On some lacunary sums of even powers of binomial coefficients

Abstract

Let [Formula: see text] be an even integer and [Formula: see text] be an odd prime. We study Franel-like sums and alternating sums, as well as lacunary sums of [Formula: see text]th powers of binomial coefficients from the point of view of arithmetic properties. This paper complements the author’s prior work on the cases with [Formula: see text] odd although, it uses a different approach. It develops new supercongruences and determines the [Formula: see text]-adic order of these sums as well as of generalized harmonic sums restricted to particular remainder classes modulo [Formula: see text].