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A characterization of the Rogers \(q\)-Hermite polynomials. (English) Zbl 0835.33012

A family of polynomials \(\{p_n(x)\}\) is called Appell if \(Dp_n(x)=p_{n-1}(x)\) with \(={d\over dx}\), or if it has a generating function \(A(t)\exp(xt)\). The only set of Appell polynomials which are orthogonal are Hermite polynomials.
The operator \(D\) can be replaced by a number of different operators. The finite difference operator \(f(x + 1) - f(x)\) was done directly by Carlitz, with Charlier polynomials the only orthogonal polynomials satisfying \(p_n (x + 1) - p_n(x) = p_{n - 1} (x)\). This is also a corollary of a more general theorem of Hahn finding all orthogonal polynomials \(\{p_n (x)\}\) when \(D\) is the operator \(f(x) \to [f(qx + h) - f(x)]/[(q - 1) x + h]\).
The most general of the divided difference operators which lead to orthogonal polynomials is considered here, the divided difference operator which works for the continuous \(q\)-Hermite polynomials of L. J. Rogers. These polynomials are shown to be the only \(D_q\)-Appell polynomials which are orthogonal. The associated generating function is also given. The exponential like function is the new one discovered by Ismail and Zhang. A number of very attractive formulas are obtained in the course of obtaining this theorem and finding the associated generating function. One is the analogue of \(e^{- D^2} x^n = H_n (x)\) for \(D = {d \over dx}\).
Reviewer: R.Askey (Madison)

MSC:

33D70 Other basic hypergeometric functions and integrals in several variables
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
42A65 Completeness of sets of functions in one variable harmonic analysis
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