On q-series identities related to interval orders

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 ∑n≥0(1p;1q)n=∑n≥0pqn(p;q)n(q;q)n and ∑n≥0(-1)n(1p;1q)n=∑n≥0pqn(p;q)n(-q;q)n=∑n≥0(qp)n(p;q2)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.