J. Korean Math. Soc. 1997; 34(4): 973-1008
Printed December 1, 1997
Copyright © The Korean Mathematical Society.
Kil H. Kwon and Lance L. Littlejohn
KAIST and Utah State University
We reconsider the problem of classifying all classical orthogonal polynomial sequences which are solutions to a second-order differential equation of the form $$\ell_2(x)y''(x)+\ell_1(x)y'(x)=\lambda_ny(x).$$ We first obtain new (algebraic) necessary and sufficient conditions on the coefficients $\ell_1(x)$ and $\ell_2(x)$ for the above differential equation to have orthogonal polynomial solutions. Using this result, we then obtain a complete classification of all classical orthogonal polynomials~: up to a $\italic real$ linear change of variable, there are the six distinct orthogonal polynomial sets of Jacobi, Bessel, Laguerre, Hermite, twisted Hermite, and twisted Jacobi.
Keywords: classical orthogonal polynomials, second-order differential equations
MSC numbers: Primary 33A65
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