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On a differential equation for Koornwinder’s generalized Laguerre polynomials. (English) Zbl 0737.33003
The aim of the authors is to establish a differential equation of the form $N\sum_{i=0}^ \infty a_ i(x)y^{(i)}(x)+xy''(x)+(\alpha+1- x)y'(x)+ny(x)=0, \qquad \alpha>-1, \quad N\geq 0 (*)$ to be satisfied by the generalized Laguerre polynomials $$L_ n^{\alpha,N}(x)$$ ($$n=0,1,2,\dots$$). In (*) $$N$$ stands for a given constant while $$a_ i(x)$$ are continuous functions to be determined. As to the polynomials $$L_ n^{\alpha,N}(x)$$, they are orthogonal on the interval $$[0,\infty)$$ with respect to the weight function $$x^ \alpha e^{-x}/\Gamma(\alpha+1)+N\delta(x)$$. By assuming that the functions $$a_ i(x)$$ $$(i\geq 1)$$ are independent of $$n$$ they show that the degree of $$a_ 1(x)$$ is equal to $$i$$ $$(i=1,2,\dots)$$ when $$N>0$$ and $$\alpha\neq 0,1,2,\dots$$. This implies that the differential equation (*) is of infinite order. Conversely, for nonnegative integer values of $$\alpha$$ and $$N>0$$ one has $$a_ 1(x)\equiv 0$$ when $$i>2\alpha+4$$, which implies that the differential equation (*) is of order $$2\alpha+4$$.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
##### Keywords:
Laguerre polynomials
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