On the log-convexity of combinatorial sequences

Details On the log-convexity of combinatorial sequences On the log-convexity of combinatorial sequences

2006-02-28

arxiv.org/abs/math/0602672v3

Advances in Applied Mathematics, 39 (2007) 453-476

Mathematics

Combinatorics

25 pages, final version, to appear in Advances in Applied Mathematics

Scientific paper

This paper is devoted to the study of the log-convexity of combinatorial sequences. We show that the log-convexity is preserved under componentwise sum, under binomial convolution, and by the linear transformations given by the matrices of binomial coefficients and Stirling numbers of two kinds. We develop techniques for dealing with the log-convexity of sequences satisfying a three-term recurrence. We also introduce the concept of $q$-log-convexity and establish the connection with linear transformations preserving the log-convexity. As applications of our results, we prove the log-convexity and $q$-log-convexity of many famous combinatorial sequences of numbers and polynomials.

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