Construction of a new class of symmetric function of binary products of (p, q)-numbers with 2-orthogonal Chebyshev polynomials

In this paper, we give some new generating functions of the products of (p, q)-Fibonacci numbers, (p, q) -Lucas numbers, (p, q)-Pell numbers, (p, q) -Pell Lucas numbers, (p, q)-Jacobsthal numbers, and (p, q)-Jacobsthal Lucas numbers with 2-orthogonal Chebyshev polynomials and trivariate Fibonacci polynomials. © 2021, Sociedad Matemática Mexicana

On Fourier integral transforms for qq-Fibonacci and qq-Lucas polynomials

We study in detail two families of $q$-Fibonacci polynomials and $q$-Lucas polynomials, which are defined by non-conventional three-term recurrences. They were recently introduced by Cigler and have been then employed by Cigler and Zeng to construct novel $q$-extensions of classical Hermite polynomials. We show that both of these $q$-polynomial families exhibit simple transformation properties with respect to the classical Fourier integral transform

Higher order matching polynomials and d-orthogonality

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials -- the Chebyshev, Hermite, and Laguerre polynomials -- can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogonality for polynomials and we use sign-reversing involutions to show that the higher-order Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are d-orthogonal. We also investigate the moments and find generating functions of those polynomials.Comment: 21 pages, many TikZ figures; v2: minor clarifications and addition

q-Chebyshev polynomials

In this overview paper a direct approach to q-Chebyshev polynomials and their elementary properties is given. Special emphasis is placed on analogies with the classical case. There are also some connections with q-tangent and q-Genocchi numbers.Comment: enlarged version, 34 page