The orthogonality of Al-Salam-Carlitz polynomials for complex parameters. (English) Zbl 1411.33012

Cohl, Howard S.; Costas-Santos, Roberto S.; Xu, Wenqing

Nashed, M. Zuhair (ed.) et al., Frontiers in orthogonal polynomials and \(q\)-series. Papers based on the international conference, University of Central Florida, Orlando, FL, USA, May 10–12, 2015. Dedicated to Professor Mourad Ismail on his 70th birthday. Hackensack, NJ: World Scientific. Contemp. Math. Appl., Monogr. Expo. Lect. Notes 1, 155-167 (2018).

Summary: In this chapter, 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 \(\mathbf{u}\) with respect to such a 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 \(U^{(a)}_n(x;q)\) of degree \(n\) up to a constant factor. We also obtain a generalization of the unique generating function for these polynomials.
For the entire collection see [Zbl 1407.33001].

MSC:

33C47	Other special orthogonal polynomials and functions
42C05	Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis

Keywords:

\(q\)-orthogonal polynomials; \(q\)-difference operator; \(q\)-integral representation; discrete measure