Iterated Sequences and the Geometry of Zeros

We study the effect on the zeros of generating functions of sequences under certain non-linear transformations. Characterizations of Pólya– Schur type are given of the transformations that preserve the property of having only real and non-positive zeros. In particular, if a polynomial a0 + a1z + · · · + anzn has only real and non-positive zeros, then so does the polynomial a 2 0 +(a 1 − a0a2)z + · · ·+ (a n−1 − an−2an)z + a2nz n. This confirms a conjecture of Fisk, McNamara–Sagan and Stanley, respectively. A consequence is that if a polynomial has only real and non-positive zeros, then its Taylor coefficients form an infinitely log-concave sequence. We extend the results to concern entire functions in the Laguerre–Pólya class, and discuss the consequences to problems on iterated Turán inequalities, studied by Craven and Csordas. Finally, we propose a new approach to a conjecture of Boros and Moll.