Binomial coefficients, Catalan numbers and Lucas quotients

Abstract

Let $p$ be an odd prime and let $a,m$ be integers with $a>0$ and $m\not\equiv 0(\mathrm{mod}p)$. In this paper we determine ${\sum }_{k=0}^{p^a-1}\left(\genfrac{}{}{0ex}{}{2k}{k+d}\right)/m^k$ mod $p^2$ for $d=0,1$; for example, $$\sum _{k=0}^{p^a-1}\frac{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{m^k}\equiv \left(\frac{m^2-4m}{p^a}\right)+\left(\frac{m^2-4m}{p^{a-1}}\right)u_{p-(\frac{m^2-4m}{p})}(\mathrm{mod}{p}^2),$$ where $(-)$ is the Jacobi symbol, and $\{u_n{\}}_{n⩾0}$ is the Lucas sequence given by $u_0=0$, $u_1=1$ and $u_{n+1}=(m-2)u_n-u_{n-1}$ for $n=1,2,3,\dots$. As an application, we determine ${\sum }_{0<k<p^a,k\equiv r(\mathrm{mod}p-1)}{C}_k$ modulo $p^2$ for any integer $r$, where $C_k$ denotes the Catalan number $\left(\genfrac{}{}{0ex}{}{2k}{k}\right)/(k+1)$. We also pose some related conjectures.

24 pages. Correct few typos

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