Editing Binomial coefficient (section)

=== Generalized binomial coefficients ===
The [[Gamma function#Euler's definition as an infinite product|infinite product formula for the gamma function]] also gives an expression for binomial coefficients
<math display="block">(-1)^k {z \choose k}= {-z+k-1 \choose k} = \frac{1}{\Gamma(-z)} \frac{1}{(k+1)^{z+1}} \prod_{j=k+1} \frac{\left(1+\frac{1}{j}\right)^{-z-1}}{1-\frac{z+1}{j}}</math>
which yields the asymptotic formulas
<math display="block">{z \choose k} \approx \frac{(-1)^k}{\Gamma(-z) k^{z+1}} \qquad \text{and} \qquad {z+k \choose k} = \frac{k^z}{\Gamma(z+1)}\left( 1+\frac{z(z+1)}{2k}+\mathcal{O}\left(k^{-2}\right)\right)</math>
as <math>k \to \infty</math>.

This asymptotic behaviour is contained in the approximation
<math display="block">{z+k \choose k}\approx \frac{e^{z(H_k-\gamma)}}{\Gamma(z+1)}</math>
as well. (Here <math>H_k</math> is the ''k''-th [[harmonic number]] and <math>\gamma</math> is the [[Euler–Mascheroni constant]].)

Further, the asymptotic formula
<math display="block">\frac{{z+k\choose j}}{{k\choose j}}\to \left(1-\frac{j}{k}\right)^{-z}\quad\text{and}\quad \frac{{j\choose j-k}}{{j-z\choose j-k}}\to \left(\frac{j}{k}\right)^z</math>
hold true, whenever <math>k\to\infty</math> and <math>j/k \to x</math> for some complex number <math>x</math>.