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