Publication: Generalized Delta coherent pairs
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2004
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Korean Mathematical Society
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Abstract
A pair of quasi-definite linear functionals ${u_0,u_1}$ is a generalized $Delta$-coherent pair if monic orthogonal polynomials $${P_n(x)}_{n=0} nfty$$ and $${R_n(x)}_{n=0} nfty$$ 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.
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.
Description
18 pages, no figures.-- MSC2000 codes: 42C05, 33C45.
MR#: MR2095548 (2005k:33007)
Zbl#: Zbl 1058.42018
MR#: MR2095548 (2005k:33007)
Zbl#: Zbl 1058.42018
Keywords
Discrete orthogonal polynomials, Delta-coherent pairs
Bibliographic citation
Journal of the Korean Mathematical Society, 2004, vol. 41, n. 6, p. 977-994