Exceptional orthogonal polynomials are complete families of orthogonal polynomials that arise as eigenfunctions of a Sturm–Liouville problem. Antonio Durán discovered a gap in the original proof of completeness for exceptional Hermite polynomials, that has propagated to analogous results for other exceptional families. In this paper we provide an alternative proof that follows essentially the same arguments, but provides a direct proof of the key lemma on which the completeness proof is based. This direct proof makes use of the theory of trivial monodromy potentials developed by Duistermaat and Grünbaum and Oblomkov.

中文翻译：

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关于例外Hermite多项式的完整性证明的勘误

出色的正交多项式是作为Sturm-Liouville问题的本征函数出现的正交多项式的完整族。安东尼奥·杜兰（AntonioDurán）在例外Hermite多项式的原始完整性证明中发现了一个空白，这一空白已经传播到其他例外族的相似结果。在本文中，我们提供了一个替代的证明，该证明遵循基本相同的论点，但提供了完整性证明所基于的关键引理的直接证明。这种直接证明利用了Duistermaat和Grünbaum和Oblomkov提出的琐碎单峰势理论。