The Properties Of Some Combinatorial Sequences

In this thesis, we discuss the properties of hyperharmonic numbers, generalized Cauchy numbers of two kinds, hyperfibonacci numbers, hyperlucas numbers, generalized hyperfibonacci numbers, and generalized hyperlucas numbers. By applying various combinatorial methods such as generating function method, difference method and asymptotic enumeration method, we discuss sums related to the above sequences, and give asymptotic approximations of certain sums involving hyperharmonic numbers. We investigate the log-concavity (log-convexity) of hyperharmonic numbers, hyperfibonacci numbers, hyperlucas numbers, generalized hyperfibonacci numbers, generalized hyperlucas numbers. We discuss the log-concavity (log-convexity) of generalized Cauchy numbers under some conditions, and the q-log-concavity of some polynomial related to hyperharmonic numbers. The results of this thesis can be summarized as follows:In chapter1, we discuss some properties of hyperharmonic by applying generating function method and asymptotic enumeration method. Some identities for hyperharmonic are established, and the asymptotic approximations of certain sums related to hyperharmonic numbers are given. In addition, we discuss the log-concavity (log-convexity) of hyperharmonic numbers.In chapter2, we discuss some properties of generalized Cauchy numbers of two kinds. We give some identities for generalized Cauchy numbers of two kinds, and we divestigate the log-concavity (log-convexity) of generalized Cauchy numbers under some conditions.In chapter3, we discuss the properties of hyperfibonacci numbers, hyperlucas numbers, generalized hyperfibonacci numbers, and generalized hyperlucas numbers. We investigate the log-concavity (log-convexity) of these sequences by a new and simple method. In addition, we discuss the log-balanceness of some sequences involving hyperfibonacci numbers and hyperlucas numbers.