Binomial coefficients, Catalan numbers and Lucas quotients

Abstract

Let p be an odd prime and let a,m ∈ ℤ with a > 0 and p ∤ m. In this paper we determine Σ ^pa−1_k=0 ( ^k+d_2k )/m ^k mod p ² for d 0, 1; for example,

$$ \sum\limits_{k = 0}^{p^a - 1} {\frac{{\left( {\begin{array}{*{20}c} {2k} \\ k \\ \end{array} } \right)}} {{m^k }}} \equiv \left( {\frac{{m^2 - 4m}} {{p^a }}} \right) + \left( {\frac{{m^2 - 4m}} {{p^{a - 1} }}} \right)u_{p - (\frac{{m^2 - 4m}} {p})} (\bmod p^2 ), $$

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} (n = 1, 2, 3, ...). As an application, we determine $\sum\nolimits_{0 < k < p^a ,k \equiv r(\bmod p - 1)} {C_k }$ modulo p ² for any integer r, where C _k denotes the Catalan number ( ^2k _k /(k+1). We also pose some related conjectures.