Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials

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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Pascal Baseilhac, A. M. Gainutdinov, Thi Thao Vu. Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials. Linear Algebra and its Applications, Elsevier, In press, ⟨10.1016/j.laa.2017.02.009⟩. ⟨hal-01341852⟩

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