Infinite log-concavity: developments and conjectures

Given a sequence (ak) = a0, a1, a2, . . . of real numbers, define a new sequence L(ak) = (bk) where bk = ak − ak−1ak+1. So (ak) is log-concave if and only if (bk) is a nonnegative sequence. Call (ak) infinitely log-concave if L(ak) is nonnegative for all i ≥ 1. Boros and Moll [3] 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 [24] 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.