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 \(S_n(z)\) and little Schröder polynomials \(s_n(z)\): for any odd prime p, nonnegative integer \(r\in {\mathbb {N}}\), \(\varepsilon \in \{-1,1\}\) and \(z\in {\mathbb {Z}}\) with \(\gcd (p,z(z+1))=1\), we have

$$\begin{aligned} \sum _{k=0}^{p-1}(2k+1)^{2r+1}\varepsilon ^k S_k(z)\equiv 1\pmod {p}\quad \text {and} \quad \sum _{k=0}^{p-1}(2k+1)^{2r+1}\varepsilon ^k s_k(z)\equiv 0\pmod {p}. \end{aligned}$$