Binomial coefficients, Catalan numbers and Lucas quotients |
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Authors: | ZhiWei Sun |
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Institution: | 1. Department of Mathematics, Nanjing University, Nanjing, 210093, China
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Abstract: | Let p be an odd prime and let a,m ∈ ? with a > 0 and p ? m. In this paper we determine Σ k=0 pa?1 ( 2k k+d )/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 ( k 2k /(k+1). We also pose some related conjectures. |
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