On a practically solvable product-type system of difference equations of second order

On a practically solvable product-type system of difference equations of second order

Stevo Stevi´c

and Dragana Rankovi´c

1Mathematical Institute of the Serbian Academy of Sciences Knez Mihailova 36/III, 11000 Beograd, Serbia

2King Abdulaziz University, Department of Mathematics, P.O. Box 80203, Jeddah 21589, Saudi Arabia

3Department of Physics and Mathematics, Faculty of Pharmacy, University of Belgrade Vojvode Stepe 450, Belgrade, Serbia

Received 20 May 2016, appeared 26 July 2016 Communicated by Leonid Berezansky

Abstract. The problem of solvability of the following second order system of difference equations

zn+1=αz^a_nw^b_n, wn+1=βw^c_nz^d_n−1, n∈_N0, wherea,b,c,d∈Z,α,β∈C\ {0},z−1,z0,w0∈C\ {0}, is studied in detail.

Keywords: system of difference equations, second order system, product-type system, practically solvable system.

2010 Mathematics Subject Classification: 39A10, 39A20.

1 Introduction

Investigation of various concrete nonlinear difference equations and systems has been of a great recent interest (see, e.g., [1–5], [10–35] and the references therein). Here we mention two subareas of interest. First, motivated by some classes of concrete nonlinear difference equations some experts started investigating the corresponding symmetric or cyclic systems of difference equations, as well as some modifications of the symmetric/cyclic systems (see, e.g., [4,10–12,14,15,19–21,23–27,29–33,35]). Second, one of the basic problems related to difference equations and systems is the problem of their solvability. Books [7–9] contain numerous classical results on the topic. Solvable difference equations and systems have appeared in the literature from time to time for a long time. It has turned out that some of recently studied concrete nonlinear difference equations and systems are solvable. For example, the solvability of systems in [4] and [21] was discovered during the investigation of the long-term behavior of their solutions. On the other hand, some papers which give solutions of quite concrete equations and systems, without any theoretical explanations, have appeared recently. One of the first papers of this type is [5]. These facts motivated some experts to work on the problem

BCorresponding author. Email: sstevic@ptt.rs

of solvability more seriously (see, e.g., [4,13,16,17], [19]-[30], [32–35]). Short note [16] can be regarded as a starting point for the serious investigation, and its importance is found in the method which is used for showing the solvability of an extension of the equation in [5].

The method was later used numerous times (see, e.g., [1,13,17,22,34]) and was essentially extended and developed in many of the other above mentioned papers on solvability.

Another fact of interest is to note that many difference equations and systems are ob- tained from product-type ones by some modifications of their right-hand sides (e.g., the equa- tions/systems in [18] and [31] are of this type). Product-type equations and systems are solvable for the case of positive coefficients and initial values. However, this is not always the case if some of them are real or complex numbers. Hence, the problem of their solvability in these cases is of some interest. An investigation in this direction was started by S. Stevi´c and some of his collaborators in [27], [29], [30] and [35], where some product-type systems are solved not only theoretically, but essentially also practically, that is, by presenting concrete formulas for solutions depending on the initial values and involving parameters.

This paper is devoted to the study of solvability of the following second order system of difference equations

zn+₁ =αz^a_nw^b_n, wn+₁= βw^c_nz^d_n−1, n∈_N0, (1.1) where a,b,c,d ∈ _Z, α,β ∈ _Cand z−1,z₀,w₀ ∈ C. Motivated by [27] system (1.1), unlike the ones in papers [29], [30] and [35], has some new parameters, namely, the coefficientsαandβ.

Since the casesα=0 orβ =0 are very simple, so of not much interest, from now on we will assume thatα,β∈_C\ {0}.

Note that the domain of undefinable solutions [24] to system (1.1) is a subset of the fol- lowing set

U = {(z−1,z₀,w₀)∈ _C³:z−1=0 or z₀ =0 or w₀=0}.

Thus we can regard that(z−1,z₀,w₀)belong toC³\ U, although in some cases C³\ U can be even equal toC³(for example, if a,b,canddare natural numbers).

For a system of difference equations of the form

zn = f(z_n−1, . . . ,z_n−k,w_n−1, . . . ,w_n−l)

wn =_g(_{zn−1, . . . ,zn−s,wn−1, . . . ,wn−t})_{, n}∈ _N0,

wherek,l,s,t ∈ N, is said that it issolvable in closed form if its general solution can be found in terms of initial valuesz−i,i=1, max{k,s},w−j,j=1, max{l,t}, delaysk,l,s,t, and index n only.

Let us also say that, as usual, the sums of the form∑^m_j=la_j, for m< l, will be regarded to have value equal to zero.

2 Auxiliary results

Several auxiliary results, which will be used in the proofs of the main results are given in this section. The first lemma is well-known (see, e.g., [9]).

Lemma 2.1. Let i∈_N0and

s⁽_nⁱ⁾(z) =1+2ⁱz+3ⁱz²+· · ·+nⁱzⁿ⁻¹, n∈_N, where z∈_C.

Then

s⁽n⁰⁾(z) = ¹−zⁿ

1−z , (2.1)

s⁽n¹⁾(_z) = ¹−(n+₁)zⁿ+nzⁿ⁺¹

(1−z)^{2 , (2.2)}

for every z∈ _C\ {1}and n ∈_N.

Remark 2.2. Since by the above mentioned conventions⁽_nⁱ⁾(z) =_∑ⁿ_j=1jⁱz^j−1=0, forn=−1, 0, a simple calculation shows that (2.1) also holds for n = 0 and z 6= 0, while (2.2) holds for n=−1 andn=0 whenz 6=0. Note also thats⁽_n¹⁾(1) = ⁿ⁽ⁿ₂⁺¹⁾.

Lemma 2.3. Letα_i,β_i, i=0,n, be complex numbers. Then

∑

α_i

n−i j

∑

β_j =

∑

β_j

n−j i

∑

α_i, (2.3)

for every n∈_N0. Proof. Let

S_n={(i,j)_{: 0}≤i≤ n, 0≤ j≤n−i}_.

From the definition of set S_n we see that variable jtakes all the values from 0 to n too, and that for a fixed j, from the inequalityj≤n−iwe have thati≤n−j, that is, the upper bound foriis equaln−j. Hence

S_n ={(i,j): 0≤ j≤n, 0≤i≤ n−j}.

Using this fact and by the changing order of summation, equality (2.3) easily follows.

Remark 2.4. Note that Lemma 2.3 is a simple sort of the Fubini theorem with the discrete measure.

Let

f_n(u,v) =

∑

uⁱ

n−i j

∑

v^j, (2.4)

whereu,v∈_Candn∈_N0, and where we in this case, as usual, regard that 0⁰=1.

Lemma 2.5. Let f_n(u,v)be defined in(2.4). Then

f_n(u,v) = f_n(v,u), (2.5)

for every u,v∈_Cand n ∈_N0.

Moreover, the following formulas hold.

(a) If u6=16=v and u 6=v, then

f_n(u,v) = ^v−u+uⁿ⁺²−vⁿ⁺²+uvⁿ⁺²−vuⁿ⁺²

(1−v)(1−u)(v−u) ^{, n}∈ _N0. (2.6)

(b) If u=v6=1, then

fn(u,u) = ¹−(n+2)uⁿ⁺¹+ (n+1)uⁿ⁺²

(1−u)^{2 , n}∈_N0. (2.7)

(c) If u6=1and v=1, then

f_n(u, 1) = ⁿ+1−(n+2)u+uⁿ⁺²

(1−u)^{2 , n}∈_N0. (2.8)

(d) If u=1and v 6=1, then

f_n(1,v) = ⁿ+1−(n+2)v+vⁿ⁺²

(₁−v)^{2 , n}∈_N0. (2.9)

(e) If u =v=1, then

f_n(1, 1) = (n+1)(n+2)

2 , n∈_N0. (2.10)

Proof. By using Lemma2.3withα_i =uⁱ,β_j =v^j,i,j=0,n, equality (2.5) follows.

(a)By using the formula for the sum of a geometric progression three times (see (2.1)), the conditionsu6=16= vandu6=v, and some simple calculation, for the casev6=0, we obtain

f_n(u,v) =

∑

uⁱ

n−i

∑

v^j =

∑

uⁱ

1−vⁿ⁻ⁱ⁺¹ 1−v

= ¹ 1−v

_n

i

∑

uⁱ−vⁿ⁺¹

∑

u v

i

= ¹ 1−v

1−uⁿ⁺¹

1−u −vvⁿ⁺¹−uⁿ⁺¹ v−u

= ^v−u+uⁿ⁺²−vⁿ⁺²+uvⁿ⁺²−vuⁿ⁺² (1−v)(1−u)(v−u) ^. Ifv=0 andu6=1, then

f_n(u, 0) =

∑

uⁱ = ¹−uⁿ⁺¹

1−u = −u+uⁿ⁺²

(1−u)(−u)^{, (2.11)}

which is nothing but formula (2.6) whenv=0.

(b) By using formula (2.1) twice, the conditions u = v 6= 1, and some simple calculation, we obtain

f_n(u,u) =

∑

uⁱ

n−i j

∑

u^j =

∑

uⁱ

1−uⁿ⁻ⁱ⁺¹ 1−u

= ¹ 1−u

_n

i

∑

uⁱ−uⁿ⁺¹

∑

1

= ¹ 1−u

1−uⁿ⁺¹

1−u −(n+1)uⁿ⁺¹

= ¹−(n+2)uⁿ⁺¹+ (n+1)uⁿ⁺² (1−u)^{2 .}

(c)By using formula (2.1), the conditionsu 6= 1 and v = 1, formula (2.2) withz = u, and some simple calculation, we obtain

f_n(u, 1) =

∑

uⁱ

n−i

∑

1=

∑

uⁱ(n−i+1)

= (n+1)

∑

uⁱ−u

∑

iuⁱ⁻¹

= (n+1)¹−uⁿ⁺¹

1−u −u1−(n+1)uⁿ+nuⁿ⁺¹ (₁−u)²

= ⁿ+1−(n+2)u+uⁿ⁺² (1−u)^{2 ,} as desired.

(d) By using equality (2.5) withu = 1, we get f_n(1,v) = f_n(v, 1). From this and by (2.8) with u→v, formula (2.9) follows.

(e)We have

fn(1, 1) =

∑

n−i

∑

1=

∑

(n−i+1) =

n+1 j

∑

j= (_n+₁)(_n+₂)

2 .

Remark 2.6. If we note that f_n(u,u) =

∑

uⁱ

n−i j

∑

u^j =

∑

n−i

∑

u^i+j =

∑

(l+1)u^l = s⁽_n¹₊⁾₁(u),

since the equation i+j=l hasl+1 nonnegative integer solutions, formula (2.7) also follows from (2.2). It is also easy to see that f_n(u, 1) = uⁿs⁽_n¹₊⁾₁(¹_u), u 6= 0, from which along with (2.2), formula (2.8) can be obtained. Note also that by using the above mentioned summing convention and some simple calculation is obtained that formulas (2.6), (2.8)–(2.10) holds for n=−1 (formula (2.7) holds forn=−1 ifu 6=0).

The following result is also known (for example, it is a consequence of the Lagrange interpolation formula).

Lemma 2.7. If all the zeros z_j, j=1,k,of the polynomial

P_k(z) =γ_kz^k+γ_k−1z^k−1+· · ·+γ₁z+γ0, are such that z_i 6=z_j, i6= j, then the following formulas hold:

∑

z^l_j P_k⁰(z_j) =0 for every l∈ {0, 1, . . . ,k−2}, and

∑

z^k_j⁻¹ P_k⁰(z_j) = ¹

γ_k.

3 Main results

The main results in this paper are proved in this section. The first one is devoted to the case b=_0.

Theorem 3.1. Assume that a,c,d ∈ _{Z, b} = 0, α,β ∈ _C\ {0} and z−1,z0,w0 ∈ _C\ {0}. Then system(1.1)is solvable in closed form.

Proof. In this case system (1.1) becomes

z_n+1 =αz^a_n, w_n+1= βw^c_nz^d_n−1, n∈_N0. (3.1) From the first equation in (3.1) it is not difficult to see that

zn =α^∑

n−1

i=0 aⁱz^a₀ⁿ, n∈_N. (3.2)

From (3.2) we easily obtain that fora6=1 zn=α

1−an

1−a z₀^an, n∈_N, (3.3)

while fora=1, we have

zn= αⁿz0, n∈_N. (3.4)

By using (3.2) into the second equation in (3.1), we get w_n= βα^d∑

n−3

i=0 aⁱz^da₀ ⁿ⁻²w^c_n−1, forn≥3. (3.5) Hence, by using (3.5) with n→n−1, we have that

w_n= βα^d∑

n−3

i=0 aⁱz^da₀ ⁿ⁻² βα^d∑

n−4

i=0 aⁱz^da₀ ⁿ⁻³w^c_n−2c

= β^1+cα^d∑

n−3

i=0 aⁱ+dc∑ⁿi=⁻0⁴aⁱz^da₀ ^{n−2+dcan−3}w^c_n²₋₂, (3.6) forn≥4.

Based on (3.5) and (3.6), assume that for some k≥2 we have proved w_n= β^∑

k−1 i=0cⁱ

α^d∑

k−1

j=0 c^j∑ⁿi=⁻0^j−3aⁱ z^d∑k

−1 i=0cⁱaⁿ⁻ⁱ⁻²

0 w^c_n^k_−k, (3.7)

forn≥ k+2.

Then by using (3.5) withn→n−kinto (3.7), we get w_n =β^∑

k−1 i=0cⁱ

α^d∑

k−1

j=0 c^j∑ⁿi=⁻0^j−3aⁱ z^d∑

k−1 i=₀cⁱaⁿ⁻ⁱ⁻²

0 βα^d∑

n−k−3

i=0 aⁱz^da₀ ^n−k−2w^c_n−k−1c^k

=β^∑

k i=0cⁱ

α^d∑

kj=0 c^j∑iⁿ=⁻0^j−3aⁱ

z₀^{d∑ki=0cian−i−2}w^c_n^k₋⁺¹_k−1 (3.8) forn≥ k+3.

From (3.5), (3.8) and the method of induction we see that (3.7) holds for all natural numbers kandnsuch that 1≤k ≤n−2.

By takingk=n−2 into (3.7) we get w_n= β^∑

n−3 i=0cⁱ

α^d∑

n−3

j=0 c^j∑ⁿi=⁻0^j−3aⁱ z^d∑n

−3 i=0cⁱaⁿ⁻ⁱ⁻²

0 w^c₂ⁿ⁻², forn≥3. (3.9)

On the other hand, from the second equation in (3.1) with n=1, we have

w₂= βw^c₁z₀^d= β(βw^c₀z^d₋₁)^cz^d₀= β^1+cw^c₀²z^cd₋₁z^d₀. (3.10) From (3.9) and (3.10), we get

w_n =β^∑

n−3 i=0 cⁱ

α^d∑

n−3

j=0 c^j∑ⁿi=⁻0^j−3aⁱ z^d∑

n−3 i=0 cⁱaⁿ⁻ⁱ⁻²

0 β^1+cw^c₀²z^cd₋₁z^d₀cⁿ⁻²

=β^∑

n−1 i=0 cⁱ

α^d∑

n−3

j=0 c^j∑ⁿi=⁻0^j−3aⁱ

w^c₀ⁿz^d∑n

−2 i=0 cⁱaⁿ⁻ⁱ⁻²

0 z^dc₋₁ⁿ⁻¹, (3.11)

forn≥3.

Now the subcasesa6=canda= cwill be considered separately.

Subcase a6= c.In this case from (3.11), we get wn= β^∑

n−1 i=₀ cⁱ

α^d∑

n−3

j=0 c^j∑ⁿi=⁻0^j−3aⁱ w^c₀ⁿz^d

an−1−cn−1 a−c

0 z^dc₋₁ⁿ⁻¹, n≥2. (3.12) Ifa6=1 andc6=1, then by formula (2.6) withn→n−3, u=_candv= a, (3.12) becomes

wn= β

1−cn 1−cα^d

a−c+cn−1−an−1+can−1−acn−1 (1−a)(1−c)(a−c) w^c₀ⁿz^d

an−1−cn−1 a−c

0 z^dc₋₁ⁿ⁻¹, n≥2. (3.13) Ifa6=canda=1, then by using formula (2.8) withn→n−3,u=candv=1, we get

w_n =β

1−cn 1−c α^d

n−2−(n−1)c+cn−1 (₁−c)2 w^c₀ⁿz^d

1−cn−1 1−c

0 z^dc₋₁ⁿ⁻¹, n≥2, (3.14) while ifa6=candc=1, then by using formula (2.9) with n→n−3,u=1 andv=a, we get

wn =βⁿα^d

n−2−(n−1)a+an−1 (1−a)² w0z^d

an−1−1 a−1

0 z^d₋₁, n≥2. (3.15)

Subcase a=c. In this case from (3.11), we get wn= β^∑

n−1 i=0 aⁱ

α^d∑

n−3

j=0 a^j∑ⁿi=⁻0^j−3aⁱ

w^a₀ⁿz^d₀^{(n−1)an−2}z^da₋₁ⁿ⁻¹, (3.16) forn≥_3.

Ifa=c6=1, then by using formula (2.7) withn→n−3 andu=v =a, (3.16) becomes w_n =β

1−an 1−a α^d

1−(n−1)an−2+(n−2)an−1

(1−a)² w^a₀ⁿz^d₀^{(n−1)an−2}z^da₋₁ⁿ⁻¹, n≥_{3, (3.17)} while ifa=c=1, then by using formula (2.10) withn→n−3 andu=v =1, we get

wn= βⁿα^d

(n−2)(n−1)

2 w0z^d₀⁽ⁿ⁻¹⁾z^d₋₁, n≥3, (3.18) finishing the proof of the theorem.

Corollary 3.2. Consider system (1.1) with a,c,d ∈ _{Z, b} = 0 and α,β ∈ _C\ {0}. Assume that z₋₁,z₀,w₀∈_C\ {₀}. Then the following statements are true.

(a) If a6=c, a 6=1and c6=1then the general solution to system(1.1)is given by(3.3)and(3.13).

(b) If a6=c and a =₁then the general solution to system(1.1)is given by(3.4)and(3.14).

(c) If a6= c and c=1then the general solution to system(1.1)is given by(3.3)and(3.15).

(d) If a=c6=1then the general solution to system(1.1)is given by(3.3)and(3.17).

(e) If a=c=1then the general solution to system(1.1)is given by(3.4)and(3.18).

Theorem 3.3. Assume that a,b,c ∈ _Z, d = 0, α,β ∈ _C\ {₀} and z₋₁,z₀,w₀ ∈ _C\ {₀}. Then system(1.1)is solvable in closed form.

Proof. A proof of the theorem was essentially given in [27], but we give a slightly modified for the completeness. In this case system (1.1) becomes

z_n+1 =αz_n^aw^b_n, w_n+1 = βw^c_n, n∈_N0. (3.19) From the second equation in (3.19) we have that

w_n=β^∑

n−1

i=0 cⁱw₀^cn, n∈_{N. (3.20)}

From (3.20) we easily obtain that forc6=1 w_n= β

1−cn

1−cw^c₀ⁿ, n∈_N, (3.21)

while forc=1, we have

wn= βⁿw₀, n∈_N. (3.22)

Employing (3.20) into the first equation in (3.19), we get z_n =αβ^b∑

n−2

i=0 cⁱw^bc₀ⁿ⁻¹z^a_n−1, (3.23) forn≥2.

Hence, by using (3.23) with n→n−1 into itself, we have zn =αβ^b∑

n−2

i=0 cⁱw^bc₀ⁿ⁻¹ αβ^b∑

n−3

i=0 cⁱw₀^bcn−2z^a_n−2a

,

=α^1+aβ^b∑

n−2

i=0 cⁱ+ba∑ⁿi=⁻0³cⁱw₀^{bcn−1+bacn−2}z^a_n²₋₂, (3.24) forn≥3.

Based on (3.23) and (3.24), assume that for somek≥2 we have proved zn= α^∑

k−1 i=0aⁱ

β^b∑

k−1

j=0 a^j∑ⁿi=⁻0^j−2cⁱ w^b∑

k−1 i=0aⁱcⁿ⁻ⁱ⁻¹

0 z^a_n^k_−k, (3.25)

forn≥ k+_1.

Then by using (3.23) with n→n−kinto (3.25), we get z_n=α^∑

k−1 i=0aⁱ

β^b∑

k−1

j=0 a^j∑ⁿi=⁻0^j−2cⁱ w^b∑k

−1 i=0aⁱcⁿ⁻ⁱ⁻¹

0 αβ^b∑

n−k−2

i=0 cⁱw^bc₀^n−k−1z^a_n−k−1a^k

=α^∑

ki=0aⁱ

β^b∑

k

j=0 a^j∑ⁿi=⁻₀^j−2cⁱ

w₀^{b∑ki=0aicn−i−1}z^a_n^k₋⁺_k¹₋₁, (3.26) forn≥ k+2.

From (3.23), (3.26) and the method of induction we see that (3.25) holds for all natural numberskandnsuch that 1≤k ≤n−_1.

By takingk=n−1 into (3.25) we get zn=α^∑

n−2 i=0 aⁱ

β^b∑

n−2

j=0 a^j∑ⁿi=⁻0^j−2cⁱ w^b∑

n−2 i=0 aⁱcⁿ⁻ⁱ⁻¹

0 z₁^an−1, (3.27)

forn≥2.

By using the relationz₁ =αz₀^aw^b₀ into (3.27) it follows that z_n =α^∑

n−2 i=0 aⁱ

β^b∑

n−2

j=0 a^j∑ⁿ_i=⁻0^j−2cⁱ w^b∑n

−2 i=0 aⁱcⁿ⁻ⁱ⁻¹

0 (αz₀^aw^b₀)^an−1

=α^∑

n−1 i=0 aⁱ

β^b∑

n−2

j=0 a^j∑ⁿi=⁻0^j−2cⁱ w^b∑

n−1 i=0 aⁱcⁿ⁻ⁱ⁻¹

0 z^a₀ⁿ, (3.28)

forn≥2.

Now the subcasesa6=canda= cwill be considered separately.

Subcase a6= c.In this case from (3.28), we get zn =α^∑

n−1 i=0 aⁱ

β^b∑

n−2

j=0 a^j∑ⁿi=⁻0^j−2cⁱ z^a₀ⁿw^b

an−cn a−c

0 , n∈ _N. (3.29)

Ifa6=1 andc6=1, then by formula (2.6) withn→n−2, u= aandv=cand (2.5), (3.29) becomes

zn =α

1−an 1−a β^b

a−c+cn−an+can−acn (1−a)(1−c)(a−c) z^a₀ⁿw^b

an−cn a−c

0 , n∈_N. (3.30)

Ifa6=canda=1, then by using formula (2.9) withn→n−2,u=1 andv=c, we get zn= αⁿβ^b

n−1−nc+cn (1−c)² z0w^b

cn−1 c−1

0 , n ∈_N, (3.31)

while ifa6=candc=1, then by using formula (2.8) with n→n−2,u= aandv=1, we get z_n= α

1−an 1−a β^b

n−1−na+an

(1−a)² z^a₀ⁿw^ban

−1 a−1

0 , n ∈_{N. (3.32)}

Subcase a=c. In this case from (3.28), we get z_n= α^∑

n−1 i=0 aⁱ

β^b∑

n−2

j=0 a^j∑ⁿi=⁻0^j−2aⁱ

z^a₀ⁿw₀^bnan−1, n≥2. (3.33) Ifa=c6=1, then by using formula (2.7) withn→n−2,u=v= a, (3.33) becomes

z_n=α

1−an 1−a β^b

1−nan−1+(n−1)an

(1−a)² z₀^anw^bna₀ ⁿ⁻¹, n≥2, (3.34) while ifa=c=1, then by using formula (2.10) withn→n−_2, u=v=_{1, we get}

z_n= αⁿβ^b

(n−1)n

2 z₀w^bn₀ , n∈_N, (3.35)

completing the proof.

Corollary 3.4. Consider system (1.1) with a,b,c ∈ _{Z, d} = 0 and α,β ∈ _C\ {0}. Assume that z−1,z₀,w₀∈_C\ {0}. Then the following statements are true.

(a) If a6=c, a 6=1and c6=1then the general solution to system(1.1)is given by(3.21)and(3.30).

(b) If a6=c and a =₁then the general solution to system(1.1)is given by(3.21)and(3.31).

(c) If a6= c and c=1then the general solution to system(1.1)is given by(3.22)and(3.32).

(d) If a=c6=1, then the general solution to system(1.1)is given by(3.21)and(3.34).

(e) If a=c=1, then the general solution to system(1.1)is given by(3.22)and(3.35).

Theorem 3.5. Assume that a,b,c,d ∈ _{Z, bd} 6= 0, α,β ∈ _C\ {0}and z−1,z₀,w₀ ∈ _C\ {0}. Then system(1.1)is solvable in closed form.

Proof. A simple inductive argument shows that the conditionsα,β∈_C\ {0}andz−1,z₀,w₀∈ C\ {0}, along with the equations in (1.1) imply zn 6= 0 forn ≥ −1 and wn 6= 0 forn ∈ _N0. Further, from the first equation in (1.1), for every well-defined solution, we have that

w^b_n= ^zn+1

αz_n^a , n∈_N0, (3.36)

while by taking the second equation in (1.1) to theb-th power, is obtained

w^b_n+1 = β^bw^bc_nz^bd_n−1, n∈_N0. (3.37) Using (3.36) into (3.37) we obtain

z_n+2

αz^a_n+1 = β^bz^c_n+1

α^cz^ac_n z^bd_n−1, n∈_N0, that is,

z_n+2 =α^1−cβ^bz^a_n⁺₊^c₁z⁻_n^acz^bd_n−1, n∈_N0. (3.38) Let

a₁ =a+c, b₁ =−ac, c₁= bd, (3.39)

x₁=1−c, y₁= b. (3.40)

Then equation (3.38) can be written as

zn+2 =α^x1β^y1z^a_n¹₊₁z^b_n¹z^c_n¹₋₁, n∈_N0. (3.41) By using recurrent relation (3.41) withn →n−1 into itself, we have

z_n+2=α^x1β^y1(α^x1β^y1z^a_n¹z^b_n¹₋₁z^c_n¹₋₂)^a1z^b_n¹z^c_n¹₋₁,

=α^x1a1+x1β^y1a1+y1z^a_n^1a1+b1z^b_n¹₋^a1₁^+c1z^c_n¹₋^a1₂

=α^x2β^y2z^a_n²z^b_n²₋₁z^c_n²₋₂, (3.42) forn∈ _{N, where}

a₂ := a₁a₁+b₁, b₂ :=b₁a₁+c₁, c₂:=c₁a₁, (3.43) x₂:=x₁a₁+x₁, y₂:= y₁a₁+y₁. (3.44) Assume that for ak ∈_Nsuch that 2≤k ≤n+1, we have proved that

z_n+2 =α^xkβ^ykz^a_n^k_+2−kz^b_n^k_+1−kz^c_n^k_−k, (3.45)