q-Super Catalan numbers: Combinatorial identities, generating functions, and Narayana refinements

We derive a number of combinatorial identities satisfied by the q-super Catalan numbers. In particular, we extend some of the known combinatorial identities (Touchard, Koshy, Reed Dawson) to the q-super Catalan numbers.

Next, we introduce some q-convolution identities involving q-central binomial and q-Catalan numbers, and derive a generating function for q-Catalan numbers.

Then we introduce Narayana-type refinements of the super Catalan numbers. We prove algebraically the γ-positivity of those refinements and give a combinatorial proof in a special case through the type B analog of noncrossing partitions. Then we introduce their natural q-analogs, prove their q-γ-positivity, and prove some identities they satisfy, generalizing identities of Kreweras [17] and Le Jen-Shoo [11]. Using yet another identity, we prove that these refinements are positive integer polynomials in q.