Expansions of one density via polynomials orthogonal with respect to the other

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:383

Article

Szablowski, Pawel J.

关键词: Orthogonal polynomials;    q-Hermite polynomials;    Al-Salam-Chihara polynomials;    Chebyshev polynomials;    Rogers polynomials;    Connection coefficients;    Positive kernels;    Kernel expansion;    Poisson-Mehler expansion;    q-Gaussian distribution;    Wigner distribution;    Kesten-McKay distribution;   

DOI  :  10.1016/j.jmaa.2011.04.087

来源: Elsevier

【 摘 要 】

We expand the Chebyshev polynomials and some of its linear combination in linear combinations of the q-Hermite, the Rogers (q-utraspherical) and the Al-Salam-Chihara polynomials and vice versa. We use these expansions to obtain expansions of some densities, including q-Normal and some related to it, in infinite series constructed of the products of the other density times polynomials orthogonal to it, allowing deeper analysis and discovering new properties. On the way we find an easy proof of expansion of the Poisson-Mehler kernel as well as its reciprocal. We also formulate simple rule relating one set of orthogonal polynomials to the other given the properties of the ratio of the respective densities of measures orthogonalizing these polynomials sets. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free