Counting Functions and Finite Differences

Any increasing function p(d) on the natural numbers has an associated counting function π(n) that yields the number of inputs d for which p(d) ≤ n. In this article we derive three formulas that relate a sequence to its finite difference sequence by way of counting functions and the technique of summation by parts. We demonstrate our formulas by using them to produce several identities for Fibonacci numbers and binomial coefficients.