Differential operators having Sobolev type Laguerre polynomials as eigenfunctions

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 $M>0.$