On p-adic approximation of sums of binomial coefficients

We propose higher-order generalizations of Jacobsthal’s $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $\left(\genfrac{}{}{0px}{}{ip}{p}\right)$ ($i=1,2,\dots$) that are divisible by arbitrarily large powers of prime $p$.