282 research outputs found
The multivariate Hahn polynomials and the singular oscillator
Karlin and McGregor's d-variable Hahn polynomials are shown to arise in the
(d+1)-dimensional singular oscillator model as the overlap coefficients between
bases associated to the separation of variables in Cartesian and hyperspherical
coordinates. These polynomials in d discrete variables depend on d+1 real
parameters and are orthogonal with respect to the multidimensional
hypergeometric distribution. The focus is put on the d=2 case for which the
connection with the three-dimensional singular oscillator is used to derive the
main properties of the polynomials: forward/backward shift operators,
orthogonality relation, generating function, recurrence relations,
bispectrality (difference equations) and explicit expression in terms of the
univariate Hahn polynomials. The extension of these results to an arbitrary
number of variables is presented at the end of the paper.Comment: 34 p
The quantum superalgebra and a -generalization of the Bannai-Ito polynomials
The Racah problem for the quantum superalgebra is
considered. The intermediate Casimir operators are shown to realize a
-deformation of the Bannai-Ito algebra. The Racah coefficients of
are calculated explicitly in terms of basic orthogonal
polynomials that -generalize the Bannai-Ito polynomials. The relation
between these -deformed Bannai-Ito polynomials and the
-Racah/Askey-Wilson polynomials is discussed.Comment: 15 page
An algebraic interpretation of the multivariate -Krawtchouk polynomials
The multivariate quantum -Krawtchouk polynomials are shown to arise as
matrix elements of "-rotations" acting on the state vectors of many
-oscillators. The focus is put on the two-variable case. The algebraic
interpretation is used to derive the main properties of the polynomials:
orthogonality, duality, structure relations, difference equations and
recurrence relations. The extension to an arbitrary number of variables is
presentedComment: 22 pages; minor correction
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