Differential operators having Sobolev type Laguerre polynomials as eigenfunctions
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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 $M>0.$References
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Additional Information
- H. Bavinck
- Affiliation: Delft University of Technology, Faculty of Technical Mathematics and Informatics, Mekelweg 4, 2628 CD Delft, The Netherlands
- Email: bavinck@twi.tudelft.nl
- Received by editor(s): June 27, 1996
- Communicated by: Christopher D. Sogge
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 3561-3567
- MSC (1991): Primary 33C45, 34A35
- DOI: https://doi.org/10.1090/S0002-9939-97-04091-4
- MathSciNet review: 1422848