Binomial coefficients with real number

Question

I am studying the book Loop, knots, gauge theories and quantum gravity by Gambini and Pullin. In chapter 2 eqn 2.24 pag 35, when dealing with the extended group of loops, the authors use the binomial coefficients with real numbers. How are binomial coefficients extended to real numbers?

Answer 1

For real $x$, or complex $x$, the formula $$\binom x k = \prod_{i=0}^{k-1} \frac {x-i} {k-i}$$ extends the usual definition of binomial coefficients. This is what's used in the "Generalized binomial theorem".

Answer 2

You can also use the gamma function $$\binom x k =\frac{\Gamma(x+1)}{\Gamma(k+1)\,\,\Gamma(x-k+1)}$$

Source: equation (2) in following link