The orthogonality of Al-Salam-Carlitz polynomials for complex parameters
Cohl, H. S., Costas-Santos, R. S. and Xu, W. Frontiers in Orthogonal Polynomials and q-Series. Contemporary Mathematics and its Applications: Monographs,Expositions and Lecture Notes, Eds. Zuhair Nashed and Xin Li, Vol. 1, Chapter 8, 155—167, World Scientific Publishing Company, Cambridge, MA, 2018
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.
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BibTeX
@incollection {MR3791613,
AUTHOR={Cohl, H. S. and Costas-Santos, R. S. and Xu, W.},
TITLE={The orthogonality of {A}l-{S}alam--{C}arlitz polynomials for complex parameters},
BOOKTITLE={Frontiers in orthogonal polynomials and {$q$}-series},
SERIES={Contemp. Math. Appl. Monogr. Expo. Lect. Notes},
VOLUME={1},
PAGES={155--167},
PUBLISHER={World Sci. Publ., Hackensack, NJ},
YEAR={2018},
MRCLASS={33C47 (42C05)},
ZBL ={3A1411.33012},
MRNUMBER={3791613},
}