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.
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