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Zbl 0886.33006
Koekoek, J.; Koekoek, R.; Bavinck, H.
On differential equations for Sobolev-type Laguerre polynomials. (English)
[J] Trans. Am. Math. Soc. 350, No.1, 347-393 (1998). ISSN 0002-9947; ISSN 1088-6850/e

Summary: 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 $$ \multline M\sum_{i=0}^{\infty}a_i(x)y^{(i)}(x)+ N\sum_{i=0}^{\infty}b_i(x)y^{(i)}(x) \\ +MN\sum_{i=0}^{\infty}c_i(x)y^{(i)}(x)+ xy''(x)+(\alpha +1-x)y'(x)+ny(x)=0,\endmultline $$ 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.

MSC 2000:: *33C45 Orthogonal polynomials and functions of hypergeometric type
34A35 ODE of infinite order

Keywords: differential equations; Sobolev-type Laguerre polynomials

Cited in: Zbl 0931.33007

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Zentralblatt MATH Berlin [Germany]