Expansions of one density via polynomials orthogonal with respect to the other
arXiv:1011.1492 · doi:10.1016/j.jfa.2011.04.002
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.
References in corpus (3)
Cited by in corpus (18)
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- Towards a q-analogue of the Kibble--Slepian formula in 3 dimensions
- On three dimensional multivariate version of q-Normal distribution and probabilistic interpretations of Askey--Wilson, Al-Salam--Chihara and q-ultraspherical polynomials
- A few remarks on orthogonal polynomials
- On summable, positive Poisson-Mehler kernels built of Al-Salam--Chihara and related polynomials
- On positivity of orthogonal series and its applications in probability
- On the generalized Kesten--McKay distributions
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- Askey--Wilson polynomials and a double -series transformation formula with twelve parameters
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