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We describe various aspects of the Al-Salam-Chihara $q$-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients. It is remarkable that the corresponding moment sequence appears also in the recent work of Postnikov and Williams on enumeration of totally positive Grassmann cells.
The higher $q,t$-Catalan polynomial $C^{(m)}_n(q,t)$ can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of $n$. This pape
This contribution deals with the sequence ${mathbb{U}_{n}^{(a)}(x;q,j)}_{ngeq 0}$ of monic polynomials, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam--Carlitz I orthogonal polynomials, and involving an arbitrary numb
In recent joint work (2021), we introduced a novel multivariate polynomial attached to every metric space - in particular, to every finite simple connected graph $G$ - and showed it has several attractive properties. First, it is multi-affine and rea
Athanasiadis conjectured that, for every positive integer $r$, the local $h$-polynomial of the $r$th edgewise subdivision of any simplex has only real zeros. In this paper, based on the theory of interlacing polynomials, we prove that a family of pol
We explore some connections between moments of rescaled little q-Jacobi polynomials, q-analogues of values at negative integers for some Dirichlet series, and the q-Eulerian polynomials of wreath products of symmetric groups.