The orthogonality of Al-Salam-Carlitz polynomials for complex parameters

TL;DR

This paper investigates the orthogonality properties of Al-Salam-Carlitz polynomials with complex parameters, establishing conditions on complex contours and generalizing their generating functions beyond classical real parameters.

Contribution

It characterizes orthogonality conditions for Al-Salam-Carlitz polynomials with complex parameters and generalizes their generating function.

Findings

Orthogonality on complex contours depending on parameters

Characterization of polynomials up to a constant factor

Generalized generating function for these polynomials

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.