A q-analogue of Catalan Hankel determinants

Details A q-analogue of Catalan Hankel determinants A q-analogue of Catalan Hankel determinants

2010-09-10

arxiv.org/abs/1009.2004v2

RIMS K\^oky\^uroku Bessatsu, B11 (2009), 19--42

Mathematics

Combinatorics

17 pages

Scientific paper

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 $\frac{(aq;q)_{n}}{(abq^{2};q)_{n}}$ as a q-analogue of Catalan numbers $C_{n}=\frac1{n+1}\binom{2n}{n}$, 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 also consider a q-analogue of Schr\"oder Hankel determinants, and give a new proof of Moztkin Hankel determinants using an addition formula for ${}_2F_{1}$.

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