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On Differential Equations for Sobolev-Type Laguerre Polynomials

J. Koekoek, R. Koekoek and H. Bavinck
Transactions of the American Mathematical Society
Vol. 350, No. 1 (Jan., 1998), pp. 347-393
Stable URL: http://www.jstor.org/stable/117674
Page Count: 47
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On Differential Equations for Sobolev-Type Laguerre Polynomials
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Abstract

The Sobolev-type 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 $M\underset i=1\to{\overset \infty \to{\Sigma}}a_{i}(x)D^{i}+xD^{2}+(\alpha +1-x)D$, 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 $M\underset i=0\to{\overset \infty \to{\Sigma}}a_{i}(x)y^{(i)}(x)+N\underset i=0\to{\overset \infty \to{\Sigma}}b_{i}(x)y^{(i)}(x)$ $+MN\underset i=0\to{\overset \infty \to{\Sigma}}c_{i}(x)y^{(i)}(x)+xy^{\prime \prime}(x)+(\alpha +1-x)y^{\prime}(x)+ny(x)=0$, 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 Sobolev-type 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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