Abstract
The Dunkl operators associated with a dihedral group are a pair of differential-difference operators that generate a commutative algebra acting on differentiable functions in \({\mathbb {R}}^2\). The intertwining operator intertwines between this algebra and the algebra of differential operators. The main result of this paper is an integral representation of the intertwining operator on a class of functions. As an application, closed formulas for the Poisson kernels of h-harmonics and sieved Gegenbauer polynomials are deduced when one of the variables is at vertices of a regular polygon, and similar formulas are also derived for several other related families of orthogonal polynomials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
I’m grateful to David Chow for bringing Dixon’s paper to my attention after my paper was posted on the arXiv. Dixon’s paper was published in 1905.
References
Al-Salam, W., Allaway, W.R., Askey, R.: Sieved ultraspherical polynomials. Trans. Am. Math. Soc. 284, 39–55 (1984)
Constales, D., De Bie, H., Lian, P.: Explicit formulas for the Dunkl dihedral kernel and the \((\kappa, a)\)-generalized Fourier kernel. J. Math. Anal. Appl. 460, 900–926 (2018)
Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics. Springer, Berlin (2013)
Deleaval, L., Demni, N., Youssfi, H.: Dunkl kernel associated with dihedral groups. J. Math. Anal. Appl. 432, 928–944 (2015)
Dixon, A.L.: On the evaluation of certain definite integrals by means of Gamma functions. Proc. Lond. Math. Soc. Ser. 2(3), 189–205 (1905)
Dijksma, A., Koornwinder, T.: Spherical harmonics and the product of two Jacobi polynomials. Indag. Math. 33, 191–196 (1971)
Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311, 167–183 (1989)
Dunkl, C.F.: Poisson and Cauchy kernels for orthogonal polynomials with dihedral symmetry. J. Math. Anal. Appl. 143, 459–470 (1989)
Dunkl, C.F.: Intertwining operators associated to the group \(S^3\). Trans. Am. Math. Soc. 347, 3347–3374 (1995)
Dunkl, C.F.: An intertwining operator for the group \(B_2\). Glasgow Math. J. 49, 291–319 (2007)
Dunkl, C.F.: Polynomials associated with dihedral groups. SIGMA: Symmetry Integr. Geom. Methods Appl. 3, Paper 052 (2007)
Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and its Applications, vol. 155, 2nd edn. Cambridge University Press, Cambridge (2014)
Koornwinder, T.: Jacobi polynomials, II. An analytic proof of the product formula. SIAM J. Math. Anal. 5, 125–137 (1974)
Sawyer, P.: A Laplace-type representation of the generalized spherical functions associated with the root systems of type A. Mediterr. J. Math. 14(4), 147 (2017)
Xu, Y.: Orthogonal polynomials for a family of product weight functions on the spheres. Can. J. Math. 49, 175–192 (1997)
Xu, Y.: Intertwining operator and h-harmonics associated with reflection group. Can. J. Math. 50, 193–209 (1998)
Xu, Y.: A Product Formula for Jacobi Polynomials. Special Functions (Hong Kong, 1999), pp. 423–430. World Scientific Publishing, River Edge (2000)
Xu, Y.: An integral identity with applications in orthogonal polynomials. Proc. Am. Math. Soc. 143, 5253–5263 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Erik Koelink.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author was supported in part by NSF Grant DMS-1510296.
Rights and permissions
About this article
Cite this article
Xu, Y. Intertwining Operators Associated with Dihedral Groups. Constr Approx 52, 395–422 (2020). https://doi.org/10.1007/s00365-019-09487-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-019-09487-w