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Linearization coefficients for orthogonal polynomials using stochastic processes

Michael Anshelevich
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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.

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Michael Anshelevich. 2005-04-06. Linearization coefficients for orthogonal polynomials using stochastic processes. https://doi.org/10.1214/009117904000000757

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