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

Paweł Szabłowski

Abstract

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.
Author Paweł Szabłowski (FMIS / DAST)
Paweł Szabłowski,,
- Department of Analysis and Sigularity Theory
Journal seriesJournal of Mathematical Analysis and Applications, ISSN 0022-247X, (A 40 pkt)
Issue year2011
Vol383
No1
Pages35-54
Keywords in EnglishAl-Salam–Chihara polynomials, Chebyshev polynomials, Connection coefficients, Kernel expansion, Kesten–McKay distribution, orthogonal polynomials, Poisson–Mehler expansion, Positive kernels, q-Gaussian distribution, q-Hermite polynomials, Rogers polynomials, Wigner distribution
DOIDOI:10.1016/j.jmaa.2011.04.087
URL http://www.sciencedirect.com/science/article/pii/S0022247X11004288
Score (nominal)40
Publication indicators WoS Impact Factor: 2011 = 1.001 (2) - 2011=1.305 (5)
Citation count*9 (2015-04-21)
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