Multidimensional Hahn polynomials, intertwining functions on the symmetric group and Clebsch-Gordan coefficients

TL;DR

This paper introduces a broad class of multidimensional Hahn polynomials and intertwining functions on the symmetric group, using a novel tree-method approach and establishing their connection to Clebsch-Gordan coefficients.

Contribution

It generalizes Dunkl's construction, develops a tree-method approach for intertwining functions, and extends the relation between Hahn polynomials and Clebsch-Gordan coefficients to multiple dimensions.

Findings

Developed a wide class of multidimensional Hahn polynomials.

Established a group-theoretic proof linking Hahn polynomials to Clebsch-Gordan coefficients.

Extended the relation between Hahn polynomials and Clebsch-Gordan coefficients to multidimensional cases.

Abstract

We generalize a construction of Dunkl, obtaining a wide class intertwining functions on the symmetric group and a related family of multidimensional Hahn polynomials. Following a suggestion of Vilenkin and Klymik, we develop a tree-method approach for those intertwining functions. We also give a group theoretic proof of the relation between Hahn polynomials and Clebesh-Gordan coefficients, given analytically by Koornwinder and by Nikiforov, Smorodinski\u{i} and Suslov. Such relation is also extended to the multidimensional case.