# On Some Solvable Difference Equations and Systems of Difference Equations

Authors: Stevo Stević; Josef Diblík; Bratislav Iričanin; Zdeněk Šmarda
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 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(1.1)xn+1=axn-1b+cynxn-1,yn+1=αyn-1β+γxnyn-1,n∈ℕ0,
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:
(2.1)xn+1=xn-3±1±xn-3yn-1,yn+1=yn-3±1±yn-3xn-1,n∈ℕ0.Now we show that the results regarding system (2.1) easily follow from known ones. Indeed, if we use the change of variables
(2.2)x2n+i=un(i),y2n+i=vn(i),n≥-2,i=1,2,
then the systems in (2.1) are reduced to the next systems
(2.3)un+1(i)=un-1(i)±1±un-1(i)vn(i),vn+1(i)=vn-1(i)±1±vn-1(i)un(i),n≥-1,i=1,2.
This means that (un(i),vn(i))n≥-2, i=1,2, are two (independent) solutions of the systems of difference equations
(2.4)xn+1=xn-1±1±xn-1yn,yn+1=yn-1±1±yn-1xn,n≥-1,i=1,2.
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:
(2.5)xn+1=axn-2k+1b+cxn-2k+1yn-k+1,yn+1=αyn-2k+1β+γyn-2k+1xn-k+1,n∈ℕ0,
where k is a fixed natural number.If we use the change of variables(2.6)xmk+i=um(i),ymk+i=vm(i),m≥-2,i=1,…,k,
then system (2.5) is reduced to the following k systems of difference equations:
(2.7)un+1(i)=aun-1(i)b+cun-1(i)vn(i),vn+1(i)=αvn-1(i)β+γvn-1(i)un(i),n≥-1,i=1,…,k. This means that (un(i),vn(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
(2.8)xn+1=xn-5±1±xn-5yn-2,yn+1=yn-5±1±yn-5xn-2,n∈ℕ0.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.


## 2.1. An Extension of Systems (2.1)

Systems (2.1) can be extended as follows:
(2.5)xn+1=axn-2k+1b+cxn-2k+1yn-k+1,yn+1=αyn-2k+1β+γyn-2k+1xn-k+1,n∈ℕ0,
where k is a fixed natural number.If we use the change of variables(2.6)xmk+i=um(i),ymk+i=vm(i),m≥-2,i=1,…,k,
then system (2.5) is reduced to the following k systems of difference equations:
(2.7)un+1(i)=aun-1(i)b+cun-1(i)vn(i),vn+1(i)=αvn-1(i)β+γvn-1(i)un(i),n≥-1,i=1,…,k. This means that (un(i),vn(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
(2.8)xn+1=xn-5±1±xn-5yn-2,yn+1=yn-5±1±yn-5xn-2,n∈ℕ0.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(3.1)xn+1=xn-1ynxn-1-1,yn+1=yn-1xnyn-1-1,zn+1=1znyn,(3.2)xn+1=xn-1ynxn-1-1,yn+1=yn-1xnyn-1-1,zn+1=zn-1ynzn-1-1,(3.3)xn+1=xn-1ynxn-1-1,yn+1=yn-1xnyn-1-1,zn+1=xnzn-1yn,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 forxn and yn, it is only needed to find explicit solutions for zn in the third equations in systems (3.1)–(3.3), that is, in all three equations, the only unknown sequence is zn. The joint feature for all three cases is that zn 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
(3.4)zn+1=1znyn=yn-1ynzn-1,n∈ℕ,
from which it follows that
(3.5)z2n+i=zi∏j=1n‍y2j+i-2y2j+i-1,n∈ℕ0,i=0,1.Case of System (3.3)
From the third equation in (3.3), we get
(3.6)zn+1=xnzn-1yn=xnyn-2xn-2ynzn-3,n≥2,
from which it follows that
(3.7)z4n+i=zi∏j=1n‍x4j+i-1y4j+i-3x4j+i-3y4j+i-1,n∈ℕ0,i=-1,0,1,2.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):
(3.8)zn=anzn-k,n∈ℕ0,
where k∈ℕ. From (3.8), it follows that
(3.9)zn=anan-kzn-2k,n≥k,
and consequently,
(3.10)z2mk+i=zi-2k∏j=0m‍a2kj+ia2kj-k+i,m∈ℕ0,i=k,k+1,…,3k-1.Case of System (3.2)
If we use the change of variablesvn=1/zn, the third equation in (3.2) becomes
(3.11)vn+1=-vn-1+yn,n∈ℕ0.
Hence,
(3.12)v2n+i=-v2(n-1)+i+y2n+i-1,n∈ℕ0,i=1,2,
from which it follows that
(3.13)v2n+i=(-1)n+1vi-2+∑j=0n‍(-1)n-jy2j+i-1,
so that
(3.14)z2n+i=zi-2(-1)n+1+zi-2∑j=0n(-1)n-jy2j+i-1.Remark 3.2.
The third equation in system (3.2) is a particular case of the following difference equation:
(3.15)zn=anzn-kbnzn-k+cn,n∈ℕ0,
which, by the change of variables zn=1/vn, is transformed into
(3.16)vn=cnanvn-k+bnan,n∈ℕ0,
from which it follows that
(3.17)vkm+i=ckm+iakm+ivk(m-1)+i+bkm+iakm+i,m∈ℕ0,i=0,1,…,k-1,
so by a well-known formula, we have that
(3.18)vkm+i=vi-k∏j=0m‍ckj+iakj+i+∑j=0m‍bkj+iakj+i∏l=j+1m‍ckl+iakl+i,
for m∈ℕ0 and i=0,1,…,k-1, and consequently,
(3.19)zmk+i=zi-k∏j=0m(ckj+i/akj+i)+zi-k∑j=0m(bkj+i/akj+i)∏l=j+1m(ckl+i/akl+i),
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
(3.20)zn=anzn-k+bn,n∈ℕ0,
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 zn, 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:(4.1)xn+1=g-1(g(xn-1)ah(yn)g(xn-1)+b),yn+1=h-1(h(yn-1)cg(xn)h(yn-1)+d),n∈ℕ0,
where g,h:ℝ→ℝ are increasing functions such that
(4.2)g(0)=h(0)=0,(4.3)g(x)x>0,h(x)x>0,forx≠0.Now we will find formulae for all well-defined solutions of system (4.1), that is, for the solutions (xn,yn), n≥-1, such that
(4.4)ah(yn)g(xn-1)+b≠0,cg(xn)h(yn-1)+d≠0,
for every n∈ℕ0.Ifx-1=0, then from (4.1), (4.2), and (4.3), and by the method of induction, we get x2n+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 x2n=0,n∈ℕ0. Similarly, if y-1=0, then we get y2n+1=0,n∈ℕ0, while if y0=0, then we get y2n=0,n∈ℕ0.Ifxn0=0 for some n0∈ℕ, then from (4.1)–(4.3) it follows that xn0-2k=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 yn1=0 for some n1∈ℕ, then from (4.1)–(4.3) it follows that yn1-2k=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 valuesx-1, x0, y-1, and y0 is equal to zero. Then, for every well-defined solution of system (4.1), we have that xn≠0 and yn≠0, for every n≥-1, and consequently g(xn)≠0 and h(yn)≠0, for every n≥-1.Let(4.5)vn=1g(xn)h(yn-1),un=1g(xn-1)h(yn),
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(yn) and the second by g(xn), system (4.1) is transformed into:
(4.6)vn+1=bun+a,un+1=dvn+c,n∈ℕ0,
from which it follows that
(4.7)vn+1=bdvn-1+bc+a,n∈ℕ,un+1=bdun-1+ad+c,n∈ℕ.Hence, ifbd≠1, we have that
(4.8)v2n=(bd)ng(x0)h(y-1)+(bc+a)1-(bd)n1-bd,n∈ℕ0,v2n+1=(bd)ng(x1)h(y0)+(bc+a)1-(bd)n1-bd,n∈ℕ0,
while if bd=1, we have that
(4.9)v2n=1g(x0)h(y-1)+(bc+a)n,n∈ℕ0,v2n+1=1g(x1)h(y0)+(bc+a)n,n∈ℕ0.We have also that(4.10)u2n=(bd)nh(y0)g(x-1)+(ad+c)1-(bd)n1-bd,n∈ℕ0,u2n+1=(bd)nh(y1)g(x0)+(ad+c)1-(bd)n1-bd,n∈ℕ0,
while if bd=1, we have that
(4.11)u2n=1h(y0)g(x-1)+(ad+c)n,n∈ℕ0,u2n+1=1h(y1)g(x0)+(ad+c)n,n∈ℕ0.From (4.5), we have that
(4.12)g(xn)=1vnh(yn-1)=un-1vng(xn-2),n∈ℕ,h(yn)=1ung(xn-1)=vn-1unh(yn-2),n∈ℕ.Using the relations (4.12), we get
(4.13)x2n=g-1(g(x0)∏j=1n‍u2j-1v2j),n∈ℕ0,x2n+1=g-1(g(x-1)∏j=0n‍u2jv2j+1),n∈ℕ0,y2n=h-1(h(y0)∏j=1n‍v2j-1u2j),n∈ℕ0,y2n+1=h-1(h(y-1)∏j=0n‍v2ju2j+1),n∈ℕ0.Example 4.1.
If we chooseg(t)=t2k+1 and h(t)=t2l+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
(4.14)xn+1=(xn-12k+1ayn2l+1xn-12k+1+b)1/(2k+1),yn+1=(yn-12l+1cxn2k+1yn-12l+1+d)1/(2l+1),n∈ℕ0,
and from (4.13), we have that its solutions are given by
(4.15)x2n=x0∏j=1n‍(u2j-1v2j)1/(2k+1),n∈ℕ0,x2n+1=x-1∏j=0n‍(u2jv2j+1)1/(2k+1),n∈ℕ0,y2n=y0∏j=1n‍(v2j-1u2j)1/(2l+1),n∈ℕ0,y2n+1=y-1∏j=0n‍(v2ju2j+1)1/(2l+1),n∈ℕ0.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:
(4.16)xn+1=g-1(g(xn-2k+1)ah(yn-k+1)g(xn-2k+1)+b),yn+1=h-1(h(yn-2k+1)cg(xn-k+1)h(yn-2k+1)+d),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:
(4.17)un+1(i)=g-1(g(un-1(i))ah(vn(i))g(un-1(i))+b),vn+1(i)=h-1(h(vn-1(i))cg(un(i))h(vn-1(i))+d),n≥-1,
for i=1,…,k.Remark 4.3.
Well-defined solutions of the following system of difference equations:(4.18)xn+1=g-1(g(xn-1)anh(yn)g(xn-1)+bn),yn+1=h-1(h(yn-1)cng(xn)h(yn-1)+dn),n∈ℕ0,
where g,h:ℝ→ℝ are increasing functions satisfying conditions (4.2) and (4.3) and an, bn, cn, and dn, 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:
(5.1)xn+1=xnxn-kxn-k+1(a+bxnxn-k),n∈ℕ0,
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:
(5.2)xn+1=xnxn-kxn-k+1(an+bnxnxn-k),n∈ℕ0,
where k∈ℕ and the sequences (an)n∈ℕ0,  (bn)n∈ℕ0, as well as initial values x-i, i=0,k¯, are real numbers.Employing the change of variables(5.3)yn=1xnxn-k,n∈ℕ0,
Equation (5.2) is transformed into the linear first-order difference equation
(5.4)yn+1=anyn+bn,n∈ℕ0,
whose general solution is
(5.5)yn=y0∏l=0n-1‍al+∑s=0n-1‍bs∏l=s+1n-1‍al,n∈ℕ0.From (5.3), we have that
(5.6)xn=1ynxn-k=yn-kynxn-2k,
for n≥k, which yields
(5.7)x2km+i=xi-2k∏j=0m‍y(2j-1)k+iy2jk+i,
for every m∈ℕ0 and i∈{k,k+1,…,3k-1}.Using (5.5) in (5.7), we get
(5.8)x2km+i=xi-2k∏j=0m‍y0∏l=0(2j-1)k+i-1al+∑i=s(2j-1)k+i-1bs∏l=s+1(2j-1)k+i-1aly0∏l=02jk+i-1al+∑s=02jk+i-1bs∏l=s+12jk+i-1al,
for every m∈ℕ0 and i∈{k,k+1,…,3k-1}.Formula (5.8) generalizes the main formulae obtained in our paper [24].



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