INFINITE LOG-CONCAVITY FOR POLYNOMIAL POLYA FREQUENCY SEQUENCES

McNamara and Sagan conjectured that if a(0), a(1), a(2), ... is a Polya frequency (PF) sequence, then so is a(0)(2), a(1)(2) - a(0)a(2), a(2)(2) - a(1)a(3), .... 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.