Some Results On A Class Of Recurrence Sequences

The study on recurrence sequences began long ago, and in this paper, we investigate the properties of a class of recurrence sequences, which are called hyperfibonacci numbers and hy-perlucas numbers. By applying generating function method, asymptotic enumeration method, and difference method, we discuss sums related to hyperfibonacci numbers and hyperlucas numbers, and give asymptotic approximations of certain sums involving hyperfibonacci num-bers and hyperlucas numbers. We consider the log-concavity (log-convexity) of hyperfibonacci numbers and hyperlucas numbers. In addition, we also discuss the q-log-concavity (q-log-convexity) of some polynomials related to hyperfibonacci numbers or hyperlucas numbers. The work we have done can be summarized as follows:In chapter1, we introduce the development of recurrence sequences. We recall some definitions and notations as well.In chapter2, we discuss properties of generalized hyperfibonacci numbers and hyperlu-cas numbers by generating function method, asymptotic enumeration method and difference method. We derive a serie of identities for generalized hyperfibonacci numbers and hyperlucas numbers by generating function method and difference method. In addition, we give asymp-totic expansions for certain sums involving generalized hyperfibonacci numbers and hyperlucas numbers by applying asymptotic enumeration method.In chapter3, we first discuss the sums of reciprocal hyperfibonacci numbers and hyperlucas numbers. Further more, we give some identities related to reciprocal hyperfibonacci numbers and hyperlucas numbers.In chapter4, we mainly investigate the log-concavity (log-convexity) of hyperfibonacci numbers and hyperlucas numbers. In addition, we discuss q-concavity (q-convexity) of certain polynomials related to hyperfibanacci numbers and hyperlucas numbers.