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.
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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.
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Communicated by Erik Koelink.
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The author was supported in part by NSF Grant DMS-1510296.
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Xu, Y. Intertwining Operators Associated with Dihedral Groups. Constr Approx 52, 395–422 (2020). https://doi.org/10.1007/s00365-019-09487-w
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DOI: https://doi.org/10.1007/s00365-019-09487-w