A q-analogue of Catalan Hankel determinants

In this article we shall survey the various methods of evaluating Hankel determinants and as an illustration we evaluate some Hankel determinants of a q-analogue of Catalan numbers. Here we consider (aq;q)n (abq2;q)n as a q-analogue of Catalan numbers Cn = 1 n+1 2n n , which is known as the moments of the little q-Jacobi polynomials. We also give several proofs of this q-analogue, in which we use lattice paths, the orthogonal polynomials, or the basic hypergeometric series. We