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Doubling (dual) Hahn polynomials: classification and applications. (English) Zbl 1333.81141
Summary: We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest comes from an example appearing in a finite oscillator model [E. I. Jafarov et al., J. Phys. A, Math. Theor. 44, No. 26, Article ID 265203, 15 p. (2011; Zbl 1220.81091)]. Our classification shows there exist three dual Hahn doubles and four Hahn doubles. The same technique is then applied to Racah polynomials, yielding also four doubles. Each dual Hahn (Hahn, Racah) double gives rise to an explicit new set of symmetric orthogonal polynomials related to the Christoffel and Geronimus transformations. For each case, we also have an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. This extends the class of Sylvester-Kac matrices by remarkable new test matrices. We examine also the algebraic relations underlying the dual Hahn doubles, and discuss their usefulness for the construction of new finite oscillator models.

##### MSC:
 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 22E70 Applications of Lie groups to the sciences; explicit representations 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 33C80 Connections of hypergeometric functions with groups and algebras, and related topics 81R05 Finite-dimensional groups and algebras motivated by physics and their representations 81Q65 Alternative quantum mechanics (including hidden variables, etc.)
testmatrix
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