We prove that the polynomials generated by the relation $\displaystyle{\sum_{m=0}^{\infty} H_m(z)t^m=\frac{1}{P(t)+z t^r Q(t)}}$ are hyperbolic for $m \gg 1$ given that the zeros of the real polynomials $P$ and $Q$ are real and sufficiently separated. The paper also contains a result on a certain family of exponential polynomials, which are demonstrated to have infinitely many real zeros.

中文翻译：

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双曲多项式和线性型生成函数

我们证明 $\displaystyle{\sum_{m=0}^{\infty} H_m(z)t^m=\frac{1}{P(t)+zt^r Q(t )}}$ 对于 $m \gg 1$ 是双曲线的，因为实数多项式 $P$ 和 $Q$ 的零点是实数并且充分分离。该论文还包含某个指数多项式族的结果，这些多项式被证明具有无限多个实零。