437 research outputs found
Equilibrium Positions, Shape Invariance and Askey-Wilson Polynomials
We show that the equilibrium positions of the Ruijsenaars-Schneider-van
Diejen systems with the trigonometric potential are given by the zeros of the
Askey-Wilson polynomials with five parameters. The corresponding single
particle quantum version, which is a typical example of "discrete" quantum
mechanical systems with a q-shift type kinetic term, is shape invariant and the
eigenfunctions are the Askey-Wilson polynomials. This is an extension of our
previous study [1,2], which established the "discrete analogue" of the
well-known fact; The equilibrium positions of the Calogero systems are
described by the Hermite and Laguerre polynomials, whereas the corresponding
single particle quantum versions are shape invariant and the eigenfunctions are
the Hermite and Laguerre polynomials.Comment: 14 pages, 1 figure. The outline of derivation of the result in
section 2 is adde
A model for the continuous q-ultraspherical polynomials
We provide an algebraic interpretation for two classes of continuous
-polynomials. Rogers' continuous -Hermite polynomials and continuous
-ultraspherical polynomials are shown to realize, respectively, bases for
representation spaces of the -Heisenberg algebra and a -deformation of
the Euclidean algebra in these dimensions. A generating function for the
continuous -Hermite polynomials and a -analog of the Fourier-Gegenbauer
expansion are naturally obtained from these models
Eigenvalue Integro-Differential Equations for Orthogonal Polynomials on the Real Line
The one-dimensional harmonic oscillator wave functions are solutions to a
Sturm-Liouville problem posed on the whole real line. This problem generates
the Hermite polynomials. However, no other set of orthogonal polynomials can be
obtained from a Sturm-Liouville problem on the whole real line. In this paper
we show how to characterize an arbitrary set of polynomials orthogonal on
in terms of a system of integro-differential equations of
Hartree-Fock type. This system replaces and generalizes the linear differential
equation associated with a Sturm-Liouville problem. We demonstrate our results
for the special case of Hahn-Meixner polynomials.Comment: 28 pages, Latex, U. Texas at Austin/ Washington University preprin
The q-harmonic oscillator and an analog of the Charlier polynomials
A model of a q-harmonic oscillator based on q-Charlier polynomials of
Al-Salam and Carlitz is discussed. Simple explicit realization of q-creation
and q-annihilation operators, q-coherent states and an analog of the Fourier
transformation are found. A connection of the kernel of this transform with
biorthogonal rational functions is observed
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