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

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

ISSN-13: 9789813228870

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},
}