Doubling (Dual) Hahn Polynomials: Classification and Applications

TL;DR

This paper classifies pairs of recurrence relations involving Hahn, dual Hahn, and Racah polynomials, revealing new symmetric orthogonal polynomials, explicit matrices, and potential applications in finite oscillator models.

Contribution

It provides a complete classification of dual Hahn and Hahn doubles, introduces new symmetric orthogonal polynomials, and extends Sylvester-Kac matrices with applications to oscillator models.

Findings

Identified three dual Hahn doubles and four Hahn doubles.

Discovered four doubles for Racah polynomials.

Constructed new test matrices with explicit eigenvalues.

Abstract

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 [Jafarov E.I., Stoilova N.I., Van der Jeugt J., J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310]. 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.