Power-partible Reduction and Congruences for Schröder Polynomials

Abstract

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 \[ \sum_{k=0}^{p-1}(2k+1)^{2r+1}\varepsilon^k S_k(z)\equiv 1\pmod {p}\quad \text{and} \quad \sum_{k=0}^{p-1}(2k+1)^{2r+1}\varepsilon^k s_k(z)\equiv 0\pmod {p}. \]

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