Differentiating the binomial coefficient

Question

I took a lecture in combinatorics this semester and the professor did the following step in a proof: He showed that function $f: x \mapsto \binom{x}{r}$ is convex for $x > r - 1$ (in order to use Jensen's inequality on $f$) and did this in the following way:

"By the product-rule we have $$f''(x) = \frac{2}{r!} \sum_{0 \leq i < j \leq r - 1} \prod_{l = 0}^{r - 1} ( x - l) \frac{1}{(x - i ) (x - j)} \geq 0$$ for all $ x > r - 1$." I am a bit confused on his definition: How would one extend the binomial coefficient to $x \notin \mathbf{N}$? I first thought about piecewise linear interpolation, but then I can't differentiate it. I also thought of plugging in the Gamma-function for the factorials, but I doubt that this is the definition he used here.

Can anyone explain to me what's happening here?

Thanks!

Answer 1

A way to write the (usual) binomial coefficient is $$\binom{n}{r}= \frac{\prod_{i=0}^{r-1}(n-i)}{r!}.$$

In this expression $n$ does not have to be an integer for it to make sense. This is the (or at least one) way to extend the definition.

So $$\binom{x}{r}= \frac{\prod_{i=0}^{r-1}(x-i)}{r!}.$$

Answer 2

As small supplement:

A generalisation of the binomial coefficient is used in the binomial series representation \begin{align*} (1+x)^\alpha=\sum_{r=0}^\infty\binom{\alpha}{r}x^r\qquad\qquad |x|<1, \,\alpha\in\mathbb{C} \end{align*} where the binomial coefficient \begin{align*} \binom{\alpha}{r}=\frac{1}{r!}\alpha(\alpha-1)\cdots(\alpha-r+1) \end{align*} can be defined even for complex $\alpha$. This implies that we can consider the binomial coefficient as real-valued (or complex-valued) polynomial of degree $r$ \begin{align*} &f:\mathbb{R}\rightarrow\mathbb{R}\\ &f(x)=\binom{x}{r}\\ &\qquad=\frac{1}{r!}x(x-1)\cdots(x-r+1) \end{align*} which is so accessible to analytical operations (differentiation, etc.).

And you're right, it's not necessary to involve the Gamma-function here.