Linearization coefficients for orthogonal polynomials using stochastic processes

• 成果类型: Article
• 署名作者: Anshelevich, M
• 署名单位: University of California System; University of California Riverside
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000757
• 发表日期: 2005
• 页码: 114-136
• 关键词: noncrossing partitions; INTEGRALS; hermite

摘要

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.