Doubling (Dual) Hahn Polynomials: Classification and Applications
Copyright policy )Doubling (Dual) Hahn Polynomials: Classification and Applications
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.
- Ghent University Belgium
OSCILLATOR, Racah polynomial, LEONARD PAIRS, Christoffel pair, FOS: Physical sciences, Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Applications of Lie groups to the sciences; explicit representations, Classical Analysis and ODEs (math.CA), FOS: Mathematics, LIE, symmetric orthogonal polynomials, tridiagonal matrix, matrix eigenvalues, Mathematical Physics, 81Qxx, 33C45, Connections of hypergeometric functions with groups and algebras, and related topics, Mathematical Physics (math-ph), Finite-dimensional groups and algebras motivated by physics and their representations, Mathematics and Statistics, Hahn polynomial, Mathematics - Classical Analysis and ODEs, Alternative quantum mechanics (including hidden variables, etc.), Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, finite oscillator model, MATRICES
OSCILLATOR, Racah polynomial, LEONARD PAIRS, Christoffel pair, FOS: Physical sciences, Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Applications of Lie groups to the sciences; explicit representations, Classical Analysis and ODEs (math.CA), FOS: Mathematics, LIE, symmetric orthogonal polynomials, tridiagonal matrix, matrix eigenvalues, Mathematical Physics, 81Qxx, 33C45, Connections of hypergeometric functions with groups and algebras, and related topics, Mathematical Physics (math-ph), Finite-dimensional groups and algebras motivated by physics and their representations, Mathematics and Statistics, Hahn polynomial, Mathematics - Classical Analysis and ODEs, Alternative quantum mechanics (including hidden variables, etc.), Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics, finite oscillator model, MATRICES
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