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 $$D = {d \over dx}$$.

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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