Bulletin of the Belgian Mathematical Society - Simon Stevin

Clifford-Gegenbauer polynomials related to the Dunkl Dirac operator

H. De Bie and N. De Schepper

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We introduce the so-called Clifford-Gegenbauer polynomials in the framework of Dunkl operators, as well on the unit ball $B(1)$, as on the Euclidean space $\mathbb{R}^m$. In both cases we obtain several properties of these polynomials, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the Jacobi polynomials on the real line. As in the classical Clifford case, the orthogonality of the polynomials on $\mathbb{R}^m$ must be treated in a completely different way than the orthogonality of their counterparts on $B(1)$. In case of $\mathbb{R}^m$, it must be expressed in terms of a bilinear form instead of an integral. Furthermore, in this paper the theory of Dunkl monogenics is further developed.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 18, Number 2 (2011), 193-214.

First available in Project Euclid: 7 June 2011

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Zentralblatt MATH identifier

Primary: 30G35: Functions of hypercomplex variables and generalized variables
Secondary: 33C80: Connections with groups and algebras, and related topics 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

Clifford analysis Dunkl operators Clifford-Gegenbauer polynomials Dunkl monogenics


De Bie, H.; De Schepper, N. Clifford-Gegenbauer polynomials related to the Dunkl Dirac operator. Bull. Belg. Math. Soc. Simon Stevin 18 (2011), no. 2, 193--214. doi:10.36045/bbms/1307452070. https://projecteuclid.org/euclid.bbms/1307452070

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