Is Binomial:Gamma ever an integer?

Question

We consider:

$$\dfrac{\Gamma(n)}{\Gamma(k)\Gamma(n-k)}\quad\quad[1]$$

for $n,k\in\mathbb{R}$.

Is $[1]$ ever an integer, except for the obvious?

Answer 1

Yes, rather trivially so. The function you describe is the reciprocal of a beta function $B(a,b)$ for a suitable bijection between $(a,b)$ and $(n,k)$; explicitly, $$n = a+b, \quad k = a$$ is one such one-to-one mapping. Consequently, if there exist $n, k \in \mathbb R$ such that $[1]$ is a (positive) integer $m$, then there exists $a, b \in \mathbb R$ such that $B(a,b) = 1/m$. The level curves of the beta function in the first quadrant show that there exist infinitely many (non-integer) ordered pairs $(a,b)$ that satisfy this condition for each $m \in \mathbb Z^+$.