33,718 research outputs found
Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle
We derive raising and lowering operators for orthogonal polynomials on the
unit circle and find second order differential and -difference equations for
these polynomials. A general functional equation is found which allows one to
relate the zeros of the orthogonal polynomials to the stationary values of an
explicit quasi-energy and implies recurrences on the orthogonal polynomial
coefficients. We also evaluate the discriminants and quantized discriminants of
polynomials orthogonal on the unit circle.Comment: 27 pages, Latex2e plus AMS packages Fix to Eqs. (2.72) and (2.74
Spectral properties of operators using tridiagonalisation
A general scheme for tridiagonalising differential, difference or
q-difference operators using orthogonal polynomials is described. From the
tridiagonal form the spectral decomposition can be described in terms of the
orthogonality measure of generally different orthogonal polynomials. Three
examples are worked out: (1) related to Jacobi and Wilson polynomials for a
second order differential operator, (2) related to little q-Jacobi polynomials
and Askey-Wilson polynomials for a bounded second order q-difference operator,
(3) related to little q-Jacobi polynomials for an unbounded second order
q-difference operator. In case (1) a link with the Jacobi function transform is
established, for which we give a q-analogue using example (2).Comment: 14 pages, corrections, to appear in Analysis and Application
A difference-integral representation of Koornwinder polynomials
We construct new families of (q-) difference and (contour) integral operators
having nice actions on Koornwinder's multivariate orthogonal polynomials. We
further show that the Koornwinder polynomials can be constructed by suitable
sequences of these operators applied to the constant polynomial 1, giving the
difference-integral representation of the title. Macdonald's conjectures (as
proved by van Diejen and Sahi) for the principal specialization and norm follow
immediately, as does a Cauchy-type identity of Mimachi.Comment: 15 pages AMSLaTeX. To appear in proceedings of the Workshop on Jack,
Hall-Littlewood and Macdonald polynomials (September 2003, ICMS
On the Krall-type discrete polynomials
In this paper we present a unified theory for studying the so-called Krall-type discrete orthogonal polynomials. In particular, the three-term recurrence relation, lowering and raising operators as well as the second order linear difference equation that the sequences of monic orthogonal polynomials satisfy are established. Some relevant examples of q-Krall polynomials are considered in detail.http://www.sciencedirect.com/science/article/B6WK2-4CC2YF9-1/1/1bbcf94cc1184e679b497c3b8e754b2
Ladder Operators for q-orthogonal Polynomials
The q-difference analog of the classical ladder operators is derived for
those orthogonal polynomials arising from a class of indeterminate moments
problem.Comment: 15 pages, typos correcte
A difference-integral representation of Koornwinder polynomials
We construct new families of (q-) difference and (contour) integral operators having nice actions on Koornwinder's multivariate orthogonal polynomials. We further show that the Koornwinder polynomials can be constructed by suitable sequences of these operators applied to the constant polynomial 1, giving the difference-integral representation of the title. Macdonald's conjectures (as proved by van Diejen and Sahi) for the principal specialization and norm follow immediately, as does a Cauchy-type identity of Mimachi
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