Supercongruences for some lacunary sums of powers of binomial coefficients.

Let a , n , m ∈ ℤ + with (a , p) = 1 and p be an odd prime. We find a supercongruence for ∑ p | k a p n k m and related sums of powers of binomial coefficients. These results complement prior results for ∑ k ≡ i (mod   p) a p n k m with i = 1 , 2 , … , p − 1 obtained recently by the author. [ABSTRACT FROM AUTHOR]