Power-partible Reduction and Congruences for Schr\"oder Polynomials

TL;DR

This paper uses power-partible reduction to establish new arithmetic properties and congruences for large and small Schr"oder polynomials modulo primes, revealing their behavior in number theory.

Contribution

It introduces a novel application of power-partible reduction to derive congruences for Schr"oder polynomials, expanding their known arithmetic properties.

Findings

For any odd prime p, specific sums involving Schr"oder polynomials are congruent to 1 or 0 mod p.

The paper establishes congruences for sums of Schr"oder polynomials weighted by powers and signs.

Results hold under conditions on gcd and polynomial parameters.

Abstract

In this note, we apply the power-partible reduction to show the following arithmetic properties of large Schr\"oder polynomials $S_n(z)$ and little Schr\"oder 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 \[ \sum_{k=0}^{p-1}(2k+1)^{2r+1}\varepsilon^k S_k(z)\equiv 1\pmod {p}\quad \text{and} \quad \sum_{k=0}^{p-1}(2k+1)^{2r+1}\varepsilon^k s_k(z)\equiv 0\pmod {p}. \]