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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 \(2\alpha+4\).

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