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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Characterization of classical type orthogonal polynomials

Authors: K. H. Kwon, L. L. Littlejohn, J. K. Lee and B. H. Yoo
Journal: Proc. Amer. Math. Soc. 120 (1994), 485-493
MSC: Primary 33C45; Secondary 34A99
MathSciNet review: 1180465
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Abstract: We characterize the classical type orthogonal polynomials $ \{ {P_n}(x)\} _0^\infty $ satisfying a fourth-order differential equation of type

$\displaystyle \sum\limits_{i = 0}^4 {{\ell _i}(x){y^{(i)}}(x) = {\lambda _n}y(x)} $

where $ {\ell _i}(x)$ are polynomials of degree $ \leqslant i$ and $ {\lambda _n}$ is a constant. They are only the orthogonal polynomials satisfying an orthogonality of the form

$\displaystyle \langle {\tau _2},P_m^{''}P_n^{''}\rangle + \langle {\tau _1},P_m'P_n'\rangle + \langle {\tau _0},{P_m}{P_n}\rangle = 0\quad {\text{for}}\;m \ne n$

where $ {\tau _0},{\tau _1}$, and $ {\tau _2}$ are moment functionals.

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PII: S 0002-9939(1994)1180465-8
Article copyright: © Copyright 1994 American Mathematical Society