= \sum_{n\ge 0} pq^n(p;q)_n(-q;q)_n =\sum_{n\ge 0} (q/p)^n(p;q^2)_n$, where the equalities apply to the (purely formal) power series expansions of the above expressions at $p=q=1$, as well as at other suitable roots of unity.