Annals of Probability

Linearization coefficients for orthogonal polynomials using stochastic processes

Michael Anshelevich

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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.

Article information

Ann. Probab., Volume 33, Number 1 (2005), 114-136.

First available in Project Euclid: 11 February 2005

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05E35
Secondary: 05A18: Partitions of sets 05A30: $q$-calculus and related topics [See also 33Dxx] 46L53: Noncommutative probability and statistics 60G51: Processes with independent increments; Lévy processes

Linearization coefficients stochastic measures continuous big q-Hermite polynomials free probability


Anshelevich, Michael. Linearization coefficients for orthogonal polynomials using stochastic processes. Ann. Probab. 33 (2005), no. 1, 114--136. doi:10.1214/009117904000000757.

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