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arXiv · 1607.00606

Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials

P. Baseilhac·A. M. Gainutdinov·T. T. Vu
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Abstract

The concept of cyclic tridiagonal pairs is introduced, and explicit examples are given. For a fairly general class of cyclic tridiagonal pairs with cyclicity N, we associate a pair of `divided polynomials'. The properties of this pair generalize the ones of tridiagonal pairs of Racah type. The algebra generated by the pair of divided polynomials is identified as a higher-order generalization of the Onsager algebra. It can be viewed as a subalgebra of the q-Onsager algebra for a proper specialization at q the primitive 2Nth root of unity. Orthogonal polynomials beyond the Leonard duality are revisited in light of this framework. In particular, certain second-order Dunkl shift operators provide a realization of the divided polynomials at N=2 or q=i.

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BibTeXRIS

P. Baseilhac, A. M. Gainutdinov, T. T. Vu. 2016-07-03. Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials. https://arxiv.org/abs/1607.00606

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