Log concave coefficients

A few days ago I wrote about the rise and fall of binomial coefficients. There I gave a proof that binomial coefficients are log-concave, and so a local maximum has to be a global maximum.

Here I’ll give a one-line proof of the same result, taking advantage of the following useful theorem.

Let p(x) = c₀ + c₁x + c₂x² + … + c_nxⁿ be a polynomial all of whose zeros are real and negative. Then the coefficient sequence c_k is strictly log concave.

This is Theorem 4.5.2 from Generatingfunctionology, available for download here.

Now for the promised one-line proof. Binomial coefficients are the coefficients of (x + 1)ⁿ, which is clearly a polynomial with only real negative roots.

The same theorem shows that Stirling numbers of the first kind, s(n, k), are log concave for fixed n and k ≥ 1. This because these numbers are the coefficients of x^k in

(x + 1)(x + 2) … (x + n – 1).

The theorem can also show that Stirling numbers of the second kind are log-concave, but in that case the generating polynomial is not so easy to write out.