The Narayana transformation

For $m\in {\mathbb{Z}}_{\ge 0}$, let $$N_{n,m}(x)={}_2{F}_1(-n,-n-m;m+1;x),$$ which specializes to the Narayana polynomials of types $B$ and $A$ for $m=0$ and $m=1$, respectively. We prove that the associated basis transformation $$T_{N_m}\left(\sum _{k=0}^n{a}_k{x}^k\right)=\sum _{k=0}^n{a}_k{N}_{k,m}(x)$$ maps every real-rooted polynomial with nonnegative coefficients to a real-rooted polynomial. The proof is based on the rectangular additive convolution of polynomials. We then apply this result to products of lower triangular matrices and obtain a general criterion ensuring that their row generating functions remain real-rooted. As consequences, we recover this property for powers and products of several classical triangular matrices, including Pascal’s triangle, the Stirling triangles, and the Narayana triangles of types $A$ and $B$. We conclude with conjectures concerning the squares of the Eulerian and Delannoy triangles.