On q-Series Identities Related to Interval Orders

TL;DR

This paper establishes new q-series identities related to the generating functions of interval orders and self-dual interval orders, expanding the mathematical understanding of these combinatorial structures through formal power series equalities.

Contribution

It proves several novel q-series identities involving generating functions of interval orders and self-dual interval orders, connecting combinatorics with q-series theory.

Findings

Proved identities involving refined generating functions of interval orders.

Established equalities for formal power series at roots of unity.

Connected combinatorial structures with q-series identities.

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.