On some new congruences for binomial coefficients

Abstract

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 ${\sum }_{k=0}^{p^a-1}\left(\genfrac{}{}{0ex}{}{2k}{k+d}\right)$ mod $p^2$ for $d=0,...,p^a$ and ${\sum }_{k=0}^{p^a-1}\left(\genfrac{}{}{0ex}{}{2k}{k+\delta }\right)$ mod $p^3$ for $\delta =0,1$. We also show that $$C_n^{-1}\sum _{k=0}^{p^a-1}{C}_{p^{a}n+k}=1-3(n+1)((p^a-1)/3)(mod{p}^2)$$ for every n=0,1,2,..., where $C_m$ is the Catalan number $\left(\genfrac{}{}{0ex}{}{2m}{m}\right)/(m+1)$, and (-) is the Legendre symbol.

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• Products and sums divisible by central binomial coefficients
• Some curious congruences modulo primes
• q-Analogs of some congruences involving Catalan numbers
• On divisibility concerning binomial coefficients
• On binomial coefficients modulo squares of primes