The Combinatorics of q-Hermite polynomials and the Askey—Wilson Integral

Mourad E.H. Ismail, Dennis Stanton, Gérard Viennot

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Abstract

The q-Hermite polynomials are defined as a q-analogue of the matching polynomial of a complete graph. This allows a combinatorial evaluation of the integral used to prove the orthogonality of Askey and Wilson's 4φ3 polynomials. A special case of this result gives the linearization formula for q-Hermite polynomials. The moments and associated continued fraction are explicitly given. Another set of polynomials, closely related to the q-Hermite, is defined. These polynomials have a combinatorial interpretation in terms of finite vector spaces which give another proof of the linearization formula and the q-analogue of Mehler's formula.

Original languageEnglish (US)
Pages (from-to)379-392
Number of pages14
JournalEuropean Journal of Combinatorics
Volume8
Issue number4
DOIs
StatePublished - 1987

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