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On a Differential Equation for Koornwinder's Generalized Laguerre Polynomials

J. Koekoek and R. Koekoek
Proceedings of the American Mathematical Society
Vol. 112, No. 4 (Aug., 1991), pp. 1045-1054
DOI: 10.2307/2048653
Stable URL: http://www.jstor.org/stable/2048653
Page Count: 10
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On a Differential Equation for Koornwinder's Generalized Laguerre Polynomials
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Abstract

Koornwinder's generalized Laguerre polynomials $\{L^{\alpha, N}_n (x) \}^\infty_{n = 0}$ are orthogonal on the interval [ 0, ∞) with respect to the weight function $\frac{1}{\Gamma(\alpha + 1)} x^\alpha e^{-x} + N \delta (x), \alpha > -1, N \geq 0$. We show that these polynomials for $N > 0$ satisfy a unique differential equation of the form N ∑i = 0 ∞ ai(x)y(i) (x) + xy''(x) + (α + 1 - x)y'(x) + ny(x) = 0, where $\{a_i(x) \}^\infty_{i = 0}$ are continuous functions on the real line and $\{a_i(x) \}^\infty_{i = 1}$ are independent of the degree n. If $N > 0$, only in the case of nonnegative integer values of α this differential equation is of finite order.

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