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January 2005 Linearization coefficients for orthogonal polynomials using stochastic processes
Michael Anshelevich
Ann. Probab. 33(1): 114-136 (January 2005). DOI: 10.1214/009117904000000757

Abstract

Given a basis for a polynomial ring, the coefficients in the expansion of a product of some of its elements in terms of this basis are called linearization coefficients. These coefficients have combinatorial significance for many classical families of orthogonal polynomials. Starting with a stochastic process and using the stochastic measures machinery introduced by Rota and Wallstrom, we calculate and give an interpretation of linearization coefficients for a number of polynomial families. The processes involved may have independent, freely independent or q-independent increments. The use of noncommutative stochastic processes extends the range of applications significantly, allowing us to treat Hermite, Charlier, Chebyshev, free Charlier and Rogers and continuous big q-Hermite polynomials.

We also show that the q-Poisson process is a Markov process.

Citation

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Michael Anshelevich. "Linearization coefficients for orthogonal polynomials using stochastic processes." Ann. Probab. 33 (1) 114 - 136, January 2005. https://doi.org/10.1214/009117904000000757

Information

Published: January 2005
First available in Project Euclid: 11 February 2005

zbMATH: 1092.05076
MathSciNet: MR2118861
Digital Object Identifier: 10.1214/009117904000000757

Subjects:
Primary: 05E35
Secondary: 05A18 , 05A30 , 46L53 , 60G51

Keywords: continuous big q-Hermite polynomials , Free probability , Linearization coefficients , stochastic measures

Rights: Copyright © 2005 Institute of Mathematical Statistics

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Vol.33 • No. 1 • January 2005
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