Abstract
Let {u n} be a higher-order linear recursive sequence. In this paper, we use the properties of error estimation and the analytic method to study the reciprocal sums of higher power of higher-order sequences. Then we establish several new and interesting identities relating to the infinite and finite sums.
Highlights
The so-called Fibonacci zeta function and Lucas zeta function, defined by ζF (s) = ∞ ∑ n=1 ζL (1)where the Fn and Ln denote the Fibonacci numbers and Lucas numbers, have been considered in several different ways; see [1, 2]
For any real number β > 2 and positive integer 1 ≤ s < ⌊log(α/a1)αd⌋, where α, α1, . . . , αm−1 are the roots of the characteristic equation of un and d−1 = max {|α1|, |α2|, . . . , |αm−1|}, there exists a positive integer n3 such that uns −1 a1sn−s
For positive integer 1 ≤ s < ⌊log(α/a1)αd⌋, where α, α1, . . . , αm−1 are the roots of the characteristic equation of un and d−1 = max{|α1|, |α2|, . . . , |αm−1|}, there exists a positive integer n4 such that
Summary
The so-called Fibonacci zeta function and Lucas zeta function, defined by ζF (s). where the Fn and Ln denote the Fibonacci numbers and Lucas numbers, have been considered in several different ways; see [1, 2]. Ohtsuka and Nakamura [3] studied the partial infinite sums of reciprocal Fibonacci numbers and proved the following conclusions:. Some authors considered the nearest integer of the sums of reciprocal Fibonacci numbers and other wellknown sequences and obtained several meaningful results; see [14,15,16]. For any positive integer n > m, the mth-order linear recursive sequence {un} is defined as follows: un = a1un−1 + a2un−2 + ⋅ ⋅ ⋅ + am−1un−m+1 + amun−m, (5). In [18], Xu and Wang applied the method of undetermined coefficients and constructed a number of delicate inequalities in order to study the infinite sum of the cubes of reciprocal Pell numbers and obtained the following meaningful result.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
