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This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.On Differential Equations for SobolevType Laguerre Polynomials
J. Koekoek, R. Koekoek and H. Bavinck
Transactions of the American Mathematical Society
Vol. 350, No. 1 (Jan., 1998), pp. 347393
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/117674
Page Count: 47
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Abstract
The Sobolevtype Laguerre polynomials {Ln α,M,N(x)}n=0 ∞ are orthogonal with respect to the inner product $\langle f,g\rangle =\frac{1}{\Gamma (\alpha +1)}\int_{0}^{\infty}x^{\alpha}e^{x}f(x)g(x)dx+Mf(0)g(0)+Nf^{\prime}(0)g^{ \prime}(0)$, where $\alpha >1,M\geq 0$ and N≥ 0. In 1990 the first and second author showed that in the case $M>0$ and N=0 the polynomials are eigenfunctions of a unique differential operator of the form $M\underset i=1\to{\overset \infty \to{\Sigma}}a_{i}(x)D^{i}+xD^{2}+(\alpha +1x)D$, where {ai(x)}i=1 ∞ are independent of n. This differential operator is of order 2α +4 if α is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form $M\underset i=0\to{\overset \infty \to{\Sigma}}a_{i}(x)y^{(i)}(x)+N\underset i=0\to{\overset \infty \to{\Sigma}}b_{i}(x)y^{(i)}(x)$ $+MN\underset i=0\to{\overset \infty \to{\Sigma}}c_{i}(x)y^{(i)}(x)+xy^{\prime \prime}(x)+(\alpha +1x)y^{\prime}(x)+ny(x)=0$, where the coefficients {ai(x)}i=1 ∞,{bi(x)}i=1 ∞ and {ci(x)}i=1 ∞ are independent of n and the coefficients a0(x),b0(x) and c0(x) are independent of x, satisfied by the Sobolevtype Laguerre polynomials {Ln α,M,N(x)}n=0 ∞. Further, we show that in the case M=0 and $N>0$ the polynomials are eigenfunctions of a linear differential operator, which is of order 2α +8 if α is a nonnegative integer and of infinite order otherwise. Finally, we show that in the case $M>0$ and $N>0$ the polynomials are eigenfunctions of a linear differential operator, which is of order 4α +10 if α is a nonnegative integer and of infinite order otherwise.
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Transactions of the American Mathematical Society © 1998 American Mathematical Society