On some new congruences for binomial coefficients

الأصل · EN

In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let p be a prime and let a be any positive integer. We determine ∑ₖ₌₀ᵖᵃ⁻¹2kk+d mod p² for d=0,...,pᵃ and ∑ₖ₌₀ᵖᵃ⁻¹2kk+δ mod p³ for δ=0,1. We also show that Cₙ⁻¹∑ₖ₌₀ᵖᵃ⁻¹Cpᵃn+k=1-3(n+1)((pᵃ-1)/3) (mod p²) for every n=0,1,2,..., where Cₘ is the Catalan number 2mm/(m+1), and (-) is the Legendre symbol.