Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials

Pascal Baseilhac (1, 2) , A. M. Gainutdinov (1, 3, 4) , Thi Thao Vu (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 CRM - Centre de Recherches Mathématiques [Montréal]
3 DESY - Deutsches Elektronen-Synchrotron [Hamburg]
4 Fachbereich Mathematik [Hamburg]

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.

Mots clés: Tridiagonal pair; q−Onsager algebra; Tridiagonal algebra; Recurrence relations; Orthogonal polynomials; Leonard duality

Domaines: Algèbres quantiques [math.QA]; Physique mathématique [math-ph]; Analyse classique [math.CA]; Théorie des représentations [math.RT]

Pascal Baseilhac, A. M. Gainutdinov, Thi Thao Vu. Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials. Linear Algebra and its Applications, 2017, 522, pp.71-110. ⟨10.1016/j.laa.2017.02.009⟩. ⟨hal-01341852⟩