On the rate of p-adic convergence of sums of powers of binomial coefficients

Let m >= 1 be an integer and p be an odd prime. We study sums and lacunary sums of mth powers of binomial coefficients from the point of view of arithmetic properties. We develop new congruences and prove the p-adic convergence of some subsequences and that in every step we gain at least one or three more p-adic digits of the limit if m = 1 or m >= 2, respectively. These gains are exact under some explicitly given conditions. The main tools are congruential and divisibility properties of the binomial coefficients and multiple and alternating harmonic sums.