Generalized Narayana polynomials
For $m, n \geq 0$, the generalized Narayana polynomial is
$$N_{n,m}(x) = \sum_{k=0}^{n} \frac{\binom{n}{k}\binom{n+m}{k}}{\binom{m+k}{k}}\, x^k.$$
For $m = 1$ the coefficients are Narayana numbers; for example $N_{2,1}(x) = 1 + 3x + x^2$. These polynomials are Pólya-frequency, and the Narayana transform $x^n \mapsto N_{n,m}(x)$ preserves Pólya-frequency polynomials.
References
The coefficient normalization is from Jianxi Mao and Lijie Wang, “The Narayana transformation,” arXiv:2607.01572 (2026), Eq. (1.2). The root-location input is D. Dominici, S. J. Johnston, and K. Jordaan, “Real zeros of 2F1 hypergeometric polynomials,” Journal of Computational and Applied Mathematics 247 (2013), 152–161, which is Lemma 2.5 in Mao–Wang. See also the Narayana real-rootedness examples on symmetricfunctions.com.