On Some Solvable Difference Equations and Systems of Difference Equations

Abstract

Here, we give explicit formulae for solutions of some systems of difference equations, which extend some very particular recent results in the literature and give natural explanations for them, which were omitted in the previous literature.

1. Introduction

Recently, there has been a great interest in difference equations and systems (see, e.g., [1–25]), and among them in those ones which can be solved explicitly (see, e.g., [1–5, 9–11, 15, 16, 18–24] and the related references therein). For some classical results in the topic see, for example, [7].

Beside the above-mentioned papers, there are some papers which give formulae of some very particular equations and systems which are proved by induction, but without any explanation how these formulae are obtained and how these authors came across the equations and systems. Our explanation of such a formula that we gave in [10] has re-attracted attention to solvable difference equations.

Our aim here is to give theoretical explanations for some of the formulae recently appearing in the literature, as well as to give some extensions of their equations.

Before we formulate our results, we would like to say that the system of difference equations 𝑥_𝑛+1=𝑎𝑥_𝑛−1𝑏+𝑐𝑦_𝑛𝑥_𝑛−1,𝑦_𝑛+1=𝛼𝑦_𝑛−1𝛽+𝛾𝑥_𝑛𝑦_𝑛−1,𝑛∈ℕ₀, (1.1) where 𝑎,𝑏,𝑐,𝛼,𝛽,and𝛾 are real numbers, was completely solved in [15], that is, we found formulae for all well-defined solutions of system (1.1).

2. Scaling Indices

In the recent paper [25] were given some formulae for the solutions of the following systems of difference equations: 𝑥_𝑛+1=𝑥_𝑛−3±1±𝑥_𝑛−3𝑦_𝑛−1,𝑦_𝑛+1=𝑦_𝑛−3±1±𝑦_𝑛−3𝑥_𝑛−1,𝑛∈ℕ₀. (2.1)

Now we show that the results regarding system (2.1) easily follow from known ones. Indeed, if we use the change of variables 𝑥_2𝑛+𝑖=𝑢_𝑛^(𝑖),𝑦_2𝑛+𝑖=𝑣_𝑛^(𝑖),𝑛≥−2,𝑖=1,2, (2.2) then the systems in (2.1) are reduced to the next systems 𝑢^(𝑖)_𝑛+1=𝑢^(𝑖)_𝑛−1±1±𝑢^(𝑖)_𝑛−1𝑣_𝑛^(𝑖),𝑣^(𝑖)_𝑛+1=𝑣^(𝑖)_𝑛−1±1±𝑣^(𝑖)_𝑛−1𝑢_𝑛^(𝑖),𝑛≥−1,𝑖=1,2. (2.3) This means that (𝑢_𝑛^(𝑖),𝑣_𝑛^(𝑖))_𝑛≥−2, 𝑖=1,2, are two (independent) solutions of the systems of difference equations 𝑥_𝑛+1=𝑥_𝑛−1±1±𝑥_𝑛−1𝑦_𝑛,𝑦_𝑛+1=𝑦_𝑛−1±1±𝑦_𝑛−1𝑥_𝑛,𝑛≥−1,𝑖=1,2. (2.4) However, all the systems of difference equations in (2.4) are particular cases of system (1.1). Hence, formulae for the solutions of systems (2.1) given in [25] follow directly from those in [15].

2.1. An Extension of Systems (2.1)

Systems (2.1) can be extended as follows: 𝑥_𝑛+1=𝑎𝑥_{𝑛−2𝑘+1}𝑏+𝑐𝑥_{𝑛−2𝑘+1}𝑦_{𝑛−𝑘+1},𝑦_𝑛+1=𝛼𝑦_{𝑛−2𝑘+1}𝛽+𝛾𝑦_{𝑛−2𝑘+1}𝑥_{𝑛−𝑘+1},𝑛∈ℕ₀, (2.5) where 𝑘 is a fixed natural number.

If we use the change of variables 𝑥_{𝑚𝑘+𝑖}=𝑢_𝑚^(𝑖),𝑦_{𝑚𝑘+𝑖}=𝑣_𝑚^(𝑖),𝑚≥−2,𝑖=1,…,𝑘, (2.6) then system (2.5) is reduced to the following 𝑘 systems of difference equations: 𝑢^(𝑖)_𝑛+1=𝑎𝑢^(𝑖)_𝑛−1𝑏+𝑐𝑢^(𝑖)_𝑛−1𝑣_𝑛^(𝑖),𝑣^(𝑖)_𝑛+1=𝛼𝑣^(𝑖)_𝑛−1𝛽+𝛾𝑣^(𝑖)_𝑛−1𝑢_𝑛^(𝑖),𝑛≥−1, (2.7)𝑖=1,…,𝑘. This means that (𝑢_𝑛^(𝑖),𝑣_𝑛^(𝑖))_𝑛≥−2, 𝑖=1,…,𝑘, are 𝑘 (independent) solutions of system (1.1), and solutions of system (2.5) are obtained by interlacing solutions of systems (2.7), 𝑖=1,…,𝑘.

For example, a natural extension of the systems in (2.1) is obtained for taking 𝑘=3, 𝑏/𝑎=±1,  𝑐/𝑎=±1,   𝛽/𝛼=±1, and 𝛾/𝛼=±1 in (2.5), that is, the system becomes 𝑥_𝑛+1=𝑥_𝑛−5±1±𝑥_𝑛−5𝑦_𝑛−2,𝑦_𝑛+1=𝑦_𝑛−5±1±𝑦_𝑛−5𝑥_𝑛−2,𝑛∈ℕ₀. (2.8)

In this way it can be obtained countable many, at first sight different, systems of difference equations. Systems of difference equations in (2.1) are artificially obtained in this way. This method can be applied to any equation or system of difference equations, and one can get papers with putative “new” results.

3. Some Third-Order Systems of Difference Equations Related to (1.1)

The following third-order systems of difference equations 𝑥_𝑛+1=𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1−1,𝑦_𝑛+1=𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−1−1,𝑧_𝑛+1=1𝑧_𝑛𝑦_𝑛,𝑥_𝑛+1=𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1−1,𝑦_𝑛+1=𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−1−1,𝑧_𝑛+1=𝑧_𝑛−1𝑦_𝑛𝑧_𝑛−1,𝑥−1_𝑛+1=𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1−1,𝑦_𝑛+1=𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−1−1,𝑧_𝑛+1=𝑥_𝑛𝑧_𝑛−1𝑦_𝑛, (3.1)(3.2)(3.3)𝑛∈ℕ₀, have been studied recently (see [6] and the references therein).

As is directly seen, the first two equations in systems (3.1)–(3.3) are the same, and they form a particular case of system (1.1) which is solved in [15].

Since we know solutions for 𝑥_𝑛 and 𝑦_𝑛, it is only needed to find explicit solutions for 𝑧_𝑛 in the third equations in systems (3.1)–(3.3), that is, in all three equations, the only unknown sequence is 𝑧_𝑛. The joint feature for all three cases is that 𝑧_𝑛 can be solved in closed form.

Now we discuss systems of difference equations given in (3.1)–(3.3).

Case of System (3.1)
From the third equation in (3.1), we get 𝑧_𝑛+1=1𝑧_𝑛𝑦_𝑛=𝑦_𝑛−1𝑦_𝑛𝑧_𝑛−1,𝑛∈ℕ, (3.4) from which it follows that 𝑧_2𝑛+𝑖=𝑧_𝑖^𝑛_𝑗=1𝑦_{2𝑗+𝑖−2}𝑦_{2𝑗+𝑖−1},𝑛∈ℕ₀,𝑖=0,1. (3.5)

Case of System (3.3)
From the third equation in (3.3), we get 𝑧_𝑛+1=𝑥_𝑛𝑧_𝑛−1𝑦_𝑛=𝑥_𝑛𝑦_𝑛−2𝑥_𝑛−2𝑦_𝑛𝑧_𝑛−3,𝑛≥2, (3.6) from which it follows that 𝑧_4𝑛+𝑖=𝑧_𝑖^𝑛_𝑗=1𝑥_{4𝑗+𝑖−1}𝑦_{4𝑗+𝑖−3}𝑥_{4𝑗+𝑖−3}𝑦_{4𝑗+𝑖−1},𝑛∈ℕ₀,𝑖=−1,0,1,2. (3.7)

Remark 3.1. The third equations in systems (3.1) and (3.3) are particular cases of the following difference equation (up to the shifting indices): 𝑧_𝑛=𝑎_𝑛𝑧_𝑛−𝑘,𝑛∈ℕ₀, (3.8) where 𝑘∈ℕ. From (3.8), it follows that 𝑧_𝑛=𝑎_𝑛𝑎_𝑛−𝑘𝑧_𝑛−2𝑘,𝑛≥𝑘, (3.9) and consequently, 𝑧_{2𝑚𝑘+𝑖}=𝑧^𝑚_{𝑖−2𝑘𝑗=0}𝑎_{2𝑘𝑗+𝑖}𝑎_{2𝑘𝑗−𝑘+𝑖},𝑚∈ℕ₀,𝑖=𝑘,𝑘+1,…,3𝑘−1. (3.10)

Case of System (3.2)
If we use the change of variables 𝑣_𝑛=1/𝑧_𝑛, the third equation in (3.2) becomes 𝑣_𝑛+1=−𝑣_𝑛−1+𝑦_𝑛,𝑛∈ℕ₀. (3.11) Hence, 𝑣_2𝑛+𝑖=−𝑣_{2(𝑛−1)+𝑖}+𝑦_{2𝑛+𝑖−1},𝑛∈ℕ₀,𝑖=1,2, (3.12) from which it follows that 𝑣_2𝑛+𝑖=(−1)^𝑛+1𝑣_𝑖−2+^𝑛_𝑗=0(−1)^𝑛−𝑗𝑦_{2𝑗+𝑖−1}, (3.13) so that 𝑧_2𝑛+𝑖=𝑧_𝑖−2(−1)^𝑛+1+𝑧_𝑖−2∑^𝑛_𝑗=0(−1)^𝑛−𝑗𝑦_{2𝑗+𝑖−1}. (3.14)

Remark 3.2. The third equation in system (3.2) is a particular case of the following difference equation: 𝑧_𝑛=𝑎_𝑛𝑧_𝑛−𝑘𝑏_𝑛𝑧_𝑛−𝑘+𝑐_𝑛,𝑛∈ℕ₀, (3.15) which, by the change of variables 𝑧_𝑛=1/𝑣_𝑛, is transformed into 𝑣_𝑛=𝑐_𝑛𝑎_𝑛𝑣_𝑛−𝑘+𝑏_𝑛𝑎_𝑛,𝑛∈ℕ₀, (3.16) from which it follows that 𝑣_{𝑘𝑚+𝑖}=𝑐_{𝑘𝑚+𝑖}𝑎_{𝑘𝑚+𝑖}𝑣_{𝑘(𝑚−1)+𝑖}+𝑏_{𝑘𝑚+𝑖}𝑎_{𝑘𝑚+𝑖},𝑚∈ℕ₀,𝑖=0,1,…,𝑘−1, (3.17) so by a well-known formula, we have that 𝑣_{𝑘𝑚+𝑖}=𝑣^𝑚_{𝑖−𝑘𝑗=0}𝑐_{𝑘𝑗+𝑖}𝑎_{𝑘𝑗+𝑖}+^𝑚_𝑗=0𝑏_{𝑘𝑗+𝑖}𝑎^𝑚_{𝑘𝑗+𝑖𝑙=𝑗+1}𝑐_{𝑘𝑙+𝑖}𝑎_{𝑘𝑙+𝑖}, (3.18) for 𝑚∈ℕ₀ and 𝑖=0,1,…,𝑘−1, and consequently, 𝑧_{𝑚𝑘+𝑖}=𝑧_𝑖−𝑘∏^𝑚_𝑗=0𝑐_{𝑘𝑗+𝑖}/𝑎_{𝑘𝑗+𝑖}+𝑧_𝑖−𝑘∑^𝑚_𝑗=0𝑏_{𝑘𝑗+𝑖}/𝑎_{𝑘𝑗+𝑖}∏^𝑚_𝑙=𝑗+1𝑐_{𝑘𝑙+𝑖}/𝑎_{𝑘𝑙+𝑖}, (3.19) for 𝑚∈ℕ₀ and 𝑖=0,1,…,𝑘−1.

Remark 3.3. Note that (3.16) suggests that the third equation in (3.2) can be also of the form 𝑧_𝑛=𝑎_𝑛𝑧_𝑛−𝑘+𝑏_𝑛,𝑛∈ℕ₀, (3.20) where 𝑘∈ℕ is fixed, that is, to be an equation which consists of 𝑘 (independent) linear first-order difference equation, which is solvable. In fact, the third equation in systems (3.1)–(3.3) can be any difference equation which can be solved in 𝑧_𝑛, and in this way we can obtain numerous putative “new” results.

4. A Generalization of System (1.1)

Consider the following system of difference equations: 𝑥_𝑛+1=𝑔⁻¹𝑔𝑥_𝑛−1𝑎ℎ𝑦_𝑛𝑔𝑥_𝑛−1+𝑏,𝑦_𝑛+1=ℎ⁻¹ℎ𝑦_𝑛−1𝑐𝑔𝑥_𝑛ℎ𝑦_𝑛−1+𝑑,𝑛∈ℕ₀, (4.1) where 𝑔,ℎ∶ℝ→ℝ are increasing functions such that 𝑔(0)(0)𝑔(𝑥)(𝑥)=ℎ=0,𝑥>0,ℎ𝑥>0,for𝑥≠0. (4.2)(4.3)

Now we will find formulae for all well-defined solutions of system (4.1), that is, for the solutions (𝑥_𝑛,𝑦_𝑛), 𝑛≥−1, such that 𝑎ℎ𝑦_𝑛𝑔𝑥_𝑛−1+𝑏≠0,𝑐𝑔𝑥_𝑛ℎ𝑦_𝑛−1+𝑑≠0, (4.4) for every 𝑛∈ℕ₀.

If 𝑥₋₁=0, then from (4.1), (4.2), and (4.3), and by the method of induction, we get 𝑥_2𝑛+1=0,𝑛∈ℕ₀. Also, if 𝑥₀=0, then from (4.1), (4.2), and (4.3), and by the method of induction, we get 𝑥_2𝑛=0,𝑛∈ℕ₀. Similarly, if 𝑦₋₁=0, then we get 𝑦_2𝑛+1=0,𝑛∈ℕ₀, while if 𝑦₀=0, then we get 𝑦_2𝑛=0,𝑛∈ℕ₀.

If 𝑥_𝑛0=0 for some 𝑛₀∈ℕ, then from (4.1)–(4.3) it follows that 𝑥_{𝑛0−2𝑘}=0, for each 𝑘∈ℕ₀ such that 𝑛₀−2𝑘≥−1. Hence, in this case we have that 𝑥₋₁=0 or 𝑥₀=0. Similarly, if 𝑦_𝑛1=0 for some 𝑛₁∈ℕ, then from (4.1)–(4.3) it follows that 𝑦_{𝑛1−2𝑘}=0, for each 𝑘∈ℕ₀ such that 𝑛₁−2𝑘≥−1. Hence, in this case we have that 𝑦₋₁=0 or 𝑦₀=0. Thus, in both cases we arrive at a situation explained in the previous paragraph.

Hence, from now on, we assume that none of the initial values 𝑥₋₁, 𝑥₀, 𝑦₋₁, and 𝑦₀ is equal to zero. Then, for every well-defined solution of system (4.1), we have that 𝑥_𝑛≠0 and 𝑦_𝑛≠0, for every 𝑛≥−1, and consequently 𝑔(𝑥_𝑛)≠0 and ℎ(𝑦_𝑛)≠0, for every 𝑛≥−1.

Let 𝑣_𝑛=1𝑔𝑥_𝑛ℎ𝑦_𝑛−1,𝑢_𝑛=1𝑔𝑥_𝑛−1ℎ𝑦_𝑛, (4.5) then by taking function 𝑔 to the first equation in system (4.1) and function ℎ to the second one, then multiplying the first equation in such obtained system by ℎ(𝑦_𝑛) and the second by 𝑔(𝑥_𝑛), system (4.1) is transformed into: 𝑣_𝑛+1=𝑏𝑢_𝑛+𝑎,𝑢_𝑛+1=𝑑𝑣_𝑛+𝑐,𝑛∈ℕ₀, (4.6) from which it follows that 𝑣_𝑛+1=𝑏𝑑𝑣_𝑛−1𝑢+𝑏𝑐+𝑎,𝑛∈ℕ,_𝑛+1=𝑏𝑑𝑢_𝑛−1+𝑎𝑑+𝑐,𝑛∈ℕ. (4.7)

Hence, if 𝑏𝑑≠1, we have that 𝑣_2𝑛=(𝑏𝑑)^𝑛𝑔𝑥₀ℎ𝑦₋₁+()()𝑏𝑐+𝑎1−𝑏𝑑^𝑛1−𝑏𝑑,𝑛∈ℕ₀,𝑣_2𝑛+1=(𝑏𝑑)^𝑛𝑔𝑥₁ℎ𝑦₀+()()𝑏𝑐+𝑎1−𝑏𝑑^𝑛1−𝑏𝑑,𝑛∈ℕ₀, (4.8) while if 𝑏𝑑=1, we have that 𝑣_2𝑛=1𝑔𝑥₀ℎ𝑦₋₁+(𝑏𝑐+𝑎)𝑛,𝑛∈ℕ₀,𝑣_2𝑛+1=1𝑔𝑥₁ℎ𝑦₀+(𝑏𝑐+𝑎)𝑛,𝑛∈ℕ₀. (4.9)

We have also that 𝑢_2𝑛=(𝑏𝑑)^𝑛ℎ𝑦₀𝑔𝑥₋₁+()()𝑎𝑑+𝑐1−𝑏𝑑^𝑛1−𝑏𝑑,𝑛∈ℕ₀,𝑢_2𝑛+1=(𝑏𝑑)^𝑛ℎ𝑦₁𝑔𝑥₀+()()𝑎𝑑+𝑐1−𝑏𝑑^𝑛1−𝑏𝑑,𝑛∈ℕ₀, (4.10) while if 𝑏𝑑=1, we have that 𝑢_2𝑛=1ℎ𝑦₀𝑔𝑥₋₁+(𝑎𝑑+𝑐)𝑛,𝑛∈ℕ₀,𝑢_2𝑛+1=1ℎ𝑦₁𝑔𝑥₀+(𝑎𝑑+𝑐)𝑛,𝑛∈ℕ₀. (4.11)

From (4.5), we have that 𝑔𝑥_𝑛=1𝑣_𝑛ℎ𝑦_𝑛−1=𝑢_𝑛−1𝑣_𝑛𝑔𝑥_𝑛−2ℎ𝑦,𝑛∈ℕ,_𝑛=1𝑢_𝑛𝑔𝑥_𝑛−1=𝑣_𝑛−1𝑢_𝑛ℎ𝑦_𝑛−2,𝑛∈ℕ. (4.12)

Using the relations (4.12), we get 𝑥_2𝑛=𝑔⁻¹𝑔𝑥₀^𝑛_𝑗=1𝑢_2𝑗−1𝑣_2𝑗,𝑛∈ℕ₀,𝑥_2𝑛+1=𝑔⁻¹𝑔𝑥₋₁^𝑛_𝑗=0𝑢_2𝑗𝑣_2𝑗+1,𝑛∈ℕ₀,𝑦_2𝑛=ℎ⁻¹ℎ𝑦₀^𝑛_𝑗=1𝑣_2𝑗−1𝑢_2𝑗,𝑛∈ℕ₀,𝑦_2𝑛+1=ℎ⁻¹ℎ𝑦₋₁^𝑛_𝑗=0𝑣_2𝑗𝑢_2𝑗+1,𝑛∈ℕ₀. (4.13)

Example 4.1. If we choose 𝑔(𝑡)=𝑡^2𝑘+1 and ℎ(𝑡)=𝑡^2𝑙+1 for some 𝑘,𝑙∈ℕ₀, then conditions (4.2) and (4.3) are obviously satisfied, and system (4.1) can be written in the form 𝑥_𝑛+1=𝑥^2𝑘+1_𝑛−1𝑎𝑦_𝑛^2𝑙+1𝑥^2𝑘+1_𝑛−1+𝑏^1/(2𝑘+1),𝑦_𝑛+1=𝑦^2𝑙+1_𝑛−1𝑐𝑥_𝑛^2𝑘+1𝑦^2𝑙+1_𝑛−1+𝑑^1/(2𝑙+1),𝑛∈ℕ₀, (4.14) and from (4.13), we have that its solutions are given by 𝑥_2𝑛=𝑥₀^𝑛_𝑗=1𝑢_2𝑗−1𝑣_2𝑗^1/(2𝑘+1),𝑛∈ℕ₀,𝑥_2𝑛+1=𝑥^𝑛_−1𝑗=0𝑢_2𝑗𝑣_2𝑗+1^1/(2𝑘+1),𝑛∈ℕ₀,𝑦_2𝑛=𝑦₀^𝑛_𝑗=1𝑣_2𝑗−1𝑢_2𝑗^1/(2𝑙+1),𝑛∈ℕ₀,𝑦_2𝑛+1=𝑦^𝑛_−1𝑗=0𝑣_2𝑗𝑢_2𝑗+1^1/(2𝑙+1),𝑛∈ℕ₀. (4.15)

Remark 4.2. System (4.1) can be generalized by using the method of scaling indices from Section 2.1, that is, the following system is also solvable: 𝑥_𝑛+1=𝑔⁻¹𝑔𝑥_{𝑛−2𝑘+1}𝑎ℎ𝑦_{𝑛−𝑘+1}𝑔𝑥_{𝑛−2𝑘+1}+𝑏,𝑦_𝑛+1=ℎ⁻¹ℎ𝑦_{𝑛−2𝑘+1}𝑐𝑔𝑥_{𝑛−𝑘+1}ℎ𝑦_{𝑛−2𝑘+1}+𝑑, (4.16)𝑛∈ℕ₀, where 𝑘∈ℕ and 𝑔,ℎ∶ℝ→ℝ are increasing functions satisfying conditions (4.2) and (4.3).
It is easy to see that the change of variables in (2.6) leads to the following 𝑘 systems of difference equations: 𝑢^(𝑖)_𝑛+1=𝑔⁻¹𝑔𝑢^(𝑖)_𝑛−1𝑎ℎ𝑣_𝑛^(𝑖)𝑔𝑢^(𝑖)_𝑛−1+𝑏,𝑣^(𝑖)_𝑛+1=ℎ⁻¹ℎ𝑣^(𝑖)_𝑛−1𝑐𝑔𝑢_𝑛^(𝑖)ℎ𝑣^(𝑖)_𝑛−1+𝑑,𝑛≥−1, (4.17) for 𝑖=1,…,𝑘.

Remark 4.3. Well-defined solutions of the following system of difference equations: 𝑥_𝑛+1=𝑔⁻¹𝑔𝑥_𝑛−1𝑎_𝑛ℎ𝑦_𝑛𝑔𝑥_𝑛−1+𝑏_𝑛,𝑦_𝑛+1=ℎ⁻¹ℎ𝑦_𝑛−1𝑐_𝑛𝑔𝑥_𝑛ℎ𝑦_𝑛−1+𝑑_𝑛,𝑛∈ℕ₀, (4.18) where 𝑔,ℎ∶ℝ→ℝ are increasing functions satisfying conditions (4.2) and (4.3) and 𝑎_𝑛, 𝑏_𝑛, 𝑐_𝑛, and 𝑑_𝑛, 𝑛∈ℕ₀, are real sequences, can be found similarly. We omit the details.

5. Solutions of a Generalization of a Recent Equation

Explaining some recent formulae appearing in the literature, in our recent paper [24], we have found formulae for well-defined solutions of the following difference equation: 𝑥_𝑛+1=𝑥_𝑛𝑥_𝑛−𝑘𝑥_{𝑛−𝑘+1}𝑎+𝑏𝑥_𝑛𝑥_𝑛−𝑘,𝑛∈ℕ₀, (5.1) where 𝑘∈ℕ and the parameters 𝑎,  𝑏 as well as initial values 𝑥_−𝑖, 𝑖=0,𝑘 are real numbers.

Equation (5.1) can be extended naturally in the following way: 𝑥_𝑛+1=𝑥_𝑛𝑥_𝑛−𝑘𝑥_{𝑛−𝑘+1}𝑎_𝑛+𝑏_𝑛𝑥_𝑛𝑥_𝑛−𝑘,𝑛∈ℕ₀, (5.2) where 𝑘∈ℕ and the sequences (𝑎_𝑛)_𝑛∈ℕ0,  (𝑏_𝑛)_𝑛∈ℕ0, as well as initial values 𝑥_−𝑖, 𝑖=0,𝑘, are real numbers.

Employing the change of variables 𝑦_𝑛=1𝑥_𝑛𝑥_𝑛−𝑘,𝑛∈ℕ₀, (5.3) Equation (5.2) is transformed into the linear first-order difference equation 𝑦_𝑛+1=𝑎_𝑛𝑦_𝑛+𝑏_𝑛,𝑛∈ℕ₀, (5.4) whose general solution is 𝑦_𝑛=𝑦₀^𝑛−1_𝑙=0𝑎_𝑙+^𝑛−1_𝑠=0𝑏_𝑠^𝑛−1_𝑙=𝑠+1𝑎_𝑙,𝑛∈ℕ₀. (5.5)

From (5.3), we have that 𝑥_𝑛=1𝑦_𝑛𝑥_𝑛−𝑘=𝑦_𝑛−𝑘𝑦_𝑛𝑥_𝑛−2𝑘, (5.6) for 𝑛≥𝑘, which yields 𝑥_{2𝑘𝑚+𝑖}=𝑥^𝑚_{𝑖−2𝑘𝑗=0}𝑦_{(2𝑗−1)𝑘+𝑖}𝑦_{2𝑗𝑘+𝑖}, (5.7) for every 𝑚∈ℕ₀ and 𝑖∈{𝑘,𝑘+1,…,3𝑘−1}.

Using (5.5) in (5.7), we get 𝑥_{2𝑘𝑚+𝑖}=𝑥^𝑚_{𝑖−2𝑘𝑗=0}𝑦₀∏^{(2𝑗−1)𝑘+𝑖−1}_𝑙=0𝑎_𝑙+∑^{(2𝑗−1)𝑘+𝑖−1}_𝑖=𝑠𝑏_𝑠∏^{(2𝑗−1)𝑘+𝑖−1}_𝑙=𝑠+1𝑎_𝑙𝑦₀∏^{2𝑗𝑘+𝑖−1}_𝑙=0𝑎_𝑙+∑^{2𝑗𝑘+𝑖−1}_𝑠=0𝑏_𝑠∏^{2𝑗𝑘+𝑖−1}_𝑙=𝑠+1𝑎_𝑙, (5.8) for every 𝑚∈ℕ₀ and 𝑖∈{𝑘,𝑘+1,…,3𝑘−1}.

Formula (5.8) generalizes the main formulae obtained in our paper [24].

Acknowledgments

The second author is supported by the Grant P201/10/1032 of the Czech Grant Agency (Prague) and the “Operational Programme Research and Development for Innovations,” no. CZ.1.05/2.1.00/03.0097, as an activity of the regional Centre AdMaS. The fourth author is supported by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology and by the Grant P201/11/0768 of the Czech Grant Agency (Prague). This paper is also supported by the Serbian Ministry of Science projects III 41025, III 44006, and OI 171007.