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

Abstract

In this contribution, we study the orthogonality conditions satisfied by Al-Salam- Carlitz polynomials U_n(a)(x; q) when the parameters a and q are not necessarily real nor ‘classical’. 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.