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This (partly expository) paper originated from the study of Hankel determinants of convolution powers of Catalan numbers and of Narayana polynomials. This led to some Hankel determinants of signed Catalan numbers whose values are multiples of Fibonacci numbers and to some Hankel determinants of signed central binomial coefficients whose values are multiples of Lucas numbers. Most proofs are computational but we also include a combinatorial one due to Christian Krattenthaler. Finally we formulate some conjectures., Comment: 31 pages, 2 figures

We give an overview of known results about Hilbert matrices from the point of view of orthogonal polynomials and compute Hankel determinants of harmonic numbers and related topics., Comment: 18 pages

We show that q-Catalan numbers, q- central binomial coefficients and q- Narayana polynomials are moments of q-analogues of Fibonacci and Lucas polynomials and related polynomials., Comment: 15 pages

The Rogers-Szeg\"o polynomials are natural q-analogues of Newton binomials. In general they have no closed expression. We consider some exceptional cases which are products of a factor with a closed formula and another one with nice values for q=1 and q=-1. These are related to normalized Rogers-Szeg\"o polynomials which have nice expansions and Hankel determinants. The Exposition of the paper is elementary and almost self-contained., Comment: Enlarged and updated Version

In the first part of this paper I give an elementary overview about some number sequences which count various sorts of lattice paths in strips along the x-axis and compute their generating functions in terms of Fibonacci and Lucas polynomials. In the second part I generalize these results by introducing suitable weights and study some special cases in more detail. In the course of this work I have been led to curious number triangles and various conjectures, Comment: Simplified and updated version

In this note we give a survey about polynomials whose moments are multiples of super Catalan numbers and explore two different kinds of q-analogues.

The moments of the Lucas polynomials and of the Chebyshev polynomials of the first kind are (multiples of) central binomial coefficients and the moments of the Fibonacci polynomials and of the Chebyshev polynomials of the second kind are Catalan numbers. In this survey paper we present some generalizations of these results together with various q-analogues.

We present a simple approach to discrete q-Hermite polynomials with special emphasis on analogies with the classical case., Comment: 26 pages

It is shown that some q-analogues of the Fibonacci and Lucas polynomials lead to q-analogues of the Chebyshev polynomials which retain most of their elementary properties., Comment: 29 pages, Rodrigues-type formulae have been added

Two well-known $q$-Hermite polynomials are the continuous and discrete $q$-Hermite polynomials. In this paper we consider a new family of $q$-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with $q$-Fibonacci and $q$-Lucas polynomials. The latter relation yields a generalization of the Touchard-Riordan formula., Comment: 17 pages