Mathematics – Combinatorics
Scientific paper
2003-01-10
Annals of Probability 2005, Vol. 33, No. 1, 114-136
Mathematics
Combinatorics
Published at http://dx.doi.org/10.1214/009117904000000757 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins
Scientific paper
10.1214/009117904000000757
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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