Journal of the Korean Mathematical Society
( Vol.41 NO.6 / 2004 )
Title
Generalized $\Delta$-coherent pairs(eng)
Author
K. H. Kwon ,J. H. Lee ,F. Marcell\'an 
MSC
33C45
Publication
Page
977-994 Page
Abstract
1. Introduction
2. Preliminaries
3. Generalized $\Delta_w$-coherency
4. Generalized $\Delta$-coherent pairs
Own Status
Keyword
discrete orthogonal polynomials, $\Delta$-coherent pairs
Note
Summary
A pair of quasi-definite linear functionals ${u_0,u_1}$
is a generalized $Delta$-coherent pair if monic orthogonal
polynomials $${P_n(x)}_{n=0}^infty$$ and
$${R_n(x)}_{n=0}^infty$$ relative to $u_0$ and $u_1$,
respectively, satisfy a relation
$$ R_n(x) = frac{1}{n+1}Delta P_{n+1}(x)-frac{sigma_n}{n}Delta P_n(x)-
frac{ au_{n-1}}{n-1}Delta P_{n-1}(x), ~~ ngeq 2,$$ where
$sigma_n$ and $ au_n$ are arbitrary constants and $Delta
p=p(x+1)-p(x)$ is the difference operator.

We show that if ${u_0,u_1}$ is a generalized $Delta$-coherent
pair, then $u_0$ and $u_1$ must be discrete-semiclassical linear
functionals. We also find conditions under which either $u_0$ or
$u_1$ is discrete-classical.
Attach
DVI  PDF