# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## On differential equations for Sobolev-type Laguerre polynomialsHTML articles powered by AMS MathViewer

by J. Koekoek, R. Koekoek and H. Bavinck
Trans. Amer. Math. Soc. 350 (1998), 347-393 Request permission

## Abstract:

The Sobolev-type Laguerre polynomials $\{L_n^{\alpha ,M,N}(x)\}_{n=0}^{\infty }$ are orthogonal with respect to the inner product $\langle f,g\rangle \;=\frac {1}{\Gamma (\alpha +1)}\int _0^{\infty }x^{\alpha }e^{-x}f(x)g(x)dx+Mf(0)g(0)+ Nf’(0)g’(0),$ 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 $M\sum _{i=1}^{\infty }a_i(x)D^i+xD^2+(\alpha +1-x)D,$ 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)\\ &\hspace {1cm}{}+MN\sum _{i=0}^{\infty }c_i(x)y^{(i)}(x)+ xy''(x)+(\alpha +1-x)y’(x)+ny(x)=0, \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.
References
Similar Articles
• Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 33C45, 34A35
• Retrieve articles in all journals with MSC (1991): 33C45, 34A35
• J. Koekoek
• Affiliation: Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 5031, 2600 GA Delft, The Netherlands
• R. Koekoek
• Affiliation: Delft University of Technology, Faculty of Technical Mathematics and Informatics, P.O. Box 5031, 2600 GA 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, 2600 GA Delft, The Netherlands
• Email: h.bavinck@twi.tudelft.nl
• Received by editor(s): August 28, 1995
• Received by editor(s) in revised form: June 24, 1996