  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.
