A q-analogue of Catalan Hankel determinants

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 ${}_2F_{1}$.Comment: 17 page

Asymptotics of a 3F2{}_3F_2 polynomial associated with the Catalan-Larcombe-French sequence

The large $n$ behaviour of the hypergeometric polynomial \FFF{-n}{\sfrac12}{\sfrac12}{\sfrac12-n}{\sfrac12-n}{-1} is considered by using integral representations of this polynomial. This ${}_3F_2$ polynomial is associated with the Catalan-Larcombe-French sequence. Several other representations are mentioned, with references to the literature, and another asymptotic method is described by using a generating function of the sequence. The results are similar to those obtained by Clark (2004) who used a binomial sum for obtaining an asymptotic expansion.Comment: 10 pages, 1 figure. Accepted for publication in {\em Analysis and Applications

A q-rious positivity

The $q$-binomial coefficients \qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i), for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the many combinatorial interpretations of \qbinom{n}{m}. In this note we conjecture an arithmetically motivated generalisation of the non-negativity property for products of ratios of $q$-factorials that happen to be polynomials.Comment: 6 page

A combinatorial identity with application to Catalan numbers

By a very simple argument, we prove that if $l,m,n$ are nonnegative integers then $$\sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m} =\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}. On the basis of this identity, for $d,r=0,1,2,...$ we construct explicit $F(d,r)$ and $G(d,r)$ such that for any prime $p>\max\{d,r\}$ we have \sum_{k=1}^{p-1}k^r C_{k+d}\equiv \cases F(d,r)(mod p)& if 3|p-1, \\G(d,r)\ (mod p)& if 3|p-2, where $C_n$ denotes the Catalan number $(n+1)^{-1}\binom{2n}{n}$. For example, when $p\geq 5$ is a prime, we have \sum_{k=1}^{p-1}k^2C_k\equiv\cases-2/3 (mod p)& if 3|p-1, \1/3 (mod p)& if 3|p-2; and \sum_{0<k<p-4}\frac{C_{k+4}}k \equiv\cases 503/30 (mod p)& if 3|p-1, -100/3 (mod p)& if 3|p-2. This paper also contains some new recurrence relations for Catalan numbers.Comment: 22 page