On The Log-Convexity Of Combinatorial Sequences

The log-convexity of sequences is one of the primary topics in Combinatorics. Although the log-convexity has "one-line" definition, it may be quite difficult to show that a sequence is log-convex by its definition. The object of this paper is to investigate the log-convexity of sequences by generalized combinatorial triangles and generating functions. The frame of the thesis is as follows.The first part of the thesis introduces Aigner-Catalan-Riordan arrays, which is the common generalization of generalized Aigner recursive triangles and Riordan arrays. Based on the method of TP matrices, we give sufficient conditions for the log-convexity of the first column of the Aigner-Catalan-Riordan arrays. As applications, we give corresponding sufficient conditions for generalized Aigner recursive triangles and Riordan arrays repectively. With these results, we can give a unified proof of the log-convexity of many well-known combinatorial sequences, including the Catalan numbers, the Bell numbers, the restricted hexagonal numbers, the large Schroder numbers, the small Schroder numbers, the central binomial coefficients, the Motzkin numbers, the central Delannoy numbers, the Fine numbers, and so on.The second part of the thesis investigates the sufficient condition of the log-convexity of sequences from the view of generating functions. Furthermore, we give the sufficient conditions for the strong q-log-convexity of polynominal sequences with several applications.The third part of the thesis studies the generalized Motzkin numbers by means of the gen-eralized Motzkin triangles. We discuss the properties of the generalized Motzkin numbers, in-cluding recurrence relation, binominal transform, the Hankel transform, the log-convexity, the continued fraction of the generating function, the total positivity of the corresponding Hankel matrix, the sufficient condition for forming Stieltjes moment seqnences and the combinatorial interpretation.