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

Abstract

Let [Formula: see text] be an integer and [Formula: see text] be an odd prime. We study sums and lacunary sums of [Formula: see text]th powers of binomial coefficients from the point of view of arithmetic properties. We develop new congruences and prove the [Formula: see text]-adic convergence of some subsequences and that in every step we gain at least one or three more [Formula: see text]-adic digits of the limit if [Formula: see text] or [Formula: see text], 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.