The Continuous Binomial Coefficient: An Elementary Approach

Abstract

In this article, we use elementary methods to investigate continuous binomial coefficients: functions of the real variable x defined by way of the gamma function with y a fixed real number. We begin with a brief qualitative description of these functions and then derive several interesting representations of them including an infinite product and Taylor series. We also prove various integral formulas involving continuous binomial coefficients, many of which remarkably mirror summation formulas of the familiar binomial coefficients. We conclude by proving a continuous analog of the binomial theorem.