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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On differential equations for Sobolev-type Laguerre polynomials


Authors: J. Koekoek, R. Koekoek and H. Bavinck
Journal: Trans. Amer. Math. Soc. 350 (1998), 347-393
MSC (1991): Primary 33C45; Secondary 34A35
MathSciNet review: 1433121
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Abstract: The Sobolev-type Laguerre polynomials $\{L_n^{\alpha,M,N}(x)\}_{n=0}^{\infty}$ are orthogonal with respect to the inner product

\begin{displaymath}\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'(0)g'(0),\end{displaymath}

where $\alpha>-1$, $M\ge 0$ and $N\ge 0$. 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

\begin{displaymath}M\sum _{i=1}^{\infty}a_i(x)D^i+xD^2+(\alpha+1-x)D,\end{displaymath}

where $\left\{a_i(x)\right\}_{i=1}^{\infty}$ are independent of $n$. This differential operator is of order $2\alpha+4$ if $\alpha$ is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form

\begin{align}&M\sum _{i=0}^{\infty}a_i(x)y^{(i)}(x)+ N\sum _{i=0}^{\infty}b_i(x)y^{(i)}(x)\nonumber \\ &\hspace{1cm}{}+MN\sum _{i=0}^{\infty}c_i(x)y^{(i)}(x)+ xy''(x)+(\alpha +1-x)y'(x)+ny(x)=0,\nonumber \end{align}

where the coefficients $\left\{a_i(x)\right\}_{i=1}^{\infty}$, $\left\{b_i(x)\right\}_{i=1}^{\infty}$ and $\left\{c_i(x)\right\}_{i=1}^{\infty}$ are independent of $n$ and the coefficients $a_0(x)$, $b_0(x)$ and $c_0(x)$ are independent of $x$, satisfied by the Sobolev-type Laguerre polynomials $\{L_n^{\alpha,M,N}(x)\}_{n=0}^{\infty}$. 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 $2\alpha+8$ if $\alpha$ 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 $4\alpha+10$ if $\alpha$ is a nonnegative integer and of infinite order otherwise.


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Additional Information

J. Koekoek
Affiliation: Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 5031, 2600GA Delft, The Netherlands

R. Koekoek
Affiliation: Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 5031, 2600GA Delft, The Netherlands
Email: r.koekoek@twi.tudelft.nl

H. Bavinck
Affiliation: Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 5031, 2600GA Delft, The Netherlands
Email: h.bavinck@twi.tudelft.nl

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01993-X
PII: S 0002-9947(98)01993-X
Keywords: Differential equations, Sobolev-type Laguerre polynomials
Received by editor(s): August 28, 1995
Received by editor(s) in revised form: June 24, 1996
Article copyright: © Copyright 1998 American Mathematical Society