Infinitely Log-monotonic Combinatorial Sequences
William Y.C. Chen, Jeremy J. F. Guo and Larry X. W. Wang
Abstract: We introduce the notion of infinitely log-monotonic sequences. By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, we show that the sequences of the Bernoulli numbers, the Catalan numbers and the central binomial coefficients are infinitely log-monotonic. In particular, if a sequence {a_{n}}_{n≥0} is log-monotonic of order two, then it is ratio log-concave in the sense that the sequence {a_{n+1}/a_{n}}_{n≥0} is log-concave. Furthermore, we prove that if a sequence {a_{n}}_{n≥k} is ratio log-concave, then the sequence is strictly logconcave subject to a certain initial condition. As consequences, we show that the sequences of the derangement numbers, the Motzkin numbers, the Fine numbers, the central Delannoy numbers, the numbers of tree-like polyhexes and the Domb numbers are ratio log-concave. For the case of the Domb numbers D_{n}, we confirm a conjecture of Sun on the log-concavity of the sequence . AMS Classification: 05A20, 11B68. Keywords: logarithmically completely monotonic function, infinitely logmonotonic sequence, ratio log-concave, Riemann zeta function Download: pdf |