Power-partible reduction and congruences for Schröder polynomials

In this paper, we apply the power-partible reduction to show the following arithmetic properties of large Schröder polynomials \(S_n(z)\) $S_n(z)$ and little Schröder polynomials \(s_n(z)\) $s_n(z)$ : for any odd prime p, nonnegative integer \(r\in {\mathbb {N}}\) $r\in \mathbb{N}$ , \(\varepsilon \in \{-1,1\}\) $\epsilon \in \{-1,1\}$ and \(z\in {\mathbb {Z}}\) $z\in \mathbb{Z}$ with \(\gcd (p,z(z+1))=1\) $gcd(p,z(z+1))=1$ , we have \(\begin{aligned} \sum _{k=0}^{p-1}(2k+1)^{2r+1}\varepsilon ^k S_k(z)\equiv 1\pmod {p}\quad \text {and} \quad \sum _{k=0}^{p-1}(2k+1)^{2r+1}\varepsilon ^k s_k(z)\equiv 0\pmod {p}. \end{aligned}\) $$\begin{array}{c}\hfill \sum _{k=0}^{p-1}{(2k+1)}^{2r+1}{\epsilon }^k{S}_k(z)\equiv 1(modp)\text{and}\sum _{k=0}^{p-1}{(2k+1)}^{2r+1}{\epsilon }^k{s}_k(z)\equiv 0(modp).\end{array}$$