Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Differential operators having Sobolev type Laguerre polynomials as eigenfunctions

Author(s): H. Bavinck
Journal: Proc. Amer. Math. Soc. 125 (1997), 3561-3567.
MSC (1991): Primary 33C45, 34A35
MathSciNet review: 1422848
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Abstract: We consider the polynomials $\left\{ L_n^{\alpha ,M}(x,l)\right\} _{n=0}^\infty$ orthogonal with respect to the Sobolev type inner product

$\begin{equation*}\left\langle p,q\right\rangle =\frac 1{\Gamma (\alpha +1)}\int _0^\infty p(x)q(x)x^\alpha e^{-x}dx+Mp^{(l)}(0)q^{(l)}(0), \end{equation*}$

where $\alpha >-1,M\geq 0$ and $l$ is a nonnegative integer. It is the purpose of this paper to show that these polynomials are eigenfunctions of a class of linear differential operators containing one that is of finite order $2\alpha +4l+4$ if $\alpha$ is a nonnegative integer and $M>0.$

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