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We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) Pólya frequency sequences are infinitely log-concave. We introduce the concept of q-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a conjecture of Chen and Xia about the infinite log-convexity of the Schröder numbers. We also list some interesting problems and conjectures about the log-convexity and the Stieltjes moment property of the (generalized) Apéry numbers.

In this paper we study unimodality problems for the independence polynomial of a graph, including unimodality, log-concavity and reality of zeros. We establish recurrence relations and give factorizations of independence polynomials for certain classes of graphs. As applications we settle some unimodality conjectures and problems.Comment: 17 pages, to appear in European Journal of Combinatoric

Let {T n,k } n,k≥0 be an array of nonnegative numbers satisfying the recurrence relationWe obtain some results for the total positivity of the matrix T n,k n,k≥0 , Pólya frequency properties of the row and column generating functions, and q-log-convexity of the row generating functions. This allows a unified treatment of the properties above for some triangular arrays of the second kind, including the Stirling triangle, Jacobi-Stirling triangle, Legendre-Stirling triangle, and central factorial numbers triangle.

We develop techniques to deal with monotonicity of sequences {z n+1 /z n } and { n √ z n }. A series of conjectures of Zhi-Wei Sun and of Amdeberhan et al. are verified in certain unified approaches.