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The orthogonality of Al-Salam-Carlitz polynomials for complex parameters (1601.08178v1)
Published 29 Jan 2016 in math.CA
Abstract: In this contribution, we study the orthogonality conditions satisfied by Al-Salam-Carlitz polynomials $U{(a)}_n(x;q)$ when the parameters $a$ and $q$ are not necessarily real nor `classical', i.e., the linear functional $\bf u$ with respect to such polynomial sequence is quasi-definite and not positive definite. We establish orthogonality on a simple contour in the complex plane which depends on the parameters. In all cases we show that the orthogonality conditions characterize the Al-Salam-Carlitz polynomials $U_n{(a)}(x;q)$ of degree $n$ up to a constant factor. We also obtain a generalization of the unique generating function for these polynomials.