The q-log-concavity of q-binomial coefficients (Q2639062)

Let \(\left[ \begin{matrix} n\\ k\end{matrix} \right]_ q\) denote the number of k- dimensional subspaces of an n-dimensional vector space over GF(q). It is shown that \(\left[ \begin{matrix} n\\ k \end{matrix} \right]^ 2_ q-\left[ \begin{matrix} n\\ k-1\end{matrix} \right]_ q\left[ \begin{matrix} n\\ k+1\end{matrix} \right]_ q\), considered as a polynomial in q, has nonnegative coefficients for \(0<k<n\), i.e. the polynomials \(\left[ \begin{matrix} n\\ k\end{matrix} \right]_ q\) form a q-log-concave sequence. This settles a conjecture posed by the author in [Proc. Am. Math. Soc. 101, 771-775 (1987; Zbl 0647.20053)].