We consider the period functions associated to the Eisenstein series for the Fricke group ${\mathrm{\Gamma }}_0^{+}(N)$, the odd parts of the period functions and certain polynomials obtained from the period functions for ${\mathrm{\Gamma }}_0^{+}(N)$, and we prove that all zeros of each of them lie on the circle $|z|=\frac{1}{\sqrt{N}}$ by applying the properties of a self-inversive polynomial. In particular, our result proves Berndt and Straub's suggested problem.

中文翻译：

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关于与 Γ0+(N) 的 Eisenstein 级数相关的周期函数的零点

我们考虑与 Fricke 群的 Eisenstein 级数相关的周期函数${\mathrm{\Gamma }}_0^{+}(ñ)$，周期函数的奇数部分和从周期函数获得的某些多项式${\mathrm{\Gamma }}_0^{+}(ñ)$, 我们证明了它们中的每一个的所有零都在圆上$|z|=\frac{1}{\sqrt{ñ}}$通过应用自反多项式的性质。特别是，我们的结果证明了 Berndt 和 Straub 提出的问题。