New class of practically solvable systems of difference equations of hyperbolic-cotangent-type

New class of practically solvable systems of difference equations of hyperbolic-cotangent-type

Dedicated to Professor Jeffrey R. L. Webb on the occasion of his 75th birthday

Stevo Stevi´c

^B

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

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China

Received 16 August 2020, appeared 21 December 2020 Communicated by Gennaro Infante

Abstract. The systems of difference equations x_n+1= ^unvn−2+a

un+vn−2 , y_n+1= ^wnsn−2+a

wn+sn−2 , n∈_N0,

wherea,u₀,w₀,v_j,s_j j=−2,−1, 0, are complex numbers, and the sequencesun,vn,wn, sn are xn or yn, are studied. It is shown that each of these sixteen systems is practi- cally solvable, complementing some recent results on solvability of related systems of difference equations.

Keywords: system of difference equations, general solution, solvability of difference equations, hyperbolic-cotangent-type system of difference equations.

2020 Mathematics Subject Classification: 39A45.

1 Introduction

Let N, Z, R, C, be the sets of natural, whole, real and complex numbers respectively, and N0 =_N∪ {0}. If p,q∈_Zand p≤q, thenj= p,qis a notation forj= p,p+1, . . . ,q.

First important results on solvability of difference equations and systems belong to de Moivre [5–7], D. Bernoulli [3], Euler [9], Lagrange [15] and Laplace [16]. They found a few methods for solving linear difference equations with constant coefficients, as well as methods for solving some linear difference equations with nonconstant coefficients and some nonlinear difference equations. Many books containing basic methods for solving difference equations and systems have appeared since (see, e.g., [4,11–13,18,19,21,22]). It is interesting to note

BEmail: sstevic@ptt.rs

that many difference equations and systems have naturally appeared as some mathematical models for problems in combinatorics, population dynamics and other branches of sciences (see, e.g., [5–7,11–13,15–17,20,21,31,49]). The fact that it is difficult to find new methods for solving difference equations and systems has influenced on a lack of considerable interest in the topic for a long time. Use of computers seems renewed some interest in the topic in the last two decades.

During the ’90s has started some interest in concrete difference equations and systems.

Papaschinopoulos and Schinas have influenced on the study of such systems (see, e.g., [23–

28,32,33]). Work [29] is on solvability, whereas [24–26,28,32,33] can be regarded as ones on solvability in a wider sense, since they are devoted to finding invariants of the systems studied therein. Beside their study, have appeared several papers by some other authors which essentially rediscovered some known results. These facts motivated us to study the solvability of difference equations and systems (see, e.g., [1,34–48] and many other related references therein).

Letk,l∈_N0,a∈_R(orC), and

z_n+1 = ^zn−kzn−l+a

z_n−k+z_n−l , n∈_N0. (1.1)

Equation (1.1) have been studied by several authors. Convergence of positive solutions to the equation follows from a result in [14] (see [2]). For some generalizations of the result in [14], see [8] and [27]. The fact that equation (1.1) resembles the hyperbolic-cotangent sum formula has been a good hint for solvability of the equation. Some special cases of the equation were studied in [30]. In [43] was presented a natural way for showing solvability of the equation.

The following systems

x_n+1= ^un−kvn−l+a u_n−k+v_n−l

, y_n+1 = ^wn−ksn−l+a w_n−k+s_n−l

, n∈_N0, (1.2)

wherek,l∈ _N0, a,u_−j,w_−j,v_−j⁰,s_−j⁰ ∈ _C, j=_0,k, j⁰ = _0,l, andu_n, v_n,w_n,s_nare x_n ory_n, are natural extensions of equation (1.1) (for studying the systems in the form, we have been also motivated by [34]).

The case k = _0, l = 1, was studied in [47] and [48], and also in [41] where we presented another method. We have also shown therein the theoretical solvability of the systems in (1.2).

The casek=1, l=2, has been recently studied in [40]. Here we study practical solvability of the systems in (1.2) in the casek = _{0 and}l = 2, continuing our research in [40,41,43,47,48].

We use and combine some methods from these, as well as the following works: [35–39,42,46].

The investigation of the case has been announced in [41].

2 Main results

First we mention two lemmas. The first one belongs to Lagrange (see, e.g., [10,13,46]), while the second one should be folklore (for a proof see [40]), and have been applied for several times recently (see, e.g., [38,39,46]).

Lemma 2.1. Let t_l, l = 1,m,be the roots of p_m(t) = α_mt^m+· · ·+α₁t+α₀, α_m 6= 0, and assume that t_l 6=t_j, when l 6= j.Then

∑

m l=1

t^j_l

p⁰_m(_tl) =0, j=0,m−2, and

∑

m l=1

t^m_l ⁻¹ p⁰_m(_tl) = ¹

α_m.

Lemma 2.2. Consider the equation

x_n =a₁x_n−1+a₂x_n−2+· · ·+a_mx_n−m, n≥l, (2.1) where l ∈ _{Z, aj} ∈ _{C, j} = 1,m, a_m 6= 0. Let t_k, k = 1,m,be the roots of q_m(t) = t^m−a₁t^m−1− a₂t^m−2− · · · −a_m,and assume that t_k 6=t_s, when k6=s.

Then, the solution to equation(2.1)satisfying the initial conditions

x_j−m =0, j= l,l+m−2, and x_l−1 =1, (2.2) is

xn =

∑

m k=1

tⁿ_k^+m−l

q⁰_m(t_k)^{, (2.3)}

for n≥l−m.

We transform the systems in (1.2) with k = 0 and l = 2 to some more suitable ones. We have

x_n+1±√

a = (u_n±√

a)(v_n−2±√ a) u_n+v_n−2

and y_n+1±√

a= (w_n±√

a)(s_n−2±√ a) w_n+s_n−2

, forn∈_N0, and consequently

x_n+1+√ a xn+₁−√

a = ^un+√ a un−√

a ·^vn−2+√ a vn−2−√

a, y_n+1+√ a yn+₁−√

a = ^wn+√ a wn−√

a·^sn−2+√ a sn−2−√

a, (2.4) forn∈_N0.

Hence, the following systems are studied x_n+1+√

a x_n+1−√

a = ^xn+√ a x_n−√

a· ^xn−2+√ a x_n−2−√

a, y_n+1+√ a y_n+1−√

a = ^xn+√ a x_n−√

a ·^xn−2+√ a x_n−2−√

a, (2.5)

x_n+₁+√ a x_n+1−√

a = ^xn+√ a xn−√

a ·^xn−2+√ a xn−2−√

a, y_n+₁+√ a y_n+1−√

a = ^yn+√ a yn−√

a ·^xn−2+√ a xn−2−√

a, (2.6)

x_n+1+√ a x_n+1−√

a = ^xn+√ a x_n−√

a ·^xn−2+√ a x_n−2−√

a, y_n+1+√ a y_n+1−√

a = ^xn+√ a x_n−√

a ·^yn−2+√ a y_n−2−√

a, (2.7)

x_n+1+√ a x_n+1−√

a = ^xn+√ a x_n−√

a ·^xn−2+√ a x_n−2−√

a, y_n+1+√ a y_n+1−√

a = ^yn+√ a y_n−√

a ·^yn−2+√ a y_n−2−√

a, (2.8)

x_n+1+√ a x_n+1−√

a = ^yn+√ a y_n−√

a· ^xn−2+√ a x_n−2−√

a, y_n+1+√ a y_n+1−√

a = ^xn+√ a x_n−√

a ·^xn−2+√ a x_n−2−√

a, (2.9)

x_n+1+√ a x_n+1−√

a = ^yn+√ a yn−√

a ·^xn−2+√ a x_n−2−√

a, y_n+1+√ a y_n+1−√

a = ^yn+√ a yn−√

a ·^xn−2+√ a x_n−2−√

a, (2.10) xn+1+√

a xn+₁−√

a = ^yn+√ a yn−√

a ·^xn−2+√ a xn−2−√

a, yn+1+√ a yn+₁−√

a = ^xn+√ a xn−√

a ·^yn−2+√ a yn−2−√

a, (2.11) x_n+1+√

a x_n+1−√

a = ^yn+√ a y_n−√

a ·^xn−2+√ a x_n−2−√

a, y_n+1+√ a y_n+1−√

a = ^yn+√ a y_n−√

a ·^yn−2+√ a y_n−2−√

a, (2.12) x_n+1+√

a x_n+1−√

a = ^xn+√ a x_n−√

a ·^yn−2+√ a y_n−2−√

a, y_n+1+√ a y_n+1−√

a = ^xn+√ a x_n−√

a ·^xn−2+√ a x_n−2−√

a, (2.13)

xn+₁+√ a x_n+1−√

a = ^xn+√ a xn−√

a·^yn−2+√ a yn−2−√

a, yn+₁+√ a y_n+1−√

a = ^yn+√ a yn−√

a ·^xn−2+√ a xn−2−√

a, (2.14) x_n+1+√

a x_n+1−√

a = ^xn+√ a x_n−√

a·^yn−2+√ a y_n−2−√

a, y_n+1+√ a y_n+1−√

a = ^xn+√ a x_n−√

a·^yn−2+√ a y_n−2−√

a, (2.15) x_n+1+√

a x_n+1−√

a = ^xn+√ a x_n−√

a ·^yn−2+√ a y_n−2−√

a, y_n+1+√ a y_n+1−√

a = ^yn+√ a y_n−√

a·^yn−2+√ a y_n−2−√

a, (2.16) x_n+1+√

a x_n+1−√

a = ^yn+√ a y_n−√

a ·^yn−2+√ a y_n−2−√

a, y_n+1+√ a y_n+1−√

a = ^xn+√ a x_n−√

a ·^xn−2+√ a x_n−2−√

a, (2.17) x_n+1+√

a x_n+1−√

a = ^yn+√ a yn−√

a·^yn−2+√ a yn−2−√

a, y_n+1+√ a y_n+1−√

a = ^yn+√ a yn−√

a ·^xn−2+√ a xn−2−√

a, (2.18) x_n+1+√

a xn+₁−√

a = ^yn+√ a yn−√

a·^yn−2+√ a yn−2−√

a, y_n+1+√ a yn+₁−√

a = ^xn+√ a xn−√

a·^yn−2+√ a yn−2−√

a, (2.19) x_n+1+√

a x_n+1−√

a = ^yn+√ a y_n−√

a ·^yn−2+√ a y_n−2−√

a, y_n+1+√ a y_n+1−√

a = ^yn+√ a y_n−√

a·^yn−2+√ a y_n−2−√

a, (2.20) forn∈ _N0.

Let

ζn = ^xn+√ a xn−√

a and ηn= ^yn+√ a yn−√

a, then

x_n= √

aζn+1

ζ_n−1 and y_n=√

aηn+1

η_n−1, (2.21)

and the systems (2.5)–(2.20) respectively become

ζ_n+1 =ζnζn−2, η_n+1=ζnζn−2, (2.22) ζ_n+₁ =ζ_nζ_n−2, η_n+₁=η_nζ_n−2, (2.23) ζ_n+1 =ζ_nζ_n−2, η_n+1=ζ_nη_n−2, (2.24) ζ_n+1 =ζ_nζ_n−2, η_n+1=η_nη_n−2, (2.25) ζ_n+1 =η_nζ_n−2, η_n+1 =ζ_nζ_n−2, (2.26) ζ_n+1 =ηnζ_n−2, η_n+1 =ηnζ_n−2, (2.27) ζ_n+1 =ηnζn−2, η_n+1 =ζnηn−2, (2.28) ζ_n+1 =η_nζ_n−2, η_n+1 =η_nη_n−2, (2.29) ζ_n+1 =ζ_nη_n−2, η_n+1 =ζ_nζ_n−2, (2.30) ζ_n+1 =ζ_nη_n−2, η_n+1 =η_nζ_n−2, (2.31) ζ_n+1 =ζ_nη_n−2, η_n+1 =ζ_nη_n−2, (2.32) ζ_n+1 =ζnηn−2, η_n+1 =ηnηn−2, (2.33) ζn+₁ =ηnηn−2, ηn+₁=ζnζn−2, (2.34) ζ_n+1 =η_nη_n−2, η_n+1=η_nζ_n−2, (2.35) ζ_n+1 =η_nη_n−2, η_n+1=ζ_nη_n−2, (2.36) ζ_n+1 =η_nη_n−2, η_n+1=η_nη_n−2, (2.37)

forn∈_N0.

To study the systems we use some ideas in [35–40,42,46]. The case a = 0 is simple (see [41]). Hence, it is omitted.

2.1 System (2.22) First, note that

ζn =ηn, n∈_N. (2.38)

Let

a₁ =1, b₁=0, c₁=1, (2.39)

then

ζ_n=ζ^a_n¹₋₁ζ_n^b1₋₂ζ_n^c1₋₃, n∈_N. (2.40) Use of (2.40) implies

ζn= (ζn−2ζ_n−4)^a1ζ^b_n¹₋₂ζ^c_n¹₋₃=ζ^a_n¹₋⁺₂^b1ζ^c_n¹₋₃ζ^a_n¹₋₄ =ζ_n^a2₋₂ζ^b_n²₋₃ζ^c_n²₋₄, forn≥2, where

a₂ :=a₁+b₁, b₂ :=c₁, c₂ := a₁. Assume

ζn= ζ^a_n^k_−kζ_n^bk_−k−1ζ^c_n^k_−k−2, (2.41) a_k = a_k−1+b_k−1, b_k =c_k−1, c_k = a_k−1, (2.42) for ak≥2 andn≥k.

If we use (2.40) in (2.41), we obtain

ζn= (ζ_n−k−1ζ_n−k−3)^akζ^b_n^k_−k−1ζ_n^ck_−k−2,

=ζ_n^ak₋⁺_k^b₋^k₁ζ^c_n^k_−k−2ζ_n^ak_−k−3

=ζ_n^ak₋⁺_k¹₋₁ζ^b_n^k₋⁺¹_k−2ζ_n^ck₋⁺¹_k−3, where

a_k+1:=a_k+b_k, b_k+1:=c_k, c_k+1 := a_k.

In this way, by using induction, we proved that (2.41) and (2.42) hold for every 2≤k ≤n.

From (2.39) and (2.42) we have

a_n=a_n−1+a_n−3, (2.43)

not only for n≥4, but even for alln ∈_{Z, and}

a₀=1, a₋₁ =a−2=0, a−3=1, a₋₄ =0. (2.44) By takingk=nin (2.41), and employing (2.42) and (2.43), it follows that

ζn= ζ^a₀ⁿζ^b₋ⁿ₁ζ^c₋ⁿ₂=ζ₀^anζ^a₋ⁿ₁⁻²ζ₋^an₂⁻¹, (2.45) not only for n∈N, but even forn≥ −_2.

Combining (2.38) and (2.45), we have

η_n =ζ^a₀ⁿζ^a₋ⁿ₁⁻²ζ^a₋ⁿ₂⁻¹, (2.46)

forn∈ _N.

Now note that the characteristic polynomial

P₃(λ) =λ³−λ²−1=0 (2.47)

is associated with (2.43), and it has three different roots, say λ_j, j = 1, 3. They are routinely found [10].

By using Lemma2.2, we see that a_n =

∑

3 j=1

λⁿ_j⁺²

P₃⁰(λ_j)^{, n}∈ _Z, (2.48)

is the solution to (2.43) satisfying the initial conditions a−2 =a−1=0 anda₀=1.

From (2.21), (2.45) and (2.46), the following corollary follows.

Corollary 2.3. If a6=0,then the general solution to(2.5)is x_n=√

a x0+√

a x₀−√

a

an

x−1+√ a x−1−√

a

an−2

x−2+√ a x−2−√

a

a_n−1

+1 x0+√

a x0−√

a

an

x−1+√ a x−1−√

a

a_n−2

x−2+√ a x−2−√

a

a_n−1

−₁

, n≥ −2,

y_n=√ a

x₀+√ a x0−√

a

an

x−1+√ a x−1−√

a

an−2

x−2+√ a x−2−√

a

an−1

+1 x0+√

a x₀−√

a

an

x−1+√ a x−1−√

a

an−2

x−2+√ a x−2−√

a

a_n−1

−1

, n∈_N,

where a_nis given by(2.48).

2.2 System (2.23)

First note that (2.45) holds, and that

ηn= η_n−1ζ_n−3, n∈_N. (2.49)

By using (2.45) in (2.49), we obtain η_n =η₀

∏

n j=1

ζ_j−3

=η₀

∏

n j=1

ζ₀^aj−3ζ₋^aj−₁⁵ζ₋^aj−₂⁴

=η₀ζ^∑

nj=1aj−3

0 ζ^∑

nj=1aj−5

−1 ζ^∑

nj=1aj−4

−2 , (2.50)

forn∈ _N0.

Employing (2.43) and (2.44), it follows that

∑

n j=1

a_j−3 =

∑

n j=1

(a_j−a_j−1) =a_n−1, (2.51)

∑

n j=1

a_j−5 =

∑

n j=1

(a_j−2−a_j−3) =an−2, (2.52)

∑

n j=1

a_j−4 =

∑

n j=1

(a_j−1−a_j−2) =a_n−1, (2.53)

forn∈_N0.

From (2.50)–(2.53), it follows that

η_n =η₀ζ₀^an−1ζ^a₋ⁿ₁⁻²ζ^a₋ⁿ₂⁻¹, n ∈_N0. (2.54) From (2.21), (2.45) and (2.54), the following corollary follows.

Corollary 2.4. If a6=0,then the general solution to(2.6)is x_n=√

a x₀+√

a x0−√

a

an

x−1+√ a x−1−√

a

an−2

x−2+√ a x−2−√

a

an−1

+1 x₀+√

a x₀−√

a

an

x−1+√ a x−1−√

a

an−2

x−2+√ a x−2−√

a

an−1

−1

, n≥ −2,

y_n=√ a

y0+√ a y0−√

a

x0+√ a x0−√

a

an−1

x−1+√ a x−1−√

a

an−2

x−2+√ a x−2−√

a

a_n−1

+1 y0+√

a y0−√

a

x₀+√ a x0−√

a

an−1

x−1+√ a x−1−√

a

an−2

x−2+√ a x−2−√

a

an−1

−1

, n∈_N0,

where a_nis given by(2.48).

2.3 System (2.24)

First note that (2.45) holds, and that

η_n=ζ_n−1η_n−3, forn∈N, that is,

η_3n+i =ζ_3n−1+iη_3(n−1)+i, (2.55) forn∈_N,i=−2,−1, 0.

From (2.45) and (2.55), we have η3n =η₀

∏

n j=1

ζ_3j−1

=η₀

∏

n j=1

ζ₀^a3j−1ζ^a₋^3j₁⁻³ζ^a₋^3j₂⁻²

=η₀ζ^∑

n j=1a3j−1

0 ζ^∑

n j=1a3j−3

−1 ζ^∑

n j=1a3j−2

−2 , (2.56)

forn∈_N0,

η_3n+1 =η−2

∏

n j=0

ζ_3j

=η−2

∏

n j=0

ζ₀^a3jζ^a₋^3j₁⁻²ζ^a₋^3j₂⁻¹

=η−2ζ^∑

nj=0a_3j

0 ζ^∑

nj=0a_3j−2

−1 ζ^∑

nj=0a_3j−1

−2 , (2.57)

forn≥ −1, and

η_3n+2 =η₋₁

∏

n j=0

ζ_3j+₁

=η−1

∏

n j=₀

ζ₀^a3j+1ζ^a₋^3j₁⁻¹ζ^a₋^3j₂

=η−1ζ^∑

nj=0a3j+1

0 ζ^∑

nj=0a3j−1

−1 ζ^∑

nj=0a3j

−2 , (2.58)

forn≥ −1.

Employing (2.43) and (2.44), it follows that

∑

n j=1

a_3j−3 =

∑

n j=1

(a_3j−2−a_3j−5) = a3n−2, (2.59)

∑

n j=1

a_3j−2 =

∑

n j=1

(a_3j−1−a_3j−4) = a_3n−1, (2.60)

∑

n j=1

a_3j−1 =

∑

n j=1

(a_3j−a_3j−3) =a3n−1, (2.61)

∑

n j=0

a_3j−2 =

∑

n j=0

(a_3j−1−a_3j−4) = a_3n−1, (2.62)

∑

n j=0

a_3j−1 =

∑

n j=0

(a_3j−a_3j−3) =a_3n−_{1, (2.63)}

∑

n j=0

a_3j =

∑

n j=0

(a_3j+1−a_3j−2) = a_3n+1, (2.64)

∑

n j=0

a_3j+₁ =

∑

n j=0

(a_3j+₂−a_3j−1) = a_3n+2, (2.65) Use of (2.59)–(2.65) in (2.56)–(2.58), yield

η_3n= η₀ζ₀^a3n−1ζ^a₋³ⁿ₁⁻²ζ₋^a3n₂⁻¹, (2.66) forn∈ _N0,

η_3n+₁= η₋₂ζ₀^a3n+1ζ^a₋³ⁿ₁⁻¹ζ₋^a3n₂⁻¹, (2.67) forn≥ −1, and

η_3n+2= η−1ζ₀^a3n+2ζ^a₋³ⁿ₁⁻¹ζ₋^a3n₂⁺¹, (2.68) forn≥ −1.

From (2.21), (2.45), (2.66)–(2.68), the following corollary follows.

Corollary 2.5. If a6=0,then the general solution to(2.7)is xn=√

a x0+√

a x0−√

a

an

x−1+√ a x−1−√

a

an−2

x−2+√ a x−2−√

a

an−1

+1 x0+√

a x0−√

a

an

x−1+√ a x−1−√

a

an−2

x−2+√ a x−2−√

a

an−1

−1

, n≥ −2,

y_3n=√ a

y0+√ a y₀−√

a

x₀+√ a x₀−√

a

a3n−1

x−1+√ a x−1−√

a

a3n−2

x−2+√ a x−2−√

a

a3n−1

+1 y0+√

a y0−√

a

x0+√ a x0−√

a

a3n−1

x−1+√ a x−1−√

a

a3n−2

x−2+√ a x−2−√

a

a3n−1

−1

, n∈_N0,

y_3n+1=√ a

y−2+√ a y−2−√

a

x0+√ a x0−√

a

a3n+1

x−1+√ a x−1−√

a

a3n−1

x−2+√ a x−2−√

a

a3n−1

+1 y−2+√

a y−2−√

a

x0+√ a x0−√

a

a3n+1

x−1+√ a x−1−√

a

a3n−1

x−2+√ a x−2−√

a

a3n−1

−1

, n≥ −1,

y_3n+2=√ a

y−1+√ a y−1−√

a

x0+√ a x0−√

a

a3n+2

x−1+√ a x−1−√

a

a3n−1

x−2+√ a x−2−√

a

a3n+1

+1 _y

−1+√ a y−1−√

a

x0+√ a x0−√

a

a3n+2

x−1+√ a x−1−√

a

a3n−1

x−2+√ a x−2−√

a

a3n+1

−1

, n ≥ −1, where sequence a_n is given by(2.48).