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Suppose that $x_{0}$ is a zero of the associated Legendre function $P_{n}^{m}(x)$ (the degree $n$ is a positive integer while the order $m$ is an integer in the range from $0$ to $n$). If there exist integers $n' > n$ and $m' \leq n'$ such that $x_{0}$ is a zero of $P_{n'}^{m'}(x)$, does it follow that $x_{0}=0$? In other words, is zero the only common root of the family $\{ P_{n}^{m}(x) \}$?

In a 2005 "Note on Common Zeros of Laplace-Beltrami Eigenfunctions" (found on arXiv) Ginchev suggests that this is still an open problem. I am wondering if anyone is familiar with the recent progress in the field.

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