Binomial coefficients, Catalan numbers and Lucas quotients

By Zhi-Wei Sun. 2009

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AbstractLet p be an odd prime and let a,m ∈ ℤ with a > 0 and p ∤ m. In this paper we determine Σk=0pa−1(2kk+d)/mk mod p2 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 {un}n⩾0 is the Lucas sequence given by u0 = 0, u1 = 1 and un+1 = (m−2)un − un − 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 p2 for any integer r, where Ck denotes the Catalan number (k2k/(k+1). We also pose some related conjectures.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b65fdefbcd38d69437750aae7f7cd1620c58a228