A q-analogue of Catalan Hankel determinants

Abstract

In this paper 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 $\frac{(aq;q{)}_n}{(ab{q}^2;q{)}_n}$ as a q-analogue of Catalan numbers $C_n=\frac{1}{n+1}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$, 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 also consider a q-analogue of Schröder Hankel determinants, and give a new proof of Moztkin Hankel determinants using an addition formula for ${}_2{F}_1$.

17 pages

References in corpus (1)

• Hankel Determinants for Some Common Lattice Paths

Cited by in corpus (1)

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