On differential equations for Sobolev-type Laguerre polynomials

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

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

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

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.