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On differential equations for Sobolev-type Laguerre polynomials. (English) Zbl 0886.33006

Summary: The Sobolev-type Laguerre polynomials ${\left\{{L}_{n}^{\alpha ,M,N}\left(x\right)\right\}}_{n=0}^{\infty }$ are orthogonal with respect to the inner product

$〈f,g〉=\frac{1}{{\Gamma }\left(\alpha +1\right)}{\int }_{0}^{\infty }{x}^{\alpha }{e}^{-x}f\left(x\right)g\left(x\right)dx+Mf\left(0\right)g\left(0\right)+N{f}^{\text{'}}\left(0\right){g}^{\text{'}}\left(0\right),$

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

$M\sum _{i=1}^{\infty }{a}_{i}\left(x\right){D}^{i}+x{D}^{2}+\left(\alpha +1-x\right)D,$

where ${\left\{{a}_{i}\left(x\right)\right\}}_{i=1}^{\infty }$ 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

$\begin{array}{c}M\sum _{i=0}^{\infty }{a}_{i}\left(x\right){y}^{\left(i\right)}\left(x\right)+N\sum _{i=0}^{\infty }{b}_{i}\left(x\right){y}^{\left(i\right)}\left(x\right)\hfill \\ \hfill +MN\sum _{i=0}^{\infty }{c}_{i}\left(x\right){y}^{\left(i\right)}\left(x\right)+x{y}^{\text{'}\text{'}}\left(x\right)+\left(\alpha +1-x\right){y}^{\text{'}}\left(x\right)+ny\left(x\right)=0,\end{array}$

where the coefficients ${\left\{{a}_{i}\left(x\right)\right\}}_{i=1}^{\infty }$, ${\left\{{b}_{i}\left(x\right)\right\}}_{i=1}^{\infty }$ and ${\left\{{c}_{i}\left(x\right)\right\}}_{i=1}^{\infty }$ are independent of $n$ and the coefficients ${a}_{0}\left(x\right)$, ${b}_{0}\left(x\right)$ and ${c}_{0}\left(x\right)$ are independent of $x$, satisfied by the Sobolev-type Laguerre polynomials ${\left\{{L}_{n}^{\alpha ,M,N}\left(x\right)\right\}}_{n=0}^{\infty }$. 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.

MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 34A35 ODE of infinite order