Linearization coefficients for orthogonal polynomials using stochastic processes
Michael Anshelevich
Source: Ann. Probab.
Volume 33, Number 1
(2005), 114-136.
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.
Primary Subjects: 05E35
Secondary Subjects: 05A18, 05A30, 46L53, 60G51
Keywords: Linearization coefficients; stochastic measures; continuous big q-Hermite polynomials; free probability
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1108141722
Digital Object Identifier: doi:10.1214/009117904000000757
Mathematical Reviews number (MathSciNet):
MR2118861
Zentralblatt MATH identifier:
1092.05076
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