ABSTRACT
Most people who have a passing familiarity with binomial identities probably do not think of calculus when they think of techniques for proving them. In this chapter we’ll show that calculus is actually a very powerful tool for proving certain kinds of binomial identities. We’ll see how simply differentiating and integrating the binomial theorem can yield quick proofs of many identities. We’ll prove Leibniz’s generalized product rule for derivatives and note that it involves the binomial coefficients, and we’ll derive some identities from the rule as well. In the last section we’ll go a little deeper by considering two special functions that can be defined in terms of integrals, the gamma and beta functions. We’ll see how the binomial coefficient can be represented in terms of the beta integral, and we’ll use this representation to prove several binomial identities we have not yet seen.
