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On a differential equation for Koornwinder's generalized Laguerre polynomials


Authors: J. Koekoek and R. Koekoek
Journal: Proc. Amer. Math. Soc. 112 (1991), 1045-1054
MSC: Primary 33C45; Secondary 42C05
MathSciNet review: 1047003
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Abstract: Koornwinder's generalized Laguerre polynomials $ \left\{ {L_n^{\alpha ,N}(x)} \right\}_{n = 0}^\infty $ are orthogonal on the interval $ [0,\infty )$ with respect to the weight function $ \frac{1}{{\Gamma (\alpha + 1)}}{x^\alpha }{e^{ - x}} + N\delta (x),\alpha > - 1,N \geq 0$. We show that these polynomials for $ N > 0$ satisfy a unique differential equation of the form

$\displaystyle N\sum\limits_{i = 0}^\infty {{a_i}(x){y^{(i)}}(x) + xy''(x) + (\alpha + 1 - x)y'(x) + ny(x)} = 0,$

where $ \left\{ {{a_i}(x)} \right\}_{i = 0}^\infty $ are continuous functions on the real line and $ \left\{ {{a_i}(x)} \right\}_{i = 1}^\infty $ are independent of the degree $ n$. If $ N > 0$, only in the case of nonnegative integer values of $ \alpha $ this differential equation is of finite order.

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DOI: https://doi.org/10.1090/S0002-9939-1991-1047003-9
Article copyright: © Copyright 1991 American Mathematical Society