Higher-order matching polynomials and d-orthogonality

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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. (C) 2010 Elsevier Inc. All rights reserved.
Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
Issue Date
2011-01
Language
English
Article Type
Article
Keywords

LAGUERRE-POLYNOMIALS; HERMITE-POLYNOMIALS; COMBINATORICS; NUMBERS

Citation

ADVANCES IN APPLIED MATHEMATICS, v.46, no.1-4, pp.226 - 246

ISSN
0196-8858
DOI
10.1016/j.aam.2009.12.008
URI
http://hdl.handle.net/10203/94731
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