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Counting functions and finite differences. (English) Zbl 1195.11023

Summary: Any increasing function \(p(d)\) on the natural numbers has an associated counting function \(\pi(n)\) that yields the number of inputs \(d\) for which \(p(d) \leq n\). 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.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11B65 Binomial coefficients; factorials; \(q\)-identities
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