The Binomial Coefficient as an (In)finite Sum of Sinc Functions

TL;DR

This paper generalizes the binomial coefficient to complex numbers using sinc functions, providing formulas for integrals involving these coefficients and deriving related identities.

Contribution

It introduces a novel representation of the complex binomial coefficient as a sum of sinc functions and develops formulas for integrals involving these coefficients.

Findings

Derived a formula expressing the binomial coefficient as a sinc sum.

Established integral formulas linking binomial coefficients and functions.

Generated new identities from the sinc-based representations.

Abstract

In this article, we give a formula for the generalization of the binomial coefficient to the complex numbers as a linear combination of $\sinc$ functions. We then give a general formula to compute the integral on the real line of the product of the binomial coefficient and a given function, which, in some cases, turns out to be equal to the series of their values on the integers. Finally, we establish a list of identities obtained by applying these formulas.