Binomial coefficients, Catalan numbers and Lucas quotients

Let $p$ be an odd prime and let $a,m$ be integers with $a>0$ and $m \not\equiv0\pmod p$\mathrm{.}Inthispaperwedetermine$\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$mod$p^2$for$d=0,1$;forexample,$${\sum }_{k=0}^{p^a-1}\frac{\left(\genfrac{}{}{0px}{}{2k}{k}\right)}{m^k}\equiv \left(\frac{m^2-4m}{p^a}\right)+\left(\frac{m^2-4m}{p^{a-1}}\right)u_{p-(\frac{m^2-4m}{p})}(\mathrm{m}\mathrm{o}\mathrm{d}{p}^2),$$where$(-)$istheJacobisymbol,and${u_n}{n\geqslant0}$ 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 $\sum{0<k<p^a,, k\equiv r\pmod{p-1}}C_k$modulo$p^2$foranyinteger$r$,where$C_k$ denotes the Catalan number $\left(\genfrac{}{}{0px}{}{2k}{k}\right)\mathrm{/}(k+1)$. We also pose some related conjectures.