On -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 ( 1 p ; 1 q ) n = ∑ n ≥ 0 p q n ( p ; q ) n ( q ; q ) n and ∑ n ≥ 0 ( − 1 ) n ( 1 p ; 1 q ) n = ∑ n ≥ 0 p q n ( p ; q ) n ( − q ; q ) n = ∑ n ≥ 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.