Abstract
Recently a lot of papers have been devoted to partial infinite reciprocal sums of a higher-order linear recursive sequence. In this paper, we continue this program by finding a sequence which is asymptotically equivalent to partial infinite sums, including a reciprocal of polynomial applied to linear higher order recurrences.
Highlights
IntroductionA sequence (un )n is a linear recurrence sequence with coefficients c0 , c1 , . . . , cs−1 , where c0 6= 0 if:
A sequencen is a linear recurrence sequence with coefficients c0, c1, . . . , cs−1, where c0 6= 0 if:u n + s = c s −1 u n + s −1 + · · · + c 1 u n +1 + c 0 u n, (1)for all non-negative integers n
Choi and Choo [13] proceeded in study of sequence from [10] and found a formula for the following sum of reciprocals of the squares of generalized Fibonacci numbers:
Summary
A sequence (un )n is a linear recurrence sequence with coefficients c0 , c1 , . . . , cs−1 , where c0 6= 0 if:. Choi and Choo [13] proceeded in study of sequence from [10] and found a formula for the following sum of reciprocals of the squares of generalized Fibonacci numbers:. Suppose that k x k = b x + 1/2c (the nearest integer formula) and let (un )n be an integer sequence satisfying the recurrence formula: un = pun−1 + qun−2 + un−3 + · · · + un−k , for any positive integer p ≥ q and n ≥ k They proved the existence of a positive integer n0 such that:. There are many generalizations of this result (see, for example, [15] and references therein) In all these cases, the authors considered sequences whose characteristic polynomial has only one root outside the closed unit disc and all the other roots lie inside this disc. The calculations performed in this paper took several minutes using software Mathematica on a 2.5 GHz Intel Core i5
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