A Note on the Dunkl-Appell Orthogonal Polynomials

Year 2020, Volume: 8 Issue: 2, 263 - 267, 27.10.2020

Mabrouk Sghaier

Abstract

This paper deals with the problem of finding all orthogonal polynomial sets which are also $T_{\mu}$-Appell where $T_{\mu}, \mu \in \mathbb{C}$ is the Dunkl operator. The resulting polynomials reduce to Generalized Hermite polynomials $\{{{H}}_n(\mu)\}_{n\geq0}$.                                                                                                                                                                                                                                                                                                            

Keywords

Orthogonal polynomials , Dual sequence , Dunkl operator

References

• [1] J. Alaya and P. Maroni, Symmetric Laguerre-Hahn forms of class s = 1. Integral Transforms Spec. Funct. 4, (4), (1996), 301-320.
• [2] W. Al-Salam, Characterization theorems for orthogonal polynomials, in: P. Nevai (Ed.), Orthogonal Polynomials: Theory and Practice, in: NATO ASI Ser. C Math. Phys. Sci., vol. 294, Kluwer Academic Publishers, Dordrecht, 1990, pp. 1-24.
• [3] A. Angelesco, Sur les polynomes orthogonaux en rapport avec d’autre polynomes, Buletinul Societˆatii din Cluj, 1(1921), 44-59.
• [4] Y. Ben Cheikh and M. Gaied, Characterization of the Dunkl-classical symmetric orthogonal polynomials. Appl. Math. Comput., 187 (2007), 105-114.
• [5] Y. Ben Cheikh and M. Gaied, Dunkl-Appell d-orthogonal polynomials. Integral Transforms Spec. Funct. 18 (8), (2007) 581-597.
• [6] L. Carlitz, Characterization of certain sequences of orthogonal polynomials, Portugaliae Math., 20 (1961), 43-46.
• [7] T. S. Chihara, An introduction to orthogonal polynomials. Gordon and Breach, New York, 1978.
• [8] C.F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213-1227.
• [9] A. Ghressi and L. Kheriji, A new characterization of the generalized Hermite form. Bull Belg Math Soc Simon Stevin. 15 (3) (2008), 561-567.
• [10] L. Kheriji, P. Maroni, The Hq-classical orthogonal polynomials,Acta Appl. Math. 71 (2002), 49-115.
• [11] P. Maroni, Une theorie algebrique des polynomes orthogonaux. Application aux polynˆomes orthogonaux semi-classiques, in: Orthogonal Polynomials and their applications. (C. Brezinski et al Editors.) IMACS, Ann. Comput. Appl. Math. 9, ( Baltzer, Basel) (1991), 95-130.
• [12] M. Sghaier, A note on the Dunkl-classical orthogonal polynomials, Integral Transforms Spec. Funct. 23 (10), (2012) 753-760.
• [13] J. Shohat, The relation of the classical orthogonal polmomials to the polmomials of Appell, Amer. J. Math., 58 (1936), 453-464.