Infinite log-concavity: Developments and conjectures

Given a sequence (ak)=a0,a1,a2,… of real numbers, define a new sequence L(ak)=(bk) where . So (ak) is log-concave if and only if (bk) is a nonnegative sequence. Call (ak) infinitely log-concave if Li(ak) is nonnegative for all i⩾1. Boros and Moll conjectured that the rows of Pascal's triangle are infinitely log-concave. Using a computer and a stronger version of log-concavity, we prove their conjecture for the nth row for all n⩽1450. We also use our methods to give a simple proof of a recent result of Uminsky and Yeats about regions of infinite log-concavity. We investigate related questions about the columns of Pascal's triangle, q-analogues, symmetric functions, real-rooted polynomials, and Toeplitz matrices. In addition, we offer several conjectures.