Binomial Coefficients , Catalan Numbers and Lucas Quotients

Let p be an odd prime and let a, m ∈ Z with a > 0 and p ∤ m. In this paper we determine p a −1 k=0 2k k+d /m k mod p 2 for d = 0, 1; for example, p a −1 k=0 2k k m k ≡ m 2 − 4m p a + m 2 − 4m p a−1 u p−(m 2 −4m p) (mod p 2), where (−) is the Jacobi symbol and {u n } n0 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 0<k<p a , k≡r (mod 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.