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A unified algebraic underpinning for the Hahn polynomials and rational functions. (English) Zbl 1459.33010

The authors introduce some algebra \(\mathfrak{mh}\) with 3 generators in which the so-called Hahn and rational Hahn algebra can be embedded; these Hahn algebras were introduced by Granovsky, Lutzenko, Tsujimoto, and the authors in several papers. The authors present representation bases of \(\mathfrak{mh}\) associated (generalized) eigenvalues of operators which are related to the 3 generators of \(\mathfrak{mh}\). This leads to connections with bispectral orthogonal polynomials and rational functions. Moreover, suitable differential and difference operators show that the underlying special functions are in fact Hahn polynomials. Furthermore, an embedding of \(\mathfrak{mh}\) into \(\mathfrak{U(sl_2)}\) is given.

MSC:

33C80 Connections of hypergeometric functions with groups and algebras, and related topics
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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