Some congruences for Schröder type polynomials

Ji-Cai Liu¹

¹ Department of Mathematics Shanghai Key Laboratory of PMMP East China Normal University 500 Dongchuan Road Shanghai 200241, People’s Republic of China

Colloquium Mathematicum, Tome 146 (2017) no. 2, pp. 187-195.

Résumé

The $n$th Schröder number is given by $S_n=\sum_{k=0}^{n}{n\choose k}{n+k\choose k}\frac{1}{k+1}.$ Motivated by these numbers, for any positive integer $\alpha$ we introduce the polynomials \begin{equation*} S_n^{(\alpha)}(x)=\sum_{k=0}^{n}\left({n\atop k}\right)^{\alpha}\left({n+k\atop k}\right)^{\alpha}\frac{x^k}{(k+1)^{\alpha}}. \end{equation*} We prove that for any positive integers $r$, $\alpha$, odd prime $p$ and any integer $m$ not divisible by $p$, and for $\varepsilon=\pm 1$, \begin{align*} \sum_{k=1}^{p-1}{\varepsilon}^k(2k+1)S_k^{(2\alpha-1)}(m)^r\equiv 0 \pmod{p},\\ \sum_{k=1}^{p-1}{\varepsilon}^k(2k+1)S_k^{(2\alpha)}(m)^r\equiv -2^r \pmod{p}. \end{align*}

Détail

DOI : 10.4064/cm7004-8-2016

Keywords: nth schr der number given sum choose choose frac motivated these numbers positive integer alpha introduce polynomials begin equation* alpha sum atop right alpha atop right alpha frac alpha end equation* prove positive integers alpha odd prime integer divisible varepsilon begin align* sum p varepsilon alpha equiv pmod sum p varepsilon alpha equiv pmod end align*