Pfaffian decomposition and a Pfaffian analogue of q-Catalan Hankel determinants

Motivated by the Hankel determinant evaluation of moment sequences, we study an analogous Pfaffian evaluation. We prove an LU-decomposition analogue for skew-symmetric matrices, called Pfaffian decomposition. We then apply this formula to evaluate Pfaffians related to some moment sequences of classical orthogonal polynomials. In particular we obtain a product formula for a kind of q-Catalan Pfaffians. We also establish a connection between our Pfaffian formulas and certain weighted enumeration of shifted reverse plane partitions.