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

Abstract

In this article, we give a formula for the generalization of the binomial coefficient to the complex numbers as a linear combination of $\text{sinc}$ 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.

MSC::

• Primary33B15
• Secondary 32A10

Acknowledgment

The author wishes to thank Francesca Aicardi for her helpful tips on the presentation.

Additional information

Notes on contributors

Lorenzo David

LORENZO DAVID is a high school student who learned calculus as autodidact during 2020’s first Covid-19 lockdown.