Sums of the Fibonacci and Lucas numbers over the binary digital sums

Abstract

Let $s(n)$ denote the binary digital sum of the positive integer $n$. Using elementary combinatorial method, we present some known identities as a new form in term of $s(n)$. The sums of the Fibonacci, Lucas and harmonic numbers over the binary digital sums are considered. Moreover, we also give the sums over the binary digital sums, which derived from the binomial theorem.