October 28, 2009
Petter Brändén (SU and KTH): Infinite log-concavity and zeros of polynomials
Abstract:
During Björner's birthday conference Richard Stanley told me about a
conjecture stating that if $\{a_k\}$ is a nonnegative sequence with
the property that its generating function is a polynomial with only
real zeros, then the same is true for the sequence
$\a_k^2-a_{k-1}a{k+1}\}$. At the time I could neither prove nor
disprove it. A few months ago I played with the conjecture again and
realized that the key to the solution is a strange symmetric function
identity involving the Catalan numbers. The other ingredient is
Grace's coincidence theorem. I will prove this conjecture due to
Stanley, McNamara-Sagan and Fisk, respectively, and also discuss
related topics such as infinite log-concavity and iterated Túran inequalities.