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arXiv · 1002.3967

Polynomial Solutions of Differential Equations

H. Azad·M. T. Mustafa
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Abstract

We show that any differential operator of the form $L(y)=\sum_{k=0}^{k=N} a_{k}(x) y^{(k)}$, where $a_k$ is a real polynomial of degree $\leq k$, has all real eigenvalues in the space of polynomials of degree at most n, for all n. The eigenvalues are given by the coefficient of $x^n$ in $L(x^{n})$. If these eigenvalues are distinct, then there is a unique monic polynomial of degree n which is an eigenfunction of the operator L- for every non-negative integer n. As an application we recover Bochner's classification of second order ODEs with polynomial coefficients and polynomial solutions, as well as a family of non-classical polynomials.

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BibTeXRIS

H. Azad, M. T. Mustafa. 2010-02-28. Polynomial Solutions of Differential Equations. https://arxiv.org/abs/1002.3967

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