## Linearization coefficients for orthogonal polynomials using stochastic processes. Annals of Probability 33 (2005)

Citations: | 6 - 0 self |

### BibTeX

@MISC{Anshelevich05linearizationcoefficients,

author = {Michael Anshelevich},

title = {Linearization coefficients for orthogonal polynomials using stochastic processes. Annals of Probability 33},

year = {2005}

}

### OpenURL

### Abstract

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. 1.

### Citations

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2 |
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Citation Context ...ocesses with freely independent increments, and gives the linearization coefficients for the Chebyshev polynomials of the 2nd kind and the free Charlier polynomials. Section 5 is based the results of =-=[Ans01]-=- about q-Lévy processes, and gives the linearization coefficients for the continuous and continuous big q-Hermite polynomials. It also requires some new results about the q-Poisson process. The proofs... |

1 |
stochastic measures and q-deformed cumulants, Doc
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Citation Context ...ocesses with freely independent increments, and gives the linearization coefficients for the Chebyshev polynomials of the 2nd kind and the free Charlier polynomials. Section 5 is based the results of =-=[2]-=- about q-Lévy processes, and gives the linearization coefficients for the continuous and continuous big q-Hermite polynomials. It also requires some new results about the q-Poisson process. The proofs... |