26,996 research outputs found
Confluent hypergeometric orthogonal polynomials related to the rational quantum Calogero system with harmonic confinement
Two families (type and type ) of confluent hypergeometric polynomials
in several variables are studied. We describe the orthogonality properties,
differential equations, and Pieri type recurrence formulas for these families.
In the one-variable case, the polynomials in question reduce to the Hermite
polynomials (type ) and the Laguerre polynomials (type ), respectively.
The multivariable confluent hypergeometric families considered here may be used
to diagonalize the rational quantum Calogero models with harmonic confinement
(for the classical root systems) and are closely connected to the (symmetric)
generalized spherical harmonics investigated by Dunkl.Comment: AMS-LaTeX v1.2 (with amssymb.sty), 34 page
Properties of some families of hypergeometric orthogonal polynomials in several variables
Limiting cases are studied of the Koornwinder-Macdonald multivariable
generalization of the Askey-Wilson polynomials. We recover recently and not so
recently introduced families of hypergeometric orthogonal polynomials in
several variables consisting of multivariable Wilson, continuous Hahn and
Jacobi type polynomials, respectively. For each class of polynomials we provide
systems of difference (or differential) equations, recurrence relations, and
expressions for the norms of the polynomials in terms of the norm of the
constant polynomial.Comment: 42 pages, AMSLaTeX 1.1 with amssym
Some examples of matrix-valued orthogonal functions having a differential and an integral operator as eigenfunctions
The aim of this paper is to show some examples of matrix-valued orthogonal
functions on the real line which are simultaneously eigenfunctions of a
second-order differential operator of Schr\"{o}dinger type and an integral
operator of Fourier type. As a consequence we derive integral representations
of these functions as well as other useful structural formulas. Some of these
functions are plotted to show the relationship with the Hermite or wave
functions
Using \D-operators to construct orthogonal polynomials satisfying higher order difference or differential equations
We introduce the concept of \D-operators associated to a sequence of
polynomials and an algebra \A of operators acting in the linear
space of polynomials. In this paper, we show that this concept is a powerful
tool to generate families of orthogonal polynomials which are eigenfunctions of
a higher order difference or differential operator. Indeed, given a classical
discrete family of orthogonal polynomials (Charlier, Meixner,
Krawtchouk or Hahn), we form a new sequence of polynomials by
considering a linear combination of two consecutive :
, \beta_n\in \RR. Using the concept of \D-operator,
we determine the structure of the sequence in order that the
polynomials are common eigenfunctions of a higher order difference
operator. In addition, we generate sequences for which the
polynomials are also orthogonal with respect to a measure. The same
approach is applied to the classical families of Laguerre and Jacobi
polynomials.Comment: 43 page
A Bochner Theorem for Dunkl Polynomials
We establish an analogue of the Bochner theorem for first order operators of
Dunkl type, that is we classify all such operators having polynomial solutions.
Under natural conditions it is seen that the only families of orthogonal
polynomials in this category are limits of little and big -Jacobi
polynomials as
Kundt spacetimes as solutions of topologically massive gravity
We obtain new solutions of topologically massive gravity. We find the general
Kundt solutions, which in three dimensions are spacetimes admitting an
expansion-free null geodesic congruence. The solutions are generically of
algebraic type II, but special cases are types III, N or D. Those of type D are
the known spacelike-squashed AdS_3 solutions, and of type N are the known AdS
pp-waves or new solutions. Those of types II and III are the first known
solutions of these algebraic types. We present explicitly the Kundt solutions
that are CSI spacetimes, for which all scalar polynomial curvature invariants
are constant, whereas for the general case we reduce the field equations to a
series of ordinary differential equations. The CSI solutions of types II and
III are deformations of spacelike-squashed AdS_3 and the round AdS_3,
respectively.Comment: 30 pages. This material has come from splitting v1 of arXiv:0906.3559
into 2 separate papers. v2: minor changes
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