On differential equations for Sobolev-type Laguerre polynomials

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.