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Probabilistic aspects of Al-Salam–Chihara polynomials. (English) Zbl 1074.33015

The Al-Salam-Chihara polynomials were introduced in [Convolutions of orthogonal polynomials, SIAM J. Math. Anal. 7, 16–28 (1976; Zbl 0323.33007)] and their weight function was found by R. Askey and M. Ismail in [Recurrence relations, continued fractions, and orthogonal polynomials, Mem. Am. Math. Soc. 300 (1984; Zbl 0548.33001)].
The authors of this paper find the connection of the (renormalized) Al-Salam-Chihara polynomials \(p_{n}(x | q, a,b)\) with a regression problem in probability. In particular, they show that if the probability distribution \(\mu=\mu(dx | \rho,y)\) satisfies \[ \int H_{n}(x| q) \mu(dx)=\rho^n \,H_{n}(y| q),\;\;n=1,2,\ldots, \] then its orthogonal polynomials are Al-Salam-Chihara polynomials \(p_{n}(x | q, a,b)\) with \(a=\rho y\), \(b=\rho^2\), where \(H_{n}(x| q)=p_{n}(x| q,0,0)\) are the corresponding continuous \(q\)-Hermite polynomials.
Using this result they give a new simple derivation of the density of some \(p_{n}(x| q, a,b)\). A key role to these results plays a connection coefficient formula between the Al-Salam-Chihara polynomials \(p_{n}(x | q, a,b)\) and the continuous \(q\)-Hermite polynomials \(H_{n}(x| q)\), obtained in this paper in a simple way. The authors also compute determinants of Hankel matrices with entries that are linear combinations of the \(q\)-Hermite polynomials.

MSC:

33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A30 \(q\)-calculus and related topics
15A15 Determinants, permanents, traces, other special matrix functions
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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References:

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