Petter Brändén: Infinite log-concavity and zeros of polynomials

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.