On the Difference Equation 𝑥_𝑛=𝑎_𝑛𝑥_𝑛−𝑘/(𝑏_𝑛+𝑐_𝑛𝑥_𝑛−1⋯𝑥_𝑛−𝑘)

Abstract

The behavior of well-defined solutions of the difference equation 𝑥_𝑛=𝑎_𝑛𝑥_𝑛−𝑘/(𝑏_𝑛+𝑐_𝑛𝑥_𝑛−1⋯𝑥_𝑛−𝑘),𝑛∈ℕ₀, where 𝑘∈ℕ is fixed, the sequences 𝑎_𝑛, 𝑏_𝑛 and 𝑐_𝑛 are real, (𝑏_𝑛,𝑐_𝑛)≠(0,0), 𝑛∈ℕ₀, and the initial values 𝑥_−𝑘,…,𝑥₋₁ are real numbers, is described.

1. Introduction

Recently there has been a huge interest in studying nonlinear difference equations and systems (see, e.g., [1–33] and the references therein). Here we study the difference equation 𝑥_𝑛=𝑥_𝑛−𝑘𝑏_𝑛+𝑐_𝑛𝑥_𝑛−1⋯𝑥_𝑛−𝑘,𝑛∈ℕ₀, (1.1) where 𝑘∈ℕ is fixed, the sequences 𝑏_𝑛 and  𝑐_𝑛 are real, (𝑏_𝑛,𝑐_𝑛)≠(0,0), 𝑛∈ℕ₀, and the initial values 𝑥_−𝑘,…,𝑥₋₁ are real numbers. Equation (1.1) is a particular case of the equation 𝑥_𝑛=𝑎_𝑛𝑥_𝑛−𝑘𝑏̂_𝑛+𝑐_𝑛̂𝑥_𝑛−1⋯𝑥_𝑛−𝑘,𝑛∈ℕ₀, (1.2) with real sequences 𝑎_𝑛, 𝑏̂_𝑛 and 𝑐_𝑛̂. For 𝑎_𝑛=0, 𝑛∈ℕ₀, the equation is trivial, and, for 𝑎_𝑛≠0,𝑛∈ℕ₀, it is reduced to equation (1.1) with 𝑏_𝑛=𝑏̂_𝑛/𝑎_𝑛 and 𝑐_𝑛=𝑐_𝑛̂/𝑎_𝑛.

Equation 𝑥_𝑛=𝑥_𝑛−𝑘𝑏+𝑐𝑥_𝑛−1⋯𝑥_𝑛−𝑘,𝑛∈ℕ₀, (1.3) where 𝑏,𝑐∈ℝ, which was treated in [32], is a particular case of equation (1.1).

As in [32], here, we employ our idea of using a change of variables in equation (1.1) which extends the one in our paper [21] and is later also used, for example, in [4]. For similar methods see also [22, 25]. Equation (1.3) in the case 𝑘=2 was also studied in [1, 2], in a different way. The case when the sequences 𝑏_𝑛 and 𝑐_𝑛 are two-periodic was studied in [31] (some related results are also announced in talk [3]). For related symmetric systems of difference equations, see [27, 29]. For some other recent results on difference equations and systems which can be solved, see, for example, [6, 7, 20–22, 30, 31, 33]. Some classical results can be found, for example, in [11].

Equation (1.1) is a particular case of the equation 𝑦_𝑛=𝑓𝑦_𝑛−1,…,𝑦_𝑛−𝑘,𝑛𝑦_𝑛−𝑘,𝑛∈ℕ₀, (1.4) where 𝑓∶ℝ^𝑘+1→ℝ is a continuous function. Numerous particular cases of (1.4) have been investigated, for example, in [9, 21, 23]. In this paper we adopt the customary notation ∏^𝑘_𝑖=𝑘+1𝑔_𝑖=1 and ∑^𝑘_𝑖=𝑘+1𝑔_𝑖=0.

2. Case 𝑐_𝑛=0,𝑛∈ℕ₀

Here we consider the case 𝑐_𝑛=0,𝑛∈ℕ₀. In this case equation (1.1) becomes 𝑥_𝑛=𝑥_𝑛−𝑘𝑏_𝑛,𝑛∈ℕ₀, (2.1)𝑏_𝑛≠0,𝑛∈ℕ₀, from which it follows that for each 𝑖∈{1,…,𝑘}𝑥_{𝑘𝑚−𝑖}=𝑥_−𝑖∏^𝑚_𝑗=1𝑏_{𝑘𝑗−𝑖},𝑚∈ℕ₀. (2.2) Using formula (2.2) the following theorem can be easily proved.

Theorem 2.1. Consider equation (1.1) with 𝑐_𝑛=0, 𝑏_𝑛≠0,𝑛∈ℕ₀. Then the following statements are true: (a) if liminf_𝑚→∞𝑏_{𝑘𝑚−𝑖}=𝑝_𝑖>1, (2.3)for some 𝑖∈{1,…,𝑘}, then 𝑥_{𝑘𝑚−𝑖}→0 as 𝑚→∞;(b) if for each 𝑖∈{1,…,𝑘}, the limits 𝑝_𝑖 in (2.3) are greater than 1, then 𝑥_𝑛→0 as 𝑛→∞;(c) if 𝑏_{𝑘𝑚−𝑖}=1, for every 𝑚∈ℕ and for some 𝑖∈{1,…,𝑘}, then 𝑥_{𝑘𝑚−𝑖}=𝑥_−𝑖, 𝑚∈ℕ₀; (d) if 𝑏_{𝑘𝑚−𝑖}=−1, for every 𝑚∈ℕ and for some 𝑖∈{1,…,𝑘}, then 𝑥_{𝑘𝑚−𝑖}=(−1)^𝑚𝑥_−𝑖, 𝑚∈ℕ₀;(e) if limsup_𝑚→∞𝑏_{𝑘𝑚−𝑖}=𝑞_𝑖∈[0,1), (2.4)and 𝑥_−𝑖≠0, for some 𝑖∈{1,…,𝑘}, then |𝑥_{𝑘𝑚−𝑖}|→∞, as 𝑚→∞;(f) if for each 𝑖∈{1,…,𝑘}, the limits 𝑞_𝑖 in (2.4) belong to the interval [0,1) and x_−𝑖≠0, then |𝑥_𝑛|→∞ as 𝑛→∞.

3. Case 𝑏_𝑛=0,𝑛∈ℕ₀

In this section we consider the case 𝑏_𝑛=0,𝑛∈ℕ₀. Note that in this case equation (1.1) becomes 𝑥_𝑛=𝑥_𝑛−𝑘𝑐_𝑛𝑥_𝑛−1⋯𝑥_{𝑛−𝑘+1}𝑥_𝑛−𝑘,𝑛∈ℕ₀, (3.1) where 𝑐_𝑛≠0,𝑛∈ℕ₀. If 𝑥_𝑛 is a well-defined solution of equation (3.1) (i.e., a solution with initial values 𝑥_−𝑖≠0, 𝑖=1,…,𝑘, which implies 𝑥_𝑛≠0, 𝑛∈ℕ₀), then 𝑥_𝑛=1𝑐_𝑛𝑥_𝑛−1𝑥_𝑛−2⋯𝑥_{𝑛−𝑘+1}=𝑐_𝑛−1𝑥_𝑛−2⋯𝑥_𝑛−𝑘𝑐_𝑛𝑥_𝑛−2⋯𝑥_{𝑛−𝑘+1}=𝑐_𝑛−1𝑐_𝑛𝑥_𝑛−𝑘,𝑛∈ℕ. (3.2)

Hence for each 𝑖∈{0,1,…,𝑘−1}𝑥_{𝑘𝑚−𝑖}=𝑥^𝑚_{−𝑖𝑗=1}𝑐_{𝑘𝑗−𝑖−1}𝑐_{𝑘𝑗−𝑖},𝑚∈ℕ₀. (3.3)

Using formula (3.3) we easily prove the next theorem.

Theorem 3.1. Consider equation (1.1) with 𝑏_𝑛=0, 𝑐_𝑛≠0, 𝑛∈ℕ₀. Then the following statements are true: (a) if liminf_𝑚→∞𝑐_{𝑘𝑚−𝑖}𝑐_{𝑘𝑚−𝑖−1}=𝑝_𝑖>1, (3.4)for some 𝑖∈{0,1,…,𝑘−1}, then 𝑥_{𝑘𝑚−𝑖}→0 as 𝑚→∞;(b) if, for each 𝑖∈{0,1,…,𝑘−1}, the limits 𝑝_𝑖 in (3.4) are greater than 1, then 𝑥_𝑛→0 as 𝑛→∞;(c) if 𝑐_{𝑘𝑚−𝑖−1}=𝑐_{𝑘𝑚−𝑖}, for every 𝑚∈ℕ and for some 𝑖∈{0,1,…,𝑘−1}, then 𝑥_{𝑘𝑚−𝑖}=𝑥_−𝑖, 𝑚∈ℕ₀; (d) if 𝑐_{𝑘𝑚−𝑖−1}=−𝑐_{𝑘𝑚−𝑖}, for every 𝑚∈ℕ and for some 𝑖∈{0,1,…,𝑘−1}, then 𝑥_{𝑘𝑚−𝑖}=(−1)^𝑚𝑥_−𝑖, 𝑚∈ℕ₀;(e) if limsup_𝑚→∞𝑐_{𝑘𝑚−𝑖}𝑐_{𝑘𝑚−𝑖−1}=𝑞_𝑖∈[0,1), (3.5)and 𝑥_−𝑖≠0 for some 𝑖∈{0,1,…,𝑘−1}, then |𝑥_{𝑘𝑚−𝑖}|→∞, as 𝑚→∞;(f) if, for each 𝑖∈{0,1,…,𝑘−1}, 𝑥_−𝑖≠0 and the limits 𝑞_𝑖 in (3.5) belong to the interval [0,1), then |𝑥_𝑛|→∞ as 𝑛→∞.

4. Case 𝑏_𝑛≠0 and 𝑐_𝑛≠0

The case when 𝑏_𝑛≠0 and 𝑐_𝑛≠0 for every 𝑛∈ℕ₀ is considered in this section.

If 𝑥_−𝑖0=0 for some 𝑖₀∈{1,…,𝑘}, then from (1.1) we have that 𝑥_{𝑘𝑚−𝑖0}=0,for𝑚∈ℕ₀. (4.1) From (4.1) and (1.1) we have that for each 𝑖∈{1,…,𝑘}⧵{𝑖₀}𝑥_{𝑘𝑚−𝑖}=𝑥_{𝑘(𝑚−1)−𝑖}𝑏_{𝑘𝑚−𝑖}=𝑥_−𝑖∏^𝑚_𝑗=1𝑏_{𝑘𝑗−𝑖},𝑚∈ℕ₀. (4.2) From (4.1) we see that, for 𝑖=𝑖₀, (4.2) also holds. Hence Theorem 2.1 can be applied in this case. Note that if 𝑥_𝑛=0 for some 𝑛∈ℕ₀, then (1.1) implies that there is an 𝑖₀∈{1,…,𝑘} such that 𝑥_−𝑖0=0, and by the previous consideration we have that (4.2) also holds.

If 𝑥_−𝑖≠0, for each 𝑖∈{1,…,𝑘}, then for every well-defined solution we have 𝑥_𝑛≠0 for 𝑛≥−𝑘 (note that there are solutions which are not well defined, that is, those for which 𝑥_𝑛−1⋯𝑥_𝑛−𝑘=−𝑏_𝑛/𝑐_𝑛, for some 𝑛∈ℕ₀).

Multiplying equation (1.1) by 𝑥_𝑛−1⋯𝑥_{𝑛−𝑘+1} and using the transformation 𝑦_𝑛=1𝑥_𝑛𝑥_𝑛−1⋯𝑥_{𝑛−𝑘+1},𝑛≥−1, (4.3) we obtain equation 𝑦_𝑛=𝑏_𝑛𝑦_𝑛−1+𝑐_𝑛,𝑛∈ℕ₀. (4.4) Note that from (4.3), for every well-defined solution (𝑥_𝑛)_{𝑛≥−𝑘} of equation (1.1) such that 𝑥_−𝑖≠0, for each 𝑖∈{1,…,𝑘}, it follows that 𝑦_𝑛≠0, 𝑛≥−1.

Since 𝑏_𝑛≠0, 𝑛∈ℕ₀, we have that 𝑦_𝑛=^𝑛_𝑖=0𝑏_𝑖𝑦₋₁+^𝑛_𝑗=0𝑐_𝑗∏^𝑗_𝑖=0𝑏_𝑖,𝑛∈ℕ₀. (4.5)

From (4.3) and (4.5) we have that 𝑥_𝑛=1𝑦_𝑛𝑥_𝑛−1⋯𝑥_{𝑛−𝑘+1}=𝑦_𝑛−1𝑦_𝑛𝑥_𝑛−𝑘=𝑦₋₁+∑^𝑛−1_𝑗=0𝑐_𝑗/∏^𝑗_𝑖=0𝑏_𝑖𝑏_𝑛𝑦₋₁+∑^𝑛_𝑗=0𝑐_𝑗/∏^𝑗_𝑖=0𝑏_𝑖𝑥_𝑛−𝑘, (4.6) for every 𝑛∈ℕ₀.

Hence, from (4.6), we obtain that 𝑥_{𝑚𝑘−𝑖}=𝑥^𝑚_{−𝑖𝑙=1}1/𝛼+∑^{𝑘𝑙−𝑖−1}_𝑗=0𝑐_𝑗/∏^𝑗_𝑖=0𝑏_𝑖𝑏_{𝑘𝑙−𝑖}1/𝛼+∑^{𝑘𝑙−𝑖}_𝑗=0𝑐_𝑗/∏^𝑗_𝑖=0𝑏_𝑖, (4.7) for every 𝑚∈ℕ₀ and each 𝑖=1,2,…,𝑘, where 𝛼=^𝑘_𝑙=1𝑥_−𝑙. (4.8)

5. Case 𝑏_𝑛=1,𝑛∈ℕ₀

Here we consider the case 𝑏_𝑛=1, 𝑛∈ℕ₀. In this case, from (4.7) we have that for each 𝑖∈{1,…,𝑘}𝑥_{𝑚𝑘−𝑖}=𝑥^𝑚_{−𝑖𝑙=1}1+𝛼∑^{𝑘𝑙−𝑖−1}_𝑗=0𝑐_𝑗1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0𝑐_𝑗,𝑚∈ℕ₀. (5.1) Note that this formula includes also the case when 𝑥_−𝑖0=0 for some 𝑖₀∈{1,…,𝑘}.

Now we formulate and prove a result in this case by using formula (5.1).

Theorem 5.1. Consider equation (1.1) with 𝑏_𝑛=1, 𝑛∈ℕ₀, sign𝑐_𝑛=sign𝑐₀, 𝑛∈ℕ, 𝛼≠0, and 𝛼^𝑛_𝑗=0𝑐_𝑗≠−1,𝑛∈ℕ₀. (5.2) Then the following statements hold: (a) if for some 𝑖∈{1,…,𝑘}^∞_𝑙=1𝛼𝑐_{𝑘𝑙−𝑖}1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0𝑐_𝑗=+∞,lim_𝑙→∞𝛼𝑐_{𝑘𝑙−𝑖}1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0𝑐_𝑗=0, (5.3)(5.4)then 𝑥_{𝑚𝑘−𝑖}→0 as 𝑚→∞;(b) if (5.3) and (5.4) hold for every 𝑖∈{1,…,𝑘}, then 𝑥_𝑛→0 as 𝑛→∞;(c) if for some 𝑖∈{1,…,𝑘} the sum ^∞_𝑙=1𝛼𝑐_{𝑘𝑙−𝑖}1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0𝑐_𝑗 (5.5)converges, then the sequence 𝑥_{𝑚𝑘−𝑖} is also convergent;(d) if the sum in (5.5) is finite for every 𝑖∈{1,…,𝑘}, then the sequences 𝑥_{𝑘𝑚−𝑖} are convergent.

Proof. Let (𝑥_𝑛)_{𝑛≥−𝑘} be a solution of equation (1.1). Using condition sign𝑐_𝑛=sign𝑐₀, 𝑛∈ℕ, it is easy to see that if (5.4) holds for some 𝑖∈{1,…,𝑘}, there is an 𝑚₀∈ℕ such that for 𝑗≥𝑚₀+1 the terms in the product in (5.1) are positive and that the following asymptotic formula ln(1+𝑥)=𝑥+𝑂𝑥² (5.6) can be used with 𝑥 being the fraction in the limit (5.4). From (5.1) and (5.6) we have that 𝑥_{𝑘𝑚−𝑖}=𝑥_−𝑖^𝑚_𝑙=11+𝛼∑^{𝑘𝑙−𝑖−1}_𝑗=0𝑐_𝑗1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0𝑐_𝑗=𝑥_−𝑖𝑐𝑚₀exp^𝑚_𝑙=𝑚0+1ln1+𝛼∑^{𝑘𝑙−𝑖−1}_𝑗=0𝑐_𝑗1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0𝑐_𝑗=𝑥_−𝑖𝑐𝑚₀exp^𝑚_𝑙=𝑚0+1ln1−𝛼𝑐_{𝑘𝑙−𝑖}1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0𝑐_𝑗=𝑥_−𝑖𝑐𝑚₀exp−^𝑚_𝑙=𝑚0+1𝛼𝑐_{𝑘𝑙−𝑖}(1+𝑜(1))1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0𝑐_𝑗, (5.7) where 𝑐𝑚₀=^𝑚0_𝑙=11+𝛼∑^{𝑘𝑙−𝑖−1}_𝑗=0𝑐_𝑗1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0𝑐_𝑗. (5.8)
Using formula (5.7), the assumptions regarding the sum ∑^∞_𝑗=𝑚0+1(𝛼𝑐_{𝑘𝑙−𝑖}/(1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0𝑐_𝑗)) and the comparison test for the series whose terms are of eventually the same sign, the results in the theorem easily follow.

6. Case 𝑏_𝑛=−1,𝑛∈ℕ₀

Here we consider the case 𝑏_𝑛=−1, 𝑛∈ℕ₀. In this case from (4.7) we have 𝑥_{𝑚𝑘−𝑖}=(−1)^𝑚𝑥^𝑚_{−𝑖𝑙=1}1+𝛼∑^{𝑘𝑙−𝑖−1}_𝑗=0(−1)^𝑗+1𝑐_𝑗1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0(−1)^𝑗+1𝑐_𝑗, (6.1) for every 𝑚∈ℕ₀ and each 𝑖=1,2,…,𝑘, where 𝛼 is defined by (4.8).

Theorem 6.1. Consider equation (1.1) with 𝛼≠0, 𝑏_𝑛=−1, 𝑛∈ℕ₀, and 𝛼^𝑛_𝑗=0(−1)^𝑗+1𝑐_𝑗≠−1,𝑛∈ℕ₀. (6.2) Then the following statements hold: (a) if for some 𝑖∈{1,…,𝑘}^∞_𝑙=1𝛼(−1)^{𝑘𝑙−𝑖+1}𝑐_{𝑘𝑙−𝑖}1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0(−1)^𝑗+1𝑐_𝑗=+∞,lim_𝑙→∞𝛼(−1)^{𝑘𝑙−𝑖+1}𝑐_{𝑘𝑙−𝑖}1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0(−1)^𝑗+1𝑐_𝑗=0,_∞𝑙=1𝑐²_{𝑘𝑙−𝑖}1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0(−1)^𝑗+1𝑐_𝑗²<+∞, (6.3)(6.4)(6.5)then 𝑥_{𝑚𝑘−𝑖}→0 as 𝑚→∞;(b) if for every 𝑖∈{1,…,𝑘}, (6.3), (6.4), and (6.5) hold, then 𝑥_𝑛→0 as 𝑛→∞;(c) if for some 𝑖∈{1,…,𝑘}^∞_𝑙=1𝛼(−1)^{𝑘𝑙−𝑖+1}𝑐_{𝑘𝑙−𝑖}1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0(−1)^𝑗+1𝑐_𝑗=−∞, (6.6)conditions (6.4) and (6.5) hold, and 𝑥_−𝑖≠0, then |𝑥_{𝑚𝑘−𝑖}|→∞ as 𝑚→∞;(d) if, for every 𝑖∈{1,…,𝑘}, conditions (6.4), (6.5), and (6.6) hold, and 𝑥_−𝑖≠0, 𝑖∈{1,…,𝑘}, then |𝑥_𝑛|→∞ as 𝑛→∞;(e) if for some 𝑖∈{1,…,𝑘} the sum ^∞_𝑙=1𝛼(−1)^{𝑘𝑙−𝑖+1}𝑐_{𝑘𝑙−𝑖}1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0(−1)^𝑗+1𝑐_𝑗 (6.7)converges and condition (6.5) holds, then the sequences 𝑥_{2𝑚𝑘−𝑖} and 𝑥_{(2𝑚+1)𝑘−𝑖} are also convergent;(f) if, for every 𝑖∈{1,…,𝑘} the sum in (6.7) converges and condition (6.5) holds, then the sequences 𝑥_{2𝑘𝑚−𝑗},𝑗=1,…,2𝑘 are convergent.

Proof. Let (𝑥_𝑛)_{𝑛≥−𝑘} be a solution of equation (1.1). By (6.4) we see that irrespectively on 𝑖∈{1,…,𝑘}, there is an 𝑚₁∈ℕ such that for 𝑗≥𝑚₁+1 the terms in the product in (6.1) belong to the interval (1/2,3/2) and that asymptotic formulae ()𝑥ln1+𝑥=𝑥−²2+𝑂𝑥³ (6.8) can be used with −𝑥 being the fraction in (6.4). From this and (6.1) we have that 𝑥_{𝑘𝑚−𝑖}=𝑥_−𝑖^𝑚_𝑙=11+𝛼∑^{𝑘𝑙−𝑖−1}_𝑗=0(−1)^𝑗+1𝑐_𝑗1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0(−1)^𝑗+1𝑐_𝑗=𝑥_−𝑖𝑐₁𝑚₁exp^𝑚_𝑙=𝑚1+1ln1+𝛼∑^{𝑘𝑙−𝑖−1}_𝑗=0(−1)^𝑗+1𝑐_𝑗1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0(−1)^𝑗+1𝑐_𝑗=𝑥_−𝑖𝑐₁𝑚₁exp^𝑚_𝑙=𝑚1+1𝛼()ln1−−1^{𝑘𝑙−𝑖+1}𝑐_{𝑘𝑙−𝑖}1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0(−1)^𝑗+1𝑐_𝑗=𝑥_−𝑖𝑐₁𝑚₁exp−^𝑚_𝑙=𝑚1+1𝛼(−1)^{𝑘𝑙−𝑖+1}𝑐_{𝑘𝑙−𝑖}1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0(−1)^𝑗+1𝑐_𝑗+𝛼²𝑐²_{𝑘𝑙−𝑖}(1+𝑜(1))21+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0(−1)^𝑗+1𝑐_𝑗², (6.9) where 𝑐₁𝑚₁=^𝑚1_𝑙=11+𝛼∑^{𝑘𝑙−𝑖−1}_𝑗=0(−1)^𝑗+1𝑐_𝑗1+𝛼∑^{𝑘𝑙−𝑖}_𝑗=0(−1)^𝑗+1𝑐_𝑗. (6.10)
Using formula (6.9), the assumptions of the theorem and some well-known convergence tests for series, the results in (a)–(f) easily follow.

7. Case 𝑏_𝑛=𝑏_𝑛+𝑘,𝑐_𝑛=𝑐_𝑛+𝑘,𝑛∈ℕ₀

In this section we consider equation (1.1) for the case 𝑏_𝑛=𝑏_𝑛+𝑘, 𝑐_𝑛=𝑐_𝑛+𝑘, 𝑛∈ℕ₀, that is, when the sequences 𝑏_𝑛 and 𝑐_𝑛 are 𝑘-periodic.

First we show the existence of 𝑘-periodic solutions of equation (4.4). If 𝑦₀,𝑦₁,…,𝑦_𝑘−1 (7.1) is such a solution, then we have that 𝑦₁=𝑏₁𝑦₀+𝑐₁,𝑦₂=𝑏₂𝑦₁+𝑐₂,…,𝑦₀=𝑏_𝑘𝑦_𝑘−1+𝑐_𝑘. (7.2) By successive elimination, or by Kronecker theorem (note that system (7.2) is linear), we get 𝑦_𝑖=∑^𝑘−1_𝑗=0𝑐_{𝜎[𝑗](𝑖)}∏^𝑗−1_𝑠=0𝑏_{𝜎[𝑠](𝑖)}1−∏^𝑘_𝑗=1𝑏_𝑗,𝑖=1,𝑘, (7.3) if ∏^𝑘_𝑗=1𝑏_𝑗≠1, where 𝜎 is the permutation defined by 𝜎(𝑖)=𝑖−1,𝑖=2,𝑘,𝜎(1)=𝑘, (7.4) and 𝜎^[𝑖]=𝜎∘𝜎^[𝑖−1],𝜎^[0]=Id, where Id denotes the identity.

It is easy to see that (4.4) along with 𝑘 periodicity of sequences 𝑏_𝑛 and 𝑐_𝑛 implies 𝑦_{𝑘𝑚+𝑖}=^𝑘_𝑗=1𝑏_𝑗𝑦_{𝑘(𝑚−1)+𝑖}+^𝑘−1_𝑗=0𝑐_{𝜎[𝑗](𝑖)}^𝑗−1_𝑠=0𝑏_{𝜎[𝑠](𝑖)}, (7.5) for every 𝑚∈ℕ₀ and 𝑖∈{1,2,…,𝑘}, such that 𝑘(𝑚−1)+𝑖≥−1.

Since (7.5) is a linear first-order difference equation, we have that when ∏^𝑘_𝑗=1𝑏_𝑗≠1, its general solution is 𝑦_{𝑘𝑚+𝑖}=^𝑘_𝑗=1𝑏_𝑗^𝑚𝑦_𝑖+1−∏^𝑘_𝑗=1𝑏_𝑗^𝑚1−∏^𝑘_𝑗=1𝑏_𝑗^𝑘−1_𝑗=0𝑐_{𝜎[𝑗](𝑖)}^𝑗−1_𝑠=0𝑏_{𝜎[𝑠](𝑖)}. (7.6)

By letting 𝑚→∞ in (7.6) we obtain the following corollary.

Corollary 7.1. Consider equation (4.4) with 𝑏_𝑛=𝑏_𝑛+𝑘, 𝑐_𝑛=𝑐_𝑛+𝑘, 𝑛∈ℕ₀. Assume that ^𝑘_𝑗=1𝑏_𝑗<1. (7.7) Then for every solution 𝑦_𝑛 of the equation we have that lim_𝑚→∞𝑦_{𝑘𝑚+𝑖}=𝑦_𝑖, (7.8) for every 𝑖∈{1,2,…,𝑘}, that is, 𝑦_𝑛 converges to the 𝑘-periodic solution in formula (7.3).

Let 𝐿_𝑖∶=^𝑘−1_𝑗=0𝑐_{𝜎[𝑗](𝑖)}^𝑗−1_𝑠=0𝑏_{𝜎[𝑠](𝑖)},𝑖=1,𝑘,𝑞∶=^𝑘_𝑗=1𝑏_𝑗. (7.9)

From now on we will use the following convention: if 𝑖,𝑗∈ℕ₀, then we regard that 𝐿_𝑗=𝐿_𝑖, if 𝑖≡𝑗(mod𝑘). Also if a sequence (𝑚_𝑗)_𝑗∈ℕ0 is defined by the relation 𝑚_𝑗=𝑓(𝐿_𝑗), where 𝑓 is a real function, then we will assume that 𝑚_𝑗=𝑚_𝑖, if 𝑖≡𝑗(mod𝑘).

Using (7.6) and notation (7.9) in the relation 𝑥_𝑛=(𝑦_𝑛−1/𝑦_𝑛)𝑥_𝑛−𝑘 (see (4.6)), for the case 𝑞≠1, we have that 𝑥_{𝑘𝑚+𝑖}=𝑥^𝑚_{𝑖−𝑘𝑗=0}𝑦_𝑖−1−𝐿_𝑖−1/1−𝑞𝑞^𝑗+𝐿_𝑖−1/1−𝑞𝑦_𝑖−𝐿_𝑖/1−𝑞𝑞^𝑗+𝐿_𝑖/1−𝑞=𝑥^𝑚_{𝑖−𝑘𝑗=0}𝐿_𝑖−1𝐿_𝑖⋅1+1−𝑞𝑦_𝑖−1/𝐿_𝑖−1−1𝑞^𝑗1+1−𝑞𝑦_𝑖/𝐿_𝑖−1𝑞^𝑗, (7.10) for every 𝑚∈ℕ₀ and each 𝑖∈{2,…,𝑘}, and 𝑥_𝑘𝑚+1=𝑥^𝑚_{1−𝑘𝑗=0}𝑦_𝑘−𝐿_𝑘/1−𝑞𝑞^𝑗−1+𝐿_𝑘/1−𝑞𝑦₁−𝐿₁/1−𝑞𝑞^𝑗+𝐿₁/1−𝑞=𝑥^𝑚_{1−𝑘𝑗=0}𝐿_𝑘𝐿₁⋅1+1−𝑞𝑦_𝑘/𝐿_𝑘−1𝑞^𝑗−11+1−𝑞𝑦₁/𝐿₁−1𝑞^𝑗. (7.11)

Now we present some results, which are applications of formulae (7.10) and (7.11).

7.1. Case 𝑞=−1

If 𝑞=−1, then by (7.5) we get 𝑦_{𝑘𝑚+𝑖}=−𝑦_{𝑘(𝑚−1)+𝑖}+𝐿_𝑖=𝐿_𝑖−𝐿_𝑖−𝑦_{𝑘(𝑚−2)+𝑖}=𝑦_{𝑘(𝑚−2)+𝑖},𝑚∈ℕ, (7.12) for 𝑘(𝑚−2)+𝑖≥−1; that is, 𝑦_{𝑘𝑚+𝑖} is two-periodic for each 𝑖∈{1,…,𝑘}. Hence 𝑦_𝑛 is a 2𝑘-periodic solution of equation (4.4), in this case.

Hence from the relation 𝑥_𝑛=(𝑦_𝑛−1/𝑦_𝑛)𝑥_𝑛−𝑘 (see (4.6)), for each 𝑖∈{1,2,…,𝑘}, we have 𝑥_{𝑘𝑚−𝑖}=𝑦_{𝑘𝑚−𝑖−1}𝑦_{𝑘𝑚−𝑖}𝑥_{𝑘𝑚−𝑖−𝑘}=𝑦_{𝑘𝑚−𝑖−1}𝑦_{𝑘𝑚−𝑖}𝑦_{𝑘(𝑚−1)−𝑖−1}𝑦_{𝑘(𝑚−1)−𝑖}𝑥_{𝑘𝑚−𝑖−2𝑘}, (7.13) for 𝑘(𝑚−1)≥𝑖.

From (7.13) and by 2𝑘 periodicity of 𝑦_𝑛, we get 𝑥_{2𝑘𝑙+𝑗}=𝑦_𝑗−1𝑦_{𝑗+𝑘−1}𝑦_𝑗𝑦_𝑗+𝑘^𝑙𝑥_𝑗,𝑙∈ℕ₀, (7.14) for each 𝑗∈{−𝑘+1,…,−1,0,1,…,𝑘}.

From (7.14), the behavior of solutions of equation (1.1), in this case, easily follows. For example, if 𝑝_𝑗𝑦∶=^𝑗−1𝑦_{𝑗+𝑘−1}𝑦_𝑗𝑦_𝑗+𝑘=1, (7.15) for each 𝑗∈{−𝑘+1,…,−1,0,1,…,𝑘}, then the solution (𝑥_𝑛)_𝑛≥𝑘 of (1.1) is 2𝑘-periodic.

7.2. Case 𝑞=1

If 𝑞=1 and 𝛼≠0, then from (7.5) we obtain 𝑦_{𝑘𝑚+𝑖}=𝑦_{𝑘(𝑚−1)+𝑖}+𝐿_𝑖,𝑚∈ℕ₀,𝑖=1,𝑘, (7.16) when 𝑘(𝑚−1)+𝑖≥−1, from which along with (4.6), it follows that 𝑥_{𝑘𝑚+𝑖}=𝑥_𝑖^𝑚_𝑗=1𝑦_𝑖−1+𝑗𝐿_𝑖−1𝑦_𝑖+𝑗𝐿_𝑖,𝑚∈ℕ,𝑖=𝑥2,𝑘,_𝑘𝑚+1=𝑥₁^𝑚_𝑗=1𝑦_𝑘+𝑗−1𝐿_𝑘𝑦₁+𝑗𝐿₁,𝑚∈ℕ. (7.17)

Corollary 7.2. Consider equation (1.1). Let 𝑞=1, 𝛼≠0, and 𝑝_𝑖∶=𝐿_𝑖−1/𝐿_𝑖, 𝑖∈{1,…,𝑘}. Then the following statements hold true. (a) If |𝑝_𝑖|<1, for some 𝑖∈{1,…,𝑘}, then 𝑥_{𝑘𝑚+𝑖}→0 as 𝑚→∞. (b) If |𝑝_𝑖|>1, or 𝐿_𝑖=0 and 𝐿_𝑖−1≠0, for some 𝑖∈{1,…,𝑘}, then |𝑥_{𝑘𝑚+𝑖}|→∞ as 𝑚→∞, if 𝑥_𝑖≠0. (c) If 𝑝_𝑖=1, for some 𝑖∈{2,…,𝑘}, and (𝑦_𝑖−1−y_𝑖)/𝐿_𝑖>0, then |𝑥_{𝑘𝑚+𝑖}|→∞ as 𝑚→∞, if 𝑥_𝑖≠0. (d) If 𝑝_𝑖=1, for some 𝑖∈{2,…,𝑘}, and (𝑦_𝑖−1−𝑦_𝑖)/𝐿_𝑖<0, then 𝑥_{𝑘𝑚+𝑖}→0 as 𝑚→∞. (e) If 𝑝_𝑖=1, for some 𝑖∈{2,…,𝑘}, and 𝑦_𝑖−1=𝑦_𝑖, then the sequence (𝑥_{𝑘𝑚+𝑖})_𝑚∈ℕ0 is convergent. (f) If 𝑝₁=1, and (𝑦_𝑘−𝐿₁−𝑦₁)/𝐿₁>0, then |𝑥_𝑘𝑚+1|→∞ as 𝑚→∞, if 𝑥₁≠0. (g) If 𝑝₁=1, and (𝑦_𝑘−𝐿₁−𝑦₁)/𝐿₁<0, then 𝑥_𝑘𝑚+1→0 as 𝑚→∞. (h) If 𝑝₁=1, and 𝑦_𝑘=𝐿₁+𝑦₁, then the sequence (𝑥_𝑘𝑚+1)_𝑚∈ℕ0 is convergent.

Proof. The statements in (a) and (b) follow from the facts that lim_𝑗→∞𝑦_𝑖−1+𝑗𝐿_𝑖−1𝑦_𝑖+𝑗𝐿_𝑖=𝑝_𝑖,𝑖∈{2,…,𝑘} (7.18) if 𝐿_𝑖≠0, lim_𝑗→∞𝑦_𝑖−1+𝑗𝐿_𝑖−1𝑦_𝑖+𝑗𝐿_𝑖=+∞,𝑖∈{2,…,𝑘} (7.19) if 𝐿_𝑖=0 and 𝐿_𝑖−1≠0, lim_𝑗→∞𝑦_𝑘+𝑗−1𝐿_𝑘𝑦₁+𝑗𝐿₁=𝑝₁, (7.20) if 𝐿₁≠0, and lim_𝑗→∞𝑦_𝑘+𝑗−1𝐿_𝑘𝑦₁+𝑗𝐿₁=+∞, (7.21) if 𝐿₁=0 and 𝐿_𝑘≠0.
Now assume that 𝑝_𝑖=1 and let (𝑥_𝑛)_{𝑛≥−𝑘} be a solution of equation (1.1). It is easy to see that there is an 𝑚₂∈ℕ such that for 𝑗≥𝑚₂+1 the terms in the products in (7.17) are positive and that the following asymptotic formulae (1+𝑥)⁻¹=1−𝑥+𝑂𝑥²,ln(1+𝑥)=𝑥+𝑂𝑥² (7.22) can be applied with 𝑥=(𝑦_𝑖−1−𝑦_𝑖)/(𝑗𝐿_𝑖), when𝑖∈{2,…,𝑘}or with 𝑥=(𝑦_𝑘−𝐿₁−𝑦₁)/(𝑗𝐿₁). Using these formulae, for the case 𝑖∈{2,…,𝑘}, we have that 𝑥_{𝑘𝑚+𝑖}=𝑥_𝑖^𝑚_𝑗=1𝑦_𝑖−1+𝑗𝐿_𝑖−1𝑦_𝑖+𝑗𝐿_𝑖=𝑥_𝑖𝑐𝑚₂exp^𝑚_𝑗=𝑚2+1𝑦ln^𝑖−1+𝑗𝐿_𝑖−1𝑦_𝑖+𝑗𝐿_𝑖=𝑥_𝑖𝑐𝑚₂exp^𝑚_𝑗=𝑚2+1𝑦ln1+^𝑖−1−𝑦_𝑖𝑗𝐿_𝑖1+𝑂𝑗²=𝑥_𝑖𝑐𝑚₂exp^𝑚_𝑗=𝑚2+1𝑦_𝑖−1−𝑦_𝑖𝑗𝐿_𝑖1+𝑂𝑗², (7.23) where 𝑐𝑚₂=^𝑚2_𝑗=1𝑦_𝑖−1+𝑗𝐿_𝑖−1𝑦_𝑖+𝑗𝐿_𝑖. (7.24)
Letting 𝑚→∞ in (7.23), using the facts that ^𝑚_𝑗=𝑚2+11𝑗⟶+∞as𝑚⟶∞ (7.25) and that the series ∑^∞_𝑗=𝑚2+1𝑂(1/𝑗²) converges, we get statements (c)–(e).
If 𝑝₁=1, that is 𝐿₁=𝐿_𝑘≠0, then by using (7.22) we get 𝑥_𝑘𝑚+1=𝑥₁^𝑚_𝑗=1𝑦_𝑘+𝑗−1𝐿_𝑘𝑦₁+𝑗𝐿₁=𝑥₁𝑑𝑚₂exp^𝑚_𝑗=𝑚2+1𝑦ln^𝑘+𝑗−1𝐿_𝑘𝑦₁+𝑗𝐿₁=𝑥₁𝑑𝑚₂exp^𝑚_𝑗=𝑚2+1𝑦ln1+^𝑘−𝐿₁−𝑦₁𝑗𝐿₁1+𝑂𝑗²=𝑥₁𝑑𝑚₂exp^𝑚_𝑗=𝑚2+1𝑦_𝑘−𝐿₁−𝑦₁𝑗𝐿₁1+𝑂𝑗², (7.26) where 𝑑𝑚₂=^𝑚2_𝑗=1𝑦_𝑘+𝑗−1𝐿_𝑘𝑦₁+𝑗𝐿₁. (7.27) Letting 𝑚→∞ in (7.26), using (7.25) and the fact that the series ∑^∞_𝑗=𝑚2+1𝑂(1/𝑗²) converges, we get statements (f)–(h), as desired.

7.3. Case 𝑞≠±1

If 𝑞≠±1, then from (7.6) we get 𝑦_{𝑘𝑚+𝑖}=𝑞^𝑚𝑠_𝑖+𝑡_𝑖,𝑚∈ℕ₀, (7.28) where 𝑠_𝑖=𝑦_𝑖+𝐿_𝑖𝑞−1,𝑡_𝑖=𝐿_𝑖1−𝑞,𝑖=1,𝑘, (7.29) from (4.6) it follows that 𝑥_{𝑘𝑚+𝑖}=𝑥_𝑖^𝑚_𝑗=1𝑞^𝑗𝑠_𝑖−1+𝑡_𝑖−1𝑞^𝑗𝑠_𝑖+𝑡_𝑖,𝑚∈ℕ₀, (7.30) for 𝑖∈{2,…,𝑘}, and 𝑥_𝑘𝑚+1=𝑥₁𝑦_𝑘𝑞𝑠₁+𝑡₁^𝑚_𝑗=2𝑞^𝑗−1𝑠_𝑘+𝑡_𝑘𝑞^𝑗𝑠₁+𝑡₁,𝑚∈ℕ₀. (7.31) Note that 𝑡_𝑖=𝑡_𝑗, if 𝑖≡𝑗(mod𝑘).

Corollary 7.3. If 0<|𝑞|<1, 𝛼≠0, and 𝑞^𝑗𝑠_𝑖+𝑡_𝑖≠0, for every 𝑗∈ℕ₀ and 𝑖∈{1,…,𝑘}, then the following statements hold true.(a) If |𝑡_𝑖−1|<|𝑡_𝑖|, for some 𝑖∈{1,…,𝑘}, we have that 𝑥_{𝑘𝑚+𝑖}→0 as 𝑚→∞. (b) If |𝑡_𝑖−1|>|𝑡_𝑖|, and 𝑠_𝑖≠0 if 𝑡_𝑖=0 for some 𝑖∈{1,…,𝑘}, we have that |𝑥_{𝑘𝑚+𝑖}|→∞ as 𝑚→∞, if 𝑥_𝑖≠0. (c) If 𝑡_𝑖−1=𝑡_𝑖≠0, for some 𝑖∈{1,…,𝑘}, then 𝑥_{𝑘𝑚+𝑖} is convergent. (d) If 𝑡_𝑖−1=𝑡_𝑖=0, and |𝑠_𝑖−1|<|𝑠_𝑖| for some 𝑖∈{2,…,𝑘}, then |𝑥_{𝑘𝑚+𝑖}|→0 as 𝑚→∞. (e) If 𝑡₁=𝑡_𝑘=0, and |𝑠_𝑘|<|𝑞𝑠₁|, then 𝑥_𝑘𝑚+1→0 as 𝑚→∞. (f) If 𝑡_𝑖−1=𝑡_𝑖=0, and |𝑠_𝑖−1|>|𝑠_𝑖| for some 𝑖∈{2,…,𝑘}, then |𝑥_{𝑘𝑚+𝑖}|→∞ as 𝑚→∞, if 𝑥_𝑖≠0. (g) If 𝑡₁=𝑡_𝑘=0, and |𝑠_𝑘|>|𝑞𝑠₁|, then |𝑥_𝑘𝑚+1|→∞ as 𝑚→∞, if 𝑥₁≠0. (h) If 𝑡_𝑖−1=𝑡_𝑖=0, and 𝑠_𝑖−1=𝑠_𝑖≠0  for some 𝑖∈{2,…,𝑘}, then 𝑥_{𝑘𝑚+𝑖} is constant. (i) If 𝑡₁=𝑡_𝑘=0, and 𝑠_𝑘=𝑞𝑠₁≠0, then 𝑥_𝑘𝑚+1 is constant. (j) If 𝑡_𝑖−1=𝑡_𝑖=0, and 𝑠_𝑖−1=−𝑠_𝑖≠0  for some 𝑖∈{2,…,𝑘}, then 𝑥_{𝑘𝑚+𝑖}=(−1)^𝑚𝑥_𝑖. (k) If 𝑡₁=𝑡_𝑘=0, and 𝑠_𝑘=−𝑞𝑠₁≠0, then 𝑥_𝑘𝑚+1=𝑥₁𝑦_𝑘(−1)^𝑚−1/(𝑞𝑠₁). (l) If 𝑡_𝑖−1=−𝑡_𝑖≠0, for some 𝑖∈{1,…,𝑘}, then the subsequences 𝑥_{2𝑘𝑚+𝑖} and 𝑥_{2𝑘𝑚+𝑘+𝑖} are convergent.

Proof. Since we have that lim_𝑗→∞𝑞^𝑗𝑠_𝑖−1+𝑡_𝑖−1𝑞^𝑗𝑠_𝑖+𝑡_𝑖=𝑡_𝑖−1𝑡_𝑖,𝑖∈{2,…,𝑘} (7.32) when 𝑡_𝑖≠0, lim_𝑗→∞𝑞^𝑗𝑠_𝑖−1+𝑡_𝑖−1𝑞^𝑗𝑠_𝑖+𝑡_𝑖=+∞,𝑖∈{2,…,𝑘} (7.33) when |𝑡_𝑖−1|>𝑡_𝑖=0 and 𝑠_𝑖≠0, lim_𝑗→∞𝑞^𝑗−1𝑠_𝑘+𝑡_𝑘𝑞^𝑗𝑠₁+𝑡₁=𝑡_𝑘𝑡₁, (7.34) when 𝑡₁≠0, and lim_j→∞𝑞^𝑗−1𝑠_𝑘+𝑡_𝑘𝑞^𝑗𝑠₁+𝑡₁=+∞, (7.35) when |𝑡_𝑘|>𝑡₁=0 and 𝑠₁≠0, the statements in (a) and (b) easily follow from (7.30)–(7.35).
(c) If 𝑡_𝑖−1=𝑡_𝑖≠0, then 𝑥_{𝑘𝑚+𝑖}=𝑥_𝑖^𝑚_𝑗=11+𝑞^𝑗𝑠_𝑖−1−𝑠_𝑖𝑡_𝑖+𝑜𝑞^𝑗, (7.36) for 𝑖∈{2,…,𝑘}, and if 𝑡₁=𝑡_𝑘≠0, then 𝑥_𝑘𝑚+1=𝑥₁𝑦_𝑘𝑞𝑠₁+𝑡₁^𝑚_𝑗=21+𝑞^𝑗−1𝑠_𝑘−𝑞𝑠₁𝑡₁+𝑜𝑞^𝑗 (7.37) from which the statement in (c) easily follows.
(d)–(k) If 𝑡_𝑖−1=𝑡_𝑖=0, then 𝑥_{𝑘𝑚+𝑖}=𝑥_𝑖^𝑚_𝑗=1𝑠_𝑖−1𝑠_𝑖, (7.38) for 𝑖∈{2,…,𝑘}, and if 𝑡₁=𝑡_𝑘=0, then 𝑥_𝑘𝑚+1=𝑥₁𝑦_𝑘𝑞𝑠₁^𝑚_𝑗=2𝑠_𝑘𝑞𝑠₁ (7.39) from which the statements in (d)–(k) easily follow.
(l) If 𝑡_𝑖−1=−𝑡_𝑖≠0, then we have that 𝑥_{𝑘𝑚+𝑖}=𝑥_𝑖^𝑚_𝑗=1−1−𝑞^𝑗𝑠_𝑖−1+𝑠_𝑖𝑡_𝑖+𝑜𝑞^𝑗 (7.40) for 𝑖∈{2,…,𝑘}, and if 𝑡₁=−𝑡_𝑘≠0, then 𝑥_𝑘𝑚+1=𝑥₁𝑦_𝑘𝑞𝑠₁+𝑡₁^𝑚_𝑗=2−1−𝑞^𝑗−1𝑠_𝑘+𝑞𝑠₁𝑡₁+𝑜𝑞^𝑗 (7.41) from which the statement in (l) easily follows.

Corollary 7.4. If |𝑞|>1 and 𝛼≠0, and 𝑞^𝑗𝑠_𝑖+𝑡_𝑖≠0, for every 𝑗∈ℕ₀ and 𝑖∈{1,…,𝑘}, then the following statements hold true. (a) If |𝑠_𝑖−1|<|𝑠_𝑖|, for some 𝑖∈{2,…,𝑘}, then 𝑥_{𝑘𝑚+𝑖}→0 as 𝑚→∞. (b) If |𝑠_𝑘|<|𝑞𝑠₁|, then 𝑥_𝑘𝑚+1→0 as 𝑚→∞. (c) If |𝑠_𝑖−1|>|𝑠_𝑖|, or 𝑠_𝑖=0, 𝑠_𝑖−1≠0 and 𝑡_𝑖≠0, for some 𝑖∈{2,…,𝑘}, then |𝑥_{𝑘𝑚+𝑖}|→∞ as 𝑚→∞, if 𝑥_𝑖≠0. (d) If |𝑠_𝑘|>|𝑞𝑠₁|, or if 𝑠₁=0, 𝑠_𝑘≠0 and 𝑡₁≠0, then |𝑥_𝑘𝑚+1|→∞ as 𝑚→∞, if 𝑥₁≠0. (e) If 𝑠_𝑖−1=𝑠_𝑖≠0, for some 𝑖∈{2,…,𝑘}, then the sequence (𝑥_{𝑘𝑚+𝑖})_𝑚∈ℕ0 is convergent. (f) If 𝑠_𝑖−1=𝑠_𝑖=0 and |𝑡_𝑖−1|<|𝑡_𝑖| for some 𝑖∈{2,…,𝑘}, then 𝑥_{𝑘𝑚+𝑖}→0 as 𝑚→∞.(g) If 𝑠₁=𝑠_𝑘=0 and |𝑡_𝑘|<|𝑡₁|, then 𝑥_𝑘𝑚+1→0 as 𝑚→∞.(h) If 𝑠_𝑖−1=𝑠_𝑖=0 and |𝑡_𝑖−1|>|𝑡_𝑖| for some 𝑖∈{2,…,𝑘}, then |𝑥_{𝑘𝑚+𝑖}|→+∞ as 𝑚→∞, if 𝑥_𝑖≠0.(i) If 𝑠₁=𝑠_𝑘=0 and |𝑡_𝑘|>|𝑡₁|, then |𝑥_𝑘𝑚+1|→+∞ as 𝑚→∞, if 𝑥₁≠0.(j) If 𝑠_𝑖−1=𝑠_𝑖=0 and 𝑡_𝑖−1=𝑡_𝑖 for some 𝑖∈{2,…,𝑘}, then the sequence 𝑥_{𝑘𝑚+𝑖} is constant.(k) If 𝑠₁=𝑠_𝑘=0 and 𝑡₁=𝑡_𝑘, then the sequence 𝑥_𝑘𝑚+1 is constant.(l) If 𝑠_𝑖−1=𝑠_𝑖=0 and 𝑡_𝑖−1=−𝑡_𝑖 for some 𝑖∈{2,…,𝑘}, then the sequence 𝑥_{𝑘𝑚+𝑖} is two-periodic.(m) If 𝑠₁=𝑠_𝑘=0 and 𝑡₁=−𝑡_𝑘, then the sequence 𝑥_𝑘𝑚+1 is two periodic.(n) If 𝑠_𝑘=𝑞𝑠₁≠0, then the sequence (𝑥_𝑘𝑚+1)_𝑚∈ℕ0 is convergent. (o) If 𝑠_𝑖−1=−𝑠_𝑖≠0, for some 𝑖∈{2,…,𝑘}, then the sequences (𝑥_{2𝑘𝑚+𝑖})_𝑚∈ℕ0 and (𝑥_{2𝑘𝑚+𝑘+𝑖})_𝑚∈ℕ0, are convergent. (p) If 𝑠_𝑘=−𝑞𝑠₁≠0, then the sequences (𝑥_2𝑘𝑚+1)_𝑚∈ℕ0 and (𝑥_{2𝑘𝑚+𝑘+1})_𝑚∈ℕ0, are convergent.

Proof. (a)–(d) These statements follow correspondingly from the next relations (which are derived using formulae (7.30) and (7.31)): lim_𝑗→∞𝑞^𝑗𝑠_𝑖−1+𝑡_𝑖−1𝑞^𝑗𝑠_𝑖+𝑡_𝑖=𝑠_𝑖−1𝑠_𝑖 (7.42) for 𝑖∈{2,…,𝑘} if 𝑠_𝑖≠0, and lim_𝑗→∞𝑞^𝑗𝑠_𝑖−1+𝑡_𝑖−1𝑞^𝑗𝑠_𝑖+𝑡_𝑖=+∞ (7.43) for 𝑖∈{2,…,𝑘} if 𝑠_𝑖=0, 𝑠_𝑖−1≠0 and 𝑡_𝑖≠0; lim_𝑗→∞𝑞^𝑗−1𝑠_𝑘+𝑡_𝑘𝑞^𝑗𝑠₁+𝑡₁=𝑠_𝑘𝑞𝑠₁, (7.44) if 𝑠₁≠0, and lim_𝑗→∞𝑞^𝑗−1𝑠_𝑘+𝑡_𝑘𝑞^𝑗𝑠₁+𝑡₁=+∞ (7.45) if 𝑠₁=0, 𝑠_𝑘≠0 and 𝑡₁≠0.
(e) If 𝑠_𝑖−1=𝑠_𝑖≠0, then from (7.30) we get 𝑥_{𝑘𝑚+𝑖}=𝑥_𝑖^𝑚_𝑗=11+𝑞^−𝑗𝑡_𝑖−1−𝑡_𝑖𝑠_𝑖−1+𝑜𝑞^−𝑗, (7.46) for 𝑖∈{2,…,𝑘}, from which (e) follows.
(f)–(m) If 𝑠_𝑖−1=𝑠_𝑖=0 for some 𝑖∈{2,…,𝑘}, then for 𝑖∈{2,…,𝑘} we have 𝑞^𝑗𝑠_𝑖−1+𝑡_𝑖−1𝑞^𝑗𝑠_𝑖+𝑡_𝑖=𝑡_𝑖−1𝑡_𝑖 (7.47) while when 𝑠₁=𝑠_𝑘=0, we have 𝑞^𝑗−1𝑠_𝑘+𝑡_𝑘𝑞^𝑗𝑠₁+𝑡₁=𝑡_𝑘𝑡₁ (7.48) from which the statements (f)–(i) easily follow.
(n) If 𝑠_𝑘=𝑞𝑠₁≠0, then we have 𝑥_𝑘𝑚+1=𝑥₁𝑦_𝑘𝑞𝑠₁+𝑡₁^𝑚_𝑗=21+𝑞^−𝑗𝑡_𝑘−𝑡₁𝑠₁+𝑜𝑞^−𝑗, (7.49) from which along with the assumption |𝑞|>1 the statement follows.
(o) and (p) If 𝑠_𝑖−1=−𝑠_𝑖≠0, then 𝑥_{𝑘𝑚+𝑖}=𝑥_𝑖^𝑚_𝑗=1−1−𝑞^−𝑗𝑡_𝑖−1+𝑡_𝑖𝑠_𝑖+𝑜𝑞^−𝑗 (7.50) for 𝑖∈{2,…,𝑘}, and 𝑥_𝑘𝑚+1=𝑥₁𝑦_𝑘𝑞𝑠₁+𝑡₁^𝑚_𝑗=2−1−𝑞^−𝑗𝑡_𝑘+𝑡₁𝑠₁+𝑜𝑞^−𝑗. (7.51)
From (7.50) and (7.51) the statements in (o) and (p) correspondingly follow.

Acknowledgment

The second author is supported by Grant P201/10/1032 of the Czech Grant Agency (Prague) and by the Council of Czech Government grant MSM 00 216 30519. The fourth author is supported by Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. This paper is partially also supported by the Serbian Ministry of Science projects III 41025, III 44006, and OI 171007.