Binomial coefficients, Catalan numbers and Lucas quotients

Details Binomial coefficients, Catalan numbers and Lucas quotients Binomial coefficients, Catalan numbers and Lucas quotients

2009-09-30

arxiv.org/abs/0909.5648v11

Sci. China Math. 53(2010), no.9, 2473-2488

Mathematics

Number Theory

24 pages

Scientific paper

Let p be an odd prime and let a,m be integers with a>0 and m not=0 (mod p). In this paper we determine $\sum_{k=0}^{p^a-1}\binom[2k,k+d]/m^k$ mod p^2 for d=0,1; for example, $$\sum_{k=0}^{p^a-1}\binom[2k,k]/m^k=(\frac{m^2-4m}{p^a})+(\frac{m^2-4m}{p^{a-1}})u_{p-(\frac{m^2-4m}{p})} (mod p^2),$$ where (-) is the Jacobi symbol, u_0=0, u_1=1 and u_{n+1}=(m-2)u_n-u_{n-1} for n=1,2,3,.... As an application, we determine $\sum_{0

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