Abstract
The main purpose of this paper is, by using elementary methods and symmetry properties of the summation procedures, to study the computational problem of a certain power series related to the Tribonacci numbers, and to give some interesting identities for these numbers.
Highlights
IntroductionTaekyun Kim et al [2] first introduced the convolved Fibonacci numbers pn ( x ), which are defined by the generating function:
For integers n ≥ 0, the Fibonacci polynomials Fn ( x ) are defined by F0 ( x ) = 0, F1 ( x ) = 1 and the second-order linear recurrence sequence: Fn+1 ( x ) = xFn ( x ) + Fn−1 ( x ), for all n ≥ 1.If we take x = 1, { Fn (1)} becomes the famous Fibonacci sequence
We present a simple identity, which is required in the proof of the theorem
Summary
Taekyun Kim et al [2] first introduced the convolved Fibonacci numbers pn ( x ), which are defined by the generating function: They used the elementary and combinatorial methods to prove a series of important conclusions, one of them is the following identity:. As an interesting corollary of [3], Chen Zhuoyu and Qi Lan proved that, for any positive integer k, one has the identity: a1 + a2 + a3 +···+ ak =n. Inspired by the ideas in [2,3], it is natural to ask, for any real number h, what are the properties of the coefficients Tn (h) of the power series of the function: In view of these problems, in this paper we carry out a preliminary discussion and prove the following main result: Theorem 1. For any positive integer n, the following identity holds: Tn
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