INFINITE LOG-CONCAVITY FOR POLYNOMIAL PÓLYA FREQUENCY SEQUENCES.

McNamara and Sagan conjectured that if a0, a1, a2, ... is a Pólya frequency (PF) sequence, then so is a0², a1²- a0a2, a2² - a1a3, ... . We prove this conjecture for a natural class of PF-sequences which are interpolated by polynomials. In particular, this proves that the columns of Pascal's triangle are infinitely log-concave, as conjectured by McNamara and Sagan. We also give counterexamples to the first mentioned conjecture. Our methods provide families of nonlinear operators that preserve the property of having only real and nonpositive zeros. [ABSTRACT FROM AUTHOR]