901 research outputs found
On Fourier integral transforms for -Fibonacci and -Lucas polynomials
We study in detail two families of -Fibonacci polynomials and -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 -extensions of classical Hermite polynomials. We
show that both of these -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
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