Power-partible Reduction and Congruences for Schröder Polynomials

In this note, we apply the power-partible reduction to show the following arithmetic properties of large Schröder polynomials $S_n(z)$ and little Schröder polynomials $s_n(z)$: for any odd prime $p$, nonnegative integer $r\in \mathbb{N}$, $\epsilon \in \{-1,1\}$ and $z\in \mathbb{Z}$ with $\gcd (p,z(z+1))=1$, we have $$\sum _{k=0}^{p-1}(2k+1{)}^{2r+1}{\epsilon }^k{S}_k(z)\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}p)\text{and}\sum _{k=0}^{p-1}(2k+1{)}^{2r+1}{\epsilon }^k{s}_k(z)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}p)\mathrm{.}$$