On the Higher Power Sums of Reciprocal Higher-Order Sequences

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.

1. Introduction

The so-called Fibonacci zeta function and Lucas zeta function, defined by 𝜁_𝐹(𝑠)=^∞_𝑛=11𝐹^𝑠_𝑛,𝜁_𝐿(𝑠)=^∞_𝑛=11𝐿^𝑠_𝑛, (1) where the 𝐹_𝑛 and 𝐿_𝑛 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: ^∞_𝑘=𝑛1𝐹_𝑘⁻¹=𝐹_𝑛−2𝐹if𝑛iseven,𝑛≥2,_𝑛−2−1if𝑛isodd,𝑛≥1,_∞𝑘=𝑛1𝐹²_𝑘⁻¹=𝐹_𝑛−1𝐹_𝑛𝐹−1if𝑛iseven,𝑛≥2,_𝑛−1𝐹_𝑛if𝑛isodd,𝑛≥1, (2) where ⌊⋅⌋ denotes the floor function.

Further, Wu and Zhang [4, 5] generalized these identities to the Fibonacci polynomials and Lucas polynomials. Various properties of the Fibonacci polynomials and Lucas polynomials have been studied by many authors; see [6–13].

Recently, some authors considered the nearest integer of the sums of reciprocal Fibonacci numbers and other well-known sequences and obtained several meaningful results; see [14–16]. In particular, in [16], Kılıç and Arıkan studied a problem which is a little different from that of [3], namely, that of determining the nearest integer to (∑^∞_𝑘=𝑛(1/𝑣_𝑘))⁻¹. Specifically, suppose that ‖𝑥‖=⌊𝑥+(1/2)⌋ (the nearest integer function) and {𝑣_𝑛}_𝑛≥0 is an integer sequence satisfying the recurrence formula 𝑣_𝑛=𝑝𝑣_𝑛−1+𝑞𝑣_𝑛−2+𝑣_𝑛−3+⋯+𝑣_𝑛−𝑘, (3) for any positive integer 𝑝≥𝑞 and 𝑛>𝑘. Then we can conclude that there exists a positive integer 𝑛₀ such that ^∞_𝑘=𝑛1𝑣_𝑘⁻¹=𝑣_𝑛−𝑣_𝑛−1, (4) for all 𝑛>𝑛₀.

In [17], Wu and Zhang unified the above results by proving the following conclusion that includes all the results, [3–8, 15, 16], as special cases.

Proposition 1. For any positive integer 𝑛>𝑚, the 𝑚th-order linear recursive sequence {𝑢_𝑛} is defined as follows: 𝑢_𝑛=𝑎₁𝑢_𝑛−1+𝑎₂𝑢_𝑛−2+⋯+𝑎_𝑚−1𝑢_{𝑛−𝑚+1}+𝑎_𝑚𝑢_𝑛−𝑚, (5) with initial values 𝑢_𝑖∈ℕ for 0≤𝑖<𝑚 and at least one of them not being zero. For any positive real number 𝛽>2 and any positive integer 𝑎₁≥𝑎₂≥⋯≥𝑎_𝑚≥1, there exists a positive integer 𝑛₁ such that ^{⌊𝛽𝑛⌋}_𝑘=𝑛1𝑢_𝑘⁻¹=𝑢_𝑛−𝑢_𝑛−1,𝑛≥𝑛₁. (6) In particular, taking 𝛽→+∞, there exists a positive integer 𝑛₂ such that ^∞_𝑘=𝑛1𝑢_𝑘⁻¹=𝑢_𝑛−𝑢_𝑛−1,𝑛≥𝑛₂. (7)

It seems difficult to deal with (∑^∞_𝑘=𝑛(1/𝑢^𝑠_𝑘))⁻¹ for all integers 𝑠≥2, because it is quite unclear a priori what the shape of the result might be. 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 then obtained the following meaningful result.

Proposition 2. For any positive integer 𝑛≥1, we have the identity ^∞_𝑘=𝑛1𝑃³_𝑘⁻¹=𝑃²_𝑛𝑃_𝑛−1+3𝑃_𝑛𝑃²_𝑛−1+−61𝑃82^𝑛−91𝑃82^𝑛−1𝑃𝑖𝑓𝑛𝑖𝑠𝑒𝑣𝑒𝑛,𝑛≥2,_2𝑛𝑃_𝑛−1+3𝑃_𝑛𝑃²_𝑛−1+61𝑃82^𝑛+91𝑃82^𝑛−1𝑖𝑓𝑛𝑖𝑠𝑜𝑑𝑑,𝑛≥1. (8)

To find and prove this result is a substantial achievement since such a complex formula would not be clear beforehand that a result would even be possible. However, there is no research considering the higher power (𝑠>2) of reciprocal sums of some recursive sequences. The main purpose of this paper is using the properties of error estimation and the analytic method to study the higher power of the reciprocal sums of {𝑢_𝑛} and obtain several new and interesting identities. The results are as follows.

Theorem 3. Let {𝑢_𝑛} be an 𝑚th-order sequence defined by (5) with the restrictions 𝑎₁,𝑎₂,…,𝑎_𝑚∈ℕ and 𝑎₁≥𝑎₂≥⋯≥𝑎_𝑚≥2. For any real number 𝛽>2 and positive integer 1≤𝑠<⌊log_(𝛼/𝑎1)𝛼𝑑⌋, where 𝛼,𝛼₁,…,𝛼_𝑚−1 are the roots of the characteristic equation of 𝑢_𝑛 and 𝑑⁻¹=max{|𝛼₁|,|𝛼₂|,…,|𝛼_𝑚−1|}, then there exists a positive integer 𝑛₃ such that ^{⌊𝛽𝑛⌋}_𝑘=𝑛𝑎₁^𝑠𝑘𝑢^𝑠_𝑘⁻¹−𝑢^𝑠_𝑛𝑎₁^𝑠𝑛−𝑢^𝑠_𝑛−1𝑎₁^{𝑠𝑛−𝑠}=0,𝑛≥𝑛₃. (9)

Taking 𝛽→+∞, from Theorem 3 we may immediately deduce the following.

Corollary 4. Let {𝑢_𝑛} be an 𝑚th-order sequence defined by (5) with the restrictions 𝑎₁,𝑎₂,…,𝑎_𝑚∈ℕ and 𝑎₁≥𝑎₂≥⋯≥𝑎_𝑚≥2. For positive integer 1≤𝑠<⌊log_(𝛼/𝑎1)𝛼𝑑⌋, where 𝛼,𝛼₁,…,𝛼_𝑚−1 are the roots of the characteristic equation of 𝑢_𝑛 and 𝑑⁻¹=max{|𝛼₁|,|𝛼₂|,…,|𝛼_𝑚−1|}, then there exists a positive integer 𝑛₄ such that ^∞_𝑘=𝑛𝑎₁^𝑠𝑘𝑢^𝑠_𝑘⁻¹−𝑢^𝑠_𝑛𝑎₁^𝑠𝑛−𝑢^𝑠_𝑛−1𝑎₁^{𝑠𝑛−𝑠}=0,𝑛≥𝑛₄. (10)

For positive real number 1<𝛽≤2, whether there exits an identity for ^{⌊𝛽𝑛⌋}_𝑘=𝑛𝑎₁^𝑠𝑘𝑢^𝑠_𝑘⁻¹ (11) is an interesting open problem.

2. Several Lemmas

To complete the proof of our theorem, we need two lemmas.

Lemma 5. Let 𝑎₁,𝑎₂,…,𝑎_𝑚∈ℕ with 𝑎₁≥𝑎₂≥⋯≥𝑎_𝑚≥1 and 𝑚∈ℕ with 𝑚≥2. Then for the polynomial 𝑓(𝑥)=𝑥^𝑚−𝑎₁𝑥^𝑚−1−𝑎₂𝑥^𝑚−2−⋯−𝑎_𝑚−1𝑥−𝑎_𝑚, (12) we have the following:(I)polynomial 𝑓(𝑥) has exactly one positive real zero 𝛼 with 𝑎₁<𝛼<𝑎₁+1;(II)other 𝑚−1 zeros of 𝑓(𝑥) lie within the unit circle in the complex plane.

Proof. See Lemma 1 of [16].

Lemma 6. Let 𝑚≥2 and {𝑢_𝑛}_𝑛≥0 be an integer sequence satisfying the recurrence formula (5). Then for any positive integer 𝑠, we have 𝑢^𝑠_𝑛=𝑐^𝑠𝛼^𝑠𝑛+𝑂𝛼^{𝑠𝑛−𝑛}𝑑^−𝑛(𝑛⟶∞), (13) where 𝑐>0, 𝑑>1, and 𝑎₁<𝛼<𝑎₁+1 is the positive real zero of 𝑓(𝑥).

Proof. From Lemma 2 of [16], the closed formula of 𝑢_𝑛 is given by 𝑢_𝑛=𝑐𝛼^𝑛+𝑂𝑑^−𝑛(𝑛⟶∞), (14) where 𝑐>0, 𝑑>1, 𝑎₁<𝛼<𝑎₁+1, and 𝛼 is the positive real zero of 𝑓(𝑥). Now we prove Lemma 6 by mathematical induction. From formula (14), we have 𝑢²_𝑛=𝑐²𝛼^2𝑛+𝑂𝑑^−2𝑛+𝑂𝛼^𝑛𝑑^−𝑛=𝑐²𝛼^2𝑛+𝑂𝛼^𝑛𝑑^−𝑛. (15) That is, the lemma holds for 𝑠=2. Suppose that for all integers 2≤𝑠≤𝑘 we have 𝑢^𝑠_𝑛=𝑐^𝑠𝛼^𝑠𝑛+𝑂𝛼^{𝑠𝑛−𝑛}𝑑^−𝑛(𝑛⟶∞). (16) Then for 𝑠=𝑘+1 we have 𝑢_𝑛^𝑘+1=𝑐^𝑘𝛼^𝑘𝑛+𝑂𝛼^{𝑘𝑛−𝑛}𝑑^−𝑛⋅𝑐𝛼^𝑛+𝑂𝑑^−𝑛=𝑐^𝑘+1𝛼^{𝑘𝑛+𝑛}+𝑂𝛼^𝑘𝑛𝑑^−𝑛+𝑂𝛼^𝑘𝑛𝑑^−𝑛+𝑂𝛼^{𝑘𝑛−𝑛}𝑑^−2𝑛=𝑐^𝑘+1𝛼^(𝑘+1)𝑛+𝑂𝛼^𝑘𝑛𝑑^−𝑛. (17)
That is, Lemma 6 also holds for 𝑠=𝑘+1. This completes the proof of Lemma 6 by mathematical induction.

3. Proof of Theorem 3

In this section, we shall complete the proof of Theorem 3. From the geometric series as 𝜖→0, we have 1𝜖1±𝜖=1∓𝜖+𝑂²=1+𝑂(𝜖). (18) Using Lemma 6, we have 𝑎₁^𝑠𝑘𝑢^𝑠_𝑘=𝑎₁^𝑠𝑘𝑐^𝑠𝛼^𝑠𝑘+𝑂𝛼^{𝑠𝑘−𝑘}𝑑^−𝑘=𝑎₁^𝑠𝑘𝑐^𝑠𝛼^𝑠𝑘1+𝑂𝛼^−𝑘𝑑^−𝑘=𝑎₁^𝑠𝑘𝑐^𝑠𝛼^𝑠𝑘1+𝑂𝛼^−𝑘𝑑^−𝑘=𝑎₁^𝑠𝑘𝑐^𝑠𝛼^𝑠𝑘𝑎+𝑂^1𝑠𝑘𝛼^{𝑠𝑘+𝑘}𝑑^𝑘. (19) Consequently, ^{⌊𝛽𝑛⌋}_𝑘=𝑛𝑎₁^𝑠𝑘𝑢^𝑠_𝑘=1𝑐^{𝑠⌊𝛽𝑛⌋}_𝑘=𝑛𝑎₁𝛼^𝑠𝑘+𝑂^{⌊𝛽𝑛⌋}_𝑘=𝑛𝑎₁^𝑠𝑘𝛼^{𝑠𝑘+𝑘}𝑑^𝑘=𝛼^𝑠𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁⋅𝑎₁𝛼^𝑠𝑛−1𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁⋅𝑎₁𝛼^{𝑠⌊𝛽𝑛⌋}𝑎+𝑂^1𝑠𝑛𝛼^{𝑠𝑛+𝑛}𝑑^𝑛=𝛼^𝑠𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁⋅𝑎₁𝛼^𝑠𝑛𝑎+𝑂^1𝑠𝑛𝛼^𝑠𝑛⋅𝑎₁^{𝑠⌊𝛽𝑛⌋−𝑠𝑛}𝛼^{𝑠⌊𝛽𝑛⌋−𝑠𝑛}𝑎+𝑂^1𝑠𝑛𝛼^𝑠𝑛⋅1𝛼^𝑛𝑑^𝑛=𝛼^𝑠𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁⋅𝑎₁𝛼^𝑠𝑛𝑎+𝑂^1𝑠𝑛𝛼^𝑠𝑛⋅ℎ, (20) where ℎ=max{(𝑎₁^{𝑠⌊𝛽𝑛⌋−𝑠𝑛}/𝛼^{𝑠⌊𝛽𝑛⌋−𝑠𝑛}),(1/𝛼^𝑛𝑑^𝑛)}.

Taking the reciprocal of this expression yields ^{⌊𝛽𝑛⌋}_𝑘=𝑛𝑎₁^𝑠𝑘𝑢^𝑠_𝑘⁻¹=(1)×𝛼^𝑠𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁⋅𝑎₁𝛼^𝑠𝑛⋅𝑎1+𝑂^1𝑠𝑛ℎ𝛼^𝑠𝑛⋅𝛼^𝑠𝑛𝑎₁^𝑠𝑛−1=𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁𝛼^𝑠⋅𝛼𝑎₁^𝑠𝑛⋅((ℎ=𝑐1+𝑂))_𝑠𝛼^𝑠𝑛𝑎₁^𝑠𝑛−𝑐^𝑠𝛼^{𝑠𝑛−𝑠}𝑎₁^{𝑠𝑛−𝑠}𝛼+𝑂^𝑠𝑛𝑎₁^𝑠𝑛=𝑢⋅ℎ_𝑠𝑛𝑎₁^𝑠𝑛−𝑢^𝑠_𝑛−1𝑎₁^{𝑠𝑛−𝑠}𝛼+𝑂^𝑠𝑛𝑎₁^𝑠𝑛⋅ℎ. (21)

Case 1. If ℎ=(𝑎₁^{𝑠⌊𝛽𝑛⌋−𝑠𝑛}/𝛼^{𝑠⌊𝛽𝑛⌋−𝑠𝑛}), then for any real number 𝛽>2 and positive integer 𝑠 we have 𝛼^𝑠𝑛𝑎₁^𝑠𝑛𝛼⋅ℎ=^𝑠𝑛𝑎₁^𝑠𝑛⋅𝑎₁^{𝑠⌊𝛽𝑛⌋−𝑠𝑛}𝛼^{𝑠⌊𝛽𝑛⌋−𝑠𝑛}=𝑎₁𝛼^{𝑠⌊𝛽𝑛⌋−2𝑠𝑛}<1. (22)

Case 2. If ℎ=(1/𝛼^𝑛𝑑^𝑛), for any positive integer 𝑎₁≥2, 1<(𝛼/𝑎₁)<𝛼𝑑 holds. Then for any positive integer 𝑠 with 1≤𝑠<log_(𝛼/𝑎1)𝛼𝑑, (23) we have 𝛼^𝑠𝑛𝑎₁^𝑠𝑛𝛼⋅ℎ=^{𝑠𝑛−𝑛}𝑎₁^𝑠𝑛𝑑^𝑛=𝛼^𝑠−1𝑎^𝑠₁𝑑^𝑛<1. (24)

In both cases, it follows that for any real number 𝛽>2 and positive integer 1≤𝑠<⌊log_(𝛼/𝑎1)𝛼𝑑⌋ there exists 𝑛≥𝑛₃ sufficiently large so that the modulus of the last error term of identity (21) becomes less than 1/2. This completes the proof of Theorem 3.

Proof of Corollary 4. From identity (19), we have 𝑎₁^𝑠𝑘𝑢^𝑠_𝑘=𝑎₁^𝑠𝑘𝑐^𝑠𝛼^𝑠𝑘𝑎+𝑂^1𝑠𝑘𝛼^{𝑠𝑘+𝑘}𝑑^𝑘. (25) Consequently, ^∞_𝑘=𝑛𝑎₁^𝑠𝑘𝑢^𝑠_𝑘=1𝑐^𝑠∞_𝑘=𝑛𝑎₁𝛼^𝑠𝑘+𝑂^∞_𝑘=𝑛𝑎₁^𝑠𝑘𝛼^{𝑠𝑘+𝑘}𝑑^𝑘=𝛼^𝑠𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁⋅𝑎₁𝛼^𝑠𝑛𝑎+𝑂^1𝑠𝑛𝛼^{𝑠𝑛+𝑛}𝑑^𝑛. (26) Taking the reciprocal of this expression yields ^∞_𝑘=𝑛𝑎₁^𝑠𝑘𝑢^𝑠_𝑘⁻¹=1𝛼^𝑠/𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁⋅𝑎₁/𝛼^𝑠𝑛⋅1+𝑂1/𝛼^𝑛𝑑^𝑛=𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁𝛼^𝑠⋅𝛼𝑎₁^𝑠𝑛⋅11+𝑂𝛼^𝑛𝑑^𝑛=𝑐^𝑠𝛼^𝑠𝑛𝑎₁^𝑠𝑛−𝑐^𝑠𝛼^{𝑠𝑛−𝑠}𝑎₁^{𝑠𝑛−𝑠}𝛼+𝑂^{𝑠𝑛−𝑛}𝑎₁^𝑠𝑛𝑑^𝑛=𝑢^𝑠_𝑛𝑎₁^𝑠𝑛−𝑢^𝑠_𝑛−1𝑎₁^{𝑠𝑛−𝑠}𝛼+𝑂^{𝑠𝑛−𝑛}𝑎₁^𝑠𝑛𝑑^𝑛. (27) For any positive integer 𝑠 with 1≤𝑠<log_(𝛼/𝑎1)𝛼𝑑, (28) we have 𝛼^{𝑠𝑛−𝑛}𝑎₁^𝑠𝑛𝑑^𝑛=𝛼^𝑠−1𝑎^𝑠₁𝑑^𝑛<1. (29) So there exists 𝑛≥𝑛₄ sufficiently large so that the modulus of the last error term of identity (27) becomes less than 1/2. This completes the proof of Corollary 4.

4. Computation

We can determine the power 𝑠 of different sequence 𝑢_𝑛 by MATHEMATICA as the following examples.

Example 7. Let 𝑢_𝑛 be the second-order linear recursive sequence (see Table 1).

Table 1 

Example 8. Let 𝑢_𝑛 be the third-order linear recursive sequence (see Table 2).

Table 2 

Example 9. Let 𝑢_𝑛 be the fifth-order linear recursive sequence (see Table 3).

Table 3 

Therefore, we may immediately deduce the following corollaries.

Corollary 10. Let 𝑃_𝑛=2𝑃_𝑛−1+𝑃_𝑛−2 be the Pell numbers. For any real number 𝛽>2 and positive integer 1≤𝑠<9, there exists a positive integer 𝑛₅ such that ^{⌊𝛽𝑛⌋}_𝑘=𝑛2^𝑠𝑘𝑃^𝑠_𝑘⁻¹−𝑃^𝑠_𝑛2^𝑠𝑛−𝑃^𝑠_𝑛−12^{𝑠𝑛−𝑠}=0,𝑛≥𝑛₅. (30)

Corollary 11. Let 𝑇_𝑛=4𝑇_𝑛−1+3𝑇_𝑛−2+2𝑇_𝑛−3 be the generalized Tribonacci numbers. For any real number 𝛽>2 and positive integer 1≤𝑠<11, there exists a positive integer 𝑛₆ such that ^{⌊𝛽𝑛⌋}_𝑘=𝑛4^𝑠𝑘𝑇^𝑠_𝑘⁻¹−𝑇^𝑠_𝑛4^𝑠𝑛−𝑇^𝑠_𝑛−14^{𝑠𝑛−𝑠}=0,𝑛≥𝑛₆. (31)

Corollary 12. Let 𝑢_𝑛=5𝑢_𝑛−1+3𝑢_𝑛−2+2𝑢_𝑛−3+𝑢_𝑛−4+𝑢_𝑛−5 be a fifth-order sequence. For any real number 𝛽>2 and positive integer 1≤𝑠<18, there exists a positive integer 𝑛₇ such that ^{⌊𝛽𝑛⌋}_𝑘=𝑛5^𝑠𝑘𝑢^𝑠_𝑘⁻¹−𝑢^𝑠_𝑛5^𝑠𝑛−𝑢^𝑠_𝑛−15^{𝑠𝑛−𝑠}=0,𝑛≥𝑛₇. (32)

5. Related Results

The following results are obtained similarly.

Theorem 13. Let {𝑢_𝑛} be an 𝑚th-order sequence defined by (5) with the restrictions 𝑎₁,𝑎₂,…,𝑎_𝑚∈ℕ and 𝑎₁≥𝑎₂≥⋯≥𝑎_𝑚≥2. Let 𝑝 and 𝑞 be positive integers with 0≤𝑞<𝑝. For any real number 𝛽>2 and positive integer 1≤𝑠<⌊log_(𝛼/𝑎1)𝛼^𝑝𝑑^𝑝⌋, where 𝛼,𝛼₁,…,𝛼_𝑚−1 are the roots of the characteristic equation of 𝑢_𝑛 and 𝑑⁻¹=max{|𝛼₁|,|𝛼₂|,…,|𝛼_𝑚−1|}, then there exist positive integers 𝑛₈, 𝑛₉, and 𝑛₁₀ depending on 𝑎₁,𝑎₂,…, and 𝑎_𝑚 such that the following hold.(a)‖(∑^{⌊𝛽𝑛⌋}_𝑘=𝑛((−𝑎₁)^𝑠𝑘/𝑢^𝑠_𝑘))⁻¹−(−1)^𝑠𝑛(𝑢^𝑠_𝑛/𝑎₁^𝑠𝑛+𝑢^𝑠_𝑛−1/𝑎₁^{𝑠𝑛−𝑠})‖=0,(𝑛≥𝑛₈). (b)‖(∑^{⌊𝛽𝑛⌋}_𝑘=𝑛(𝑎₁^{𝑠𝑝𝑘+𝑠𝑞}/𝑢^𝑠_{𝑝𝑘+𝑞}))⁻¹−(𝑢^𝑠_{𝑝𝑛+𝑞}/𝑎₁^{𝑠𝑝𝑛+𝑠𝑞}−𝑢^𝑠_{𝑝𝑛−𝑝+𝑞}/𝑎₁^{𝑠𝑝𝑛+𝑠𝑞−𝑠𝑝})‖=0,(𝑛≥𝑛₉). (c)‖(∑^{⌊𝛽𝑛⌋}_𝑘=𝑛((−𝑎₁)^{𝑠𝑝𝑘+𝑠𝑞}/𝑢^𝑠_{𝑝𝑘+𝑞}))⁻¹−(−1)^{𝑠𝑝𝑛+𝑠𝑞}(𝑢^𝑠_{𝑝𝑛+𝑞}/𝑎₁^{𝑠𝑝𝑛+𝑠𝑞}+𝑢^𝑠_{𝑝𝑛−𝑝+𝑞}/𝑎₁^{𝑠𝑝𝑛+𝑠𝑞−𝑠𝑝})‖=0,(𝑛≥𝑛₁₀). For 𝛽→+∞, we deduce the following identity of infinite sum as special case of Theorem 13.

Corollary 14. Let {𝑢_𝑛} be an 𝑚th-order sequence defined by (5) with the restrictions 𝑎₁,𝑎₂,…,𝑎_𝑚∈ℕ and 𝑎₁≥𝑎₂≥⋯≥𝑎_𝑚≥2. Let 𝑝 and 𝑞 be positive integers with 0≤𝑞<𝑝. For any positive integer 1≤𝑠<⌊log_(𝛼/𝑎1)𝛼𝑑⌋, where 𝛼,𝛼₁,…,𝛼_𝑚−1 are the roots of the characteristic equation of 𝑢_𝑛 and 𝑑⁻¹=max{|𝛼₁|,|𝛼₂|,…,|𝛼_𝑚−1|}, then there exist positive integers 𝑛₁₁, 𝑛₁₂, and 𝑛₁₃ depending on 𝑎₁, 𝑎₂,…, and 𝑎_𝑚 such that the following hold.(d)‖(∑^∞_𝑘=𝑛((−𝑎₁)^𝑠𝑘/𝑢^𝑠_𝑘))⁻¹−(−1)^𝑠𝑛(𝑢^𝑠_𝑛/𝑎₁^𝑠𝑛+𝑢^𝑠_𝑛−1/𝑎₁^{𝑠𝑛−𝑠})‖=0,(𝑛≥𝑛₁₁). (e)‖(∑^∞_𝑘=𝑛(𝑎₁^{𝑠𝑝𝑘+𝑠𝑞}/𝑢^𝑠_{𝑝𝑘+𝑞}))⁻¹−(𝑢^𝑠_{𝑝𝑛+𝑞}/𝑎₁^{𝑠𝑝𝑛+𝑠𝑞}−𝑢^𝑠_{𝑝𝑛−𝑝+𝑞}/𝑎₁^{𝑠𝑝𝑛+𝑠𝑞−𝑠𝑝})‖=0,(𝑛≥𝑛₁₂). (f)‖(∑^∞_𝑘=𝑛((−𝑎₁)^{𝑠𝑝𝑘+𝑠𝑞}/𝑢^𝑠_{𝑝𝑘+𝑞}))⁻¹−(−1)^{𝑠𝑝𝑛+𝑠𝑞}(𝑢^𝑠_{𝑝𝑛+𝑞}/𝑎₁^{𝑠𝑝𝑛+𝑠𝑞}+𝑢^𝑠_{𝑝𝑛−𝑝+𝑞}/𝑎₁^{𝑠𝑝𝑛+𝑠𝑞−𝑠𝑝})‖=0,(𝑛≥𝑛₁₃).

Proof. We shall prove only (c) in Theorem 13 and other identities are proved similarly. From identity (19), we have −𝑎₁^{𝑠𝑝𝑘+𝑠𝑞}𝑢^𝑠_{𝑝𝑘+𝑞}=−𝑎₁^{𝑠𝑝𝑘+𝑠𝑞}𝑐^𝑠𝛼^{𝑠𝑝𝑘+𝑠𝑞}1+𝑂𝛼^{−𝑝𝑘−𝑞}𝑑^{−𝑝𝑘−𝑞}. (33) Consequently, ^{⌊𝛽𝑛⌋}_𝑘=𝑛−𝑎₁^{𝑠𝑝𝑘+𝑠𝑞}𝑢^𝑠_{𝑠𝑝𝑘+𝑠𝑞}=1𝑐^{𝑠⌊𝛽𝑛⌋}_𝑘=𝑛−𝑎₁𝛼^{𝑠𝑝𝑘+𝑠𝑞}+𝑂^{⌊𝛽𝑛⌋}_𝑘=𝑛𝑎₁^{𝑠𝑝𝑘+𝑠𝑞}𝛼^{𝑠𝑝𝑘+𝑠𝑞+𝑝𝑘+𝑞}𝑑^{𝑝𝑘+𝑞}=(−1)^{𝑠𝑝𝑛+𝑠𝑞}⋅𝛼^𝑠𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁⋅𝑎₁𝛼^{𝑠𝑝𝑛+𝑠𝑞}−(−1)^{𝑠𝑝𝑛+𝑠𝑞}𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁⋅𝑎₁𝛼^{𝑠𝑝⌊𝛽𝑛⌋+𝑠𝑞}𝑎+𝑂^{1𝑠𝑝𝑛}𝛼^{𝑠𝑝𝑛+𝑝𝑛}𝑑^𝑝𝑛=(−1)^{𝑠𝑝𝑛+𝑠𝑞}⋅𝛼^𝑠𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁⋅𝑎₁𝛼^{𝑠𝑝𝑛+𝑠𝑞}𝑎+𝑂^{1𝑠𝑝𝑛}𝛼^𝑠𝑝𝑛⋅𝑎₁^{𝑠𝑝⌊𝛽𝑛⌋−𝑠𝑝𝑛}𝛼^{𝑠𝑝⌊𝛽𝑛⌋−𝑠𝑝𝑛}𝑎+𝑂^{1𝑠𝑝𝑛}𝛼^𝑠𝑝𝑛⋅1𝛼^𝑝𝑛𝑑^𝑝𝑛=(−1)^{𝑠𝑝𝑛+𝑠𝑞}⋅𝛼^𝑠𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁⋅𝑎₁𝛼^{𝑠𝑝𝑛+𝑠𝑞}𝑎+𝑂^{1𝑠𝑝𝑛}𝛼^𝑠𝑝𝑛⋅ℎ^𝑝, (34) where ℎ=max{(𝑎₁^{𝑠⌊𝛽𝑛⌋−𝑠𝑛}/𝛼^{𝑠⌊𝛽𝑛⌋−𝑠𝑛}),(1/𝛼^𝑛𝑑^𝑛)}.
Taking the reciprocal of this expression yields ^{⌊𝛽𝑛⌋}_𝑘=𝑛−𝑎₁^{𝑠𝑝𝑘+𝑠𝑞}𝑢^𝑠_{𝑝𝑘+𝑞}⁻¹=(−1)^{𝑠𝑝𝑛+𝑠𝑞}⋅𝑐^𝑠𝛼^𝑠−𝑎^𝑠₁𝛼^𝑠⋅𝛼𝑎₁^{𝑠𝑝𝑛+𝑠𝑞}⋅1+𝑂ℎ^𝑝=𝑢^𝑠_{𝑝𝑛+𝑞}𝑎₁^{𝑠𝑝𝑛+𝑠𝑞}+𝑢^𝑠_{𝑝𝑛−𝑝+𝑞}𝑎₁^{𝑠𝑝𝑛+𝑠𝑞−𝑠𝑝}𝛼+𝑂^𝑠𝑝𝑛𝑎₁^𝑠𝑝𝑛⋅ℎ^𝑝. (35)
Case  1. If ℎ=(𝑎₁^{𝑠⌊𝛽𝑛⌋−𝑠𝑛})/(𝛼^{𝑠⌊𝛽𝑛⌋−𝑠𝑛}), then for any real number 𝛽>2 and positive integer 𝑠 we have 𝛼^𝑠𝑛𝑎₁^𝑠𝑛⋅ℎ^𝑝=𝛼^𝑠𝑛𝑎₁^𝑠𝑛⋅𝑎₁^{𝑠𝑝⌊𝛽𝑛⌋−𝑠𝑝𝑛}𝛼^{𝑠𝑝⌊𝛽𝑛⌋−𝑠𝑝𝑛}=𝑎₁𝛼^{𝑠𝑝⌊𝛽𝑛⌋−𝑠𝑝𝑛−𝑠𝑛}<1. (36)
Case  2. If ℎ=(1/𝛼^𝑛𝑑^𝑛), for any positive integer 𝑎₁≥2, 1<(𝛼/𝑎₁)<𝛼𝑑 holds. Then for any positive integer 𝑠 with 1≤𝑠<log_(𝛼/𝑎1)𝛼^𝑝𝑑^𝑝, (37) we have 𝛼^𝑠𝑛𝑎₁^𝑠𝑛⋅ℎ^𝑝=𝛼^{𝑠𝑛−𝑝𝑛}𝑎₁^𝑠𝑛𝑑^𝑝𝑛=𝛼^𝑠−𝑝𝑎^𝑠₁𝑑^𝑝𝑛<1. (38)
In both cases, it follows that for any real number 𝛽>2 and positive integer 1≤𝑠<⌊log_(𝛼/𝑎1)𝛼^𝑝𝑑^𝑝⌋, there exists 𝑛≥𝑛₁₀ sufficiently large so that the modulus of the last error term of identity (35) becomes less than 1/2. This completes the proof of Theorem 13(c).

Conflict of Interests

The authors declare that there is no conflict interests regarding the publication of this paper.

Acknowledgments

The authors express their gratitude to the referee for very helpful and detailed comments. This work is supported by the N.S.F. (11371291), the S.R.F.D.P. (20136101110014), the N.S.F. (2013JZ001) of Shaanxi Province, and the G.I.C.F. (YZZ12062) of NWU, China.