S e i´
ce al.Ad ances in Diffe ence Equa ions (2020) 2020:564
h ps://doi.o g/10.1186/s13662-020-03027-8
RESEARCH Open Access
Sol abili y o a class o
hype bolic-cosine- ype di e ence equa ions
S e o S e i´
c1,2,3*, B a isla I iˇ
canin4,5, Wi old Kosmala6and Zdenˇ
ek Šma da3
*Co espondence: [email p o ec ed]
1Ma hema ical Ins i u e o he
Se bian Academy o Sciences, Knez
Mihailo a 36/III, 11000 Beog ad,
Se bia
2Depa men o Medical Resea ch,
China Medical Uni e si y Hospi al,
China Medical Uni e si y, Taichung
40402, Taiwan, Republic o China
Full lis o au ho in o ma ion is
a ailable a he end o he a icle
Abs ac
We desc ibe a me hod o cons uc ing one o he basic classes o sol able
hype bolic-cosine- ype diffe ence equa ions, gene alizing a known diffe ence
equa ion by Laplace in a na u al way.
MSC: P ima y 39A20; seconda y 39A06; 39A45
Keywo ds: Diffe ence equa ion; Sol able equa ion; Closed- o m o mula;
Hype bolic-cosine- ype diffe ence equa ion
1 In oduc ion
Le N,Z,R,Cbe he se s o na u al, whole, eal, and complex numbe s, espec i ely, and
N0=N∪{0}.I k,l∈Z, henj=k,ls ands o hese o allj∈Zsuch ha k≤j≤l.
Finding closed- o m o mulas o solu ions o diffe ence equa ions is one o he basic
p oblems in he a ea. The equa ion
yn+2 =ayn+1 +byn,n∈N0,(1)
whenb=0anda2+4b=0,wassol edbydeMoi ein[1](seealso[2]).He ounda o mula
o solu ion oequa ion(1), which is called he de Moi e o mula o solu ions o linea
homogeneous second-o de diffe ence equa ion wi h cons an coefficien s, which is one
o he fi s non i ial esul s on sol abili y o diffe ence equa ions. Be o e i , some special
cases o equa ion (1)hadbeensol edin[3].
These esul sa ac edsomea en ion,andsoona e ha Be noulliin[4] oundano he
me hod o sol ing linea diffe ence equa ions wi h cons an coefficien s. A p esen a ion
o some old esul s on sol abili y can be ound in [5]. Fo some la e esul s see [6,7], as
wellas[8],whe emanyclasses o diffe enceequa ionsandsys emswe esol ed.Fo some
wen ie h cen u y p esen a ions o he heo y, see, o example, [9–13]. Some ecen e-
sul sonsol abili yo diffe enceequa ionsandsys emsha ebeenob ainedandguessedby
compu e packages o symbolic calcula ions. They can help in ge ing o guessing some
closed- o m o mulas o solu ions o heequa ionsandsys ems,bu usingonlysuch ools
could also p oduce some issues (see some commen s, e.g., in [14–17]). This has been one
o he easonswhichmo i a edus oconduc mo ese iousin es iga ionsonsol abili yo
©The Au ho (s) 2020. This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which pe mi s use,
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diffe ence equa ions and sys ems. Recen in es iga ions show ha s ill a g ea majo i y o
pape s on sol abili y use some subs i u ions o a ious complexi y, which ans o m di -
e ence equa ions and sys ems o known sol able ones (see, o example, [14,18–26]and
he e e ences he ein).I shouldbemen ioned ha sol ablediffe enceequa ionsandsys-
emsha emanyapplica ions(see,e.g.,[2,4,7–10,12,27–29]).Fo in a ian s o diffe ence
equa ions and sys ems, and hei applica ions in sol abili y, see, o example, [30–35].
Recen ly, sol abili y o he so-called hype bolic-co angen - ype diffe ence equa ions, as
well as o he co esponding sys ems o diffe ence equa ions, has been s udied (see [22–
26]). The diffe ence equa ions and sys ems he ein esemble he hype bolic-co angen
sum o mula which has been a good hin o sol abili y o he equa ions and sys ems.
Gene ally speaking, he diffe ence equa ions and sys ems which esemble some igono-
me ic o hype bolic igonome ic o mulasa e na u alcandida es obe sol able.Thisis
an obse a ion known o ma hema icians o a long ime.
Now, as a mo i a ion o he s udy, we p esen a known example along wi h he mos
impo an de ails ela ed o sol abili y o he equa ion in he example.
Example1 The ollowing diffe ence equa ion
xn+1 =x2
n–2, n∈N0,(2)
was al eady known o Laplace [8]. He no iced ha equa ion (2)issol able.Namely,i
x0∈C, hen he e is a∈C {0}such ha
x0=a+1
a(3)
(see, e.g., [36]).
By using (3)in(2), hen epea ing he p ocedu e, he no iced ha
x1=a+1
a2–2=a2+1
a2,
x2=a2+1
a22–2=a4+1
a4,
x3=a4+1
a42–2=a8+1
a8,
and concluded
xn=a2n+1
a2n,n∈N0,(4)
which is easily p o ed by induc ion. Laplace did no conduc u he analysis o solu ions
o equa ion (2).
In wha ollows we men ion se e al simple olklo e hings ela ed o sol abili y o he
equa ionin hecasewhenx0isa ealnumbe .I x0≥2, henx0canbew i enin he o m
gi enin(3) o somea>0.Fo x0=2,weha ex1= 2, and by he me hod o induc ion,
cons an solu ion xn=2 o e e yn∈N0is easily ob ained. In his case he e is unique a
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such ha (3)holds,namelya=1.I x0>2, hen he ea e woposi i e alueso asuch ha
(3) holds. They a e he oo s o he quad a ic polynomial 2–x0 +1, ha is,
a1,2 =x0±x2
0–4
2.(5)
I one o he numbe s a1,2 is deno ed by a, by he Viè e o mulas, we see ha he second
oneis 1/a.So,since o mula(4)isin a ian unde he ans o ma iona→1/a,whiche e
o hese wo numbe s is used, he o mula is always ob ained.
By combining (4)and(5), we see ha he solu ion o equa ion (2)in hiscasecanbe
w i en as ollows:
xn=x0+x2
0–4
22n
+x0+x2
0–4
2–2n
,n∈N0.
I x0≤–2, henx1=x2
0–2≥2.Thismeans ha hiscaseis educed o hep e iousone.
Namely, i
x2
0–2=x1=b+1
b(6)
o some b>0, hen
xn=b2n–1 +1
b2n–1 ,n∈N.(7)
F om (6)weha eb2–(x2
0–2)b+1=0,so ha
b1,2 =x2
0–2±|x0|x2
0–4
2,
and b1=1/b2.Using hisin(7), we ge
xn=x2
0–2–x0x2
0–4
22n–1
+x2
0–2–x0x2
0–4
2–2n–1
,n∈N.(8)
I a>0,no e ha (4)canbew i enin he ollowing o m:
xn=e2nlna+1
e2nlna=2cosh2nlna=2cosh2n
a,n∈N0,(9)
whe e
a=lna.Thismeans ha i x0=2cosh
a, henxn=2cosh(2n
a), n∈N0.
Bea ing in mind he o m o o mula (9), we see ha equa ion (2) is closely ela ed o
he hype bolic cosine unc ion. This connec ion is no so s ange a all. Namely, by using
hechangeo a iablesxn=2˜
xn,n∈N0,inequa ion(2), i is ans o med o he ollowing
one:
˜
xn+1 =2˜
x2
n–1, n∈N0,
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which esembles he o mula
cosh2x=2cosh2x–1.
Because o his, i is na u al o say ha equa ion (2)in hecase|x0|≥2 is an example o
hype bolic-cosine- ype diffe ence equa ions.
I x0∈[–2,2], hen x0can be w i en in he o m gi en in (3) o somea=eiθ,whe e
θ∈[0,2π). In his case we ha e x0=2cosθ,whe eas om(4)wege
xn=2cos2nθ,n∈N0.
Hence, in he case x0∈[–2,2], equa ion (2) is an example o cosine- ype diffe ence equa-
ions.
A na u al p oblem is o y o find ela ed hype bolic-cosine- ype diffe ence equa ions,
whicha ealsosol able.Thisp oblemseemsclassicalone,bu wecouldno findacomple e
solu ion o he p oblemin heli e a u eso a .Beside his,i is good oha eall he hings,
someo whichseemsca e edin heli e a u e,in hesameplace.Hence,weconside he e
hep oblemin de ail. Weshow ha he eis a na u al sequenceo hype bolic-cosine- ype
diffe ence equa ionswhicha esol ableinclosed o m anddesc ibe asimplecons uc i e
way o ob aining he sequence o equa ions.
2 A basic class o sol able hype bolic-cosine- ype di e ence equa ions
In his sec ion we explain how a na u al class/sequence o hype bolic-cosine- ype diffe -
ence equa ions ela ed oequa ion (2) is ob ained, which a e also sol able.
2.1 Basic ideas and equa ions
Fi s , no e ha he main hing connec ed o sol abili y o equa ion (2) is he ac ha he
ollowing ela ion holds:
a2+1
a2=a+1
a2–2 (10)
o e e y a∈C {0}, which is a simple, bu no doub e y use ul, ela ion be ween he
quan i ies
Ik:=ak+1
ak
o k=1andk=2.
The conside a ion in Example 1sugges s ha i we can exp ess he quan i y I3in e ms
o I1in a simila way, hen we can ob ain ano he sol able diffe ence equa ion. I is no
difficul o see ha such a ela ion exis s. Namely, we ha e
a+1
a3=a3+1
a3+3a+1
a(11)
o e e y a∈C {0}.
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Le he sequence (xn)n∈N0be a solu ion o he ollowing diffe ence equa ion:
xn+1 =x3
n–3xn,n∈N0, (12)
and x0∈C.
W i e he ini ial alue x0in he o m in (3). Then we ha e
x1=a+1
a3–3a+1
a=a3+1
a3,
and by a simple induc i e a gumen , we ob ain
xn=a3n+1
a3n,n∈N0. (13)
The co esponding conside a ion in Example 1shows ha equa ion (12)when|x0|≥
2 is also an example o a hype bolic-cosine- ype diffe ence equa ion, which is sol able.
Mo eo e , we see ha he ollowing esul holds.
P oposi ion1 Conside equa ion (12). Then he ollowing s a emen s hold:
(a) I x0∈Cis gi en by (3), hen he solu ion o he equa ion is gi en by (13).
(b) I x0≥2, hen he solu ion o he equa ion is gi en by
xn=x0+x2
0–4
23n
+x0+x2
0–4
2–3n
,n∈N0.
(c) I x0≤–2, hen he solu ion o he equa ion is gi en by
xn=x2
0–2–x0x2
0–4
23n–1
+x2
0–2–x0x2
0–4
2–3n–1
,n∈N.
(d) I |x0|≤2and x0=2cosθ o some θ∈[0,2π), hen he solu ion o he equa ion is
gi en by
xn=2cos3nθ,n∈N0.
Following he abo e idea, we can y o exp ess hequan i y I4in e ms o I1ina simila
way, and hen use he ela ion in o de o ob ain ano he sol able diffe ence equa ion.
Namely, we ha e
a+1
a4=a4+1
a4+4a2+1
a2+6=a4+1
a4+4a+1
a2–2. (14)
Le he sequence (xn)n∈N0be a solu ion o he ollowing diffe ence equa ion:
xn+1 =x4
n–4x2
n+2, n∈N0, (15)
and x0∈C.

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W i e he ini ial alue x0in he o m in (3). Then we ha e
x1=a+1
a4–4a+1
a2+2=a4+1
a4,
and by a simple induc i e a gumen , we ob ain
xn=a4n+1
a4n(16)
o n∈N0.
So, equa ion (15) is also an example o a hype bolic-cosine- ype diffe ence equa ion,
which is sol able. Mo eo e , we see ha he ollowing esul holds.
P oposi ion2 Conside equa ion (15). Then he ollowing s a emen s hold:
(a) I x0∈Cis gi en by (3), hen he solu ion o he equa ion is gi en by (16).
(b) I x0≥2,gene al solu ion o he equa ion is gi en by
xn=x0+x2
0–4
24n
+x0+x2
0–4
2–4n
,n∈N0.
(c) I x0≤–2,gene al solu ion o he equa ion is gi en by he ollowing o mula:
xn=x2
0–2–x0x2
0–4
24n–1
+x2
0–2–x0x2
0–4
2–4n–1
,n∈N.
(d) I |x0|≤2and x0=2cosθ o some θ∈[0,2π), hen he solu ion o he equa ion is
gi en by
xn=2cos4nθ,n∈N0.
The co esponding ela ion be ween I5and I1is he ollowing:
a+1
a5=a5+1
a5+5a3+1
a3+10a+1
a
=a5+1
a5+5a+1
a3–5a+1
a, (17)
whe e in he las equali y we ha e used ela ion (11).
Le he sequence (xn)n∈N0be a solu ion o he ollowing diffe ence equa ion:
xn+1 =x5
n–5x3
n+5xn,n∈N0, (18)
and x0∈C.
W i e he ini ial alue x0in he o m in (3). Then we ha e
x1=a+1
a5–5a+1
a3+5a+1
a=a5+1
a5,
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and by a simple induc i e a gumen , we ob ain
xn=a5n+1
a5n,n∈N0. (19)
So, equa ion (18) is ano he example o a hype bolic-cosine- ype diffe ence equa ion,
which is sol able. Mo eo e , we see ha he ollowing esul holds.
P oposi ion3 Conside equa ion (18). Then he ollowing s a emen s hold:
(a) I x0∈Cis gi en by (3), hen he solu ion o he equa ion is gi en by (19).
(b) I x0≥2,gene al solu ion o he equa ion is gi en by
xn=x0+x2
0–4
25n
+x0+x2
0–4
2–5n
,n∈N0.
(c) I x0≤–2,gene al solu ion o he equa ion is gi en by he ollowing o mula:
xn=x2
0–2–x0x2
0–4
25n–1
+x2
0–2–x0x2
0–4
2–5n–1
,n∈N.
(d) I |x0|≤2and x0=2cosθ o some θ∈[0,2π), hen he solu ion o he equa ion is
gi en by
xn=2cos5nθ,n∈N0.
The co esponding ela ion be ween I6and I1is he ollowing:
a+1
a6=a6+1
a6+6a4+1
a4+15a2+1
a2+20
=a6+1
a6+6a+1
a4–9a+1
a2+2, (20)
whe e in he las equali y we ha e used (10)and(14).
Le he sequence (xn)n∈N0be a solu ion o he ollowing diffe ence equa ion:
xn+1 =x6
n–6x4
n+9x2
n–2, n∈N0, (21)
and x0∈C.
W i e he ini ial alue x0in he o m in (3). Then we ha e
x1=a+1
a6–6a+1
a4+9a+1
a2–2=a6+1
a6,
and by a simple induc i e a gumen , we ob ain
xn=a6n+1
a6n,n∈N0. (22)
So, equa ion (21) is ano he example o a hype bolic-cosine- ype diffe ence equa ion,
which is sol able on a domain. Mo eo e , we see ha he ollowing esul holds.
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P oposi ion4 Conside equa ion (21). Then he ollowing s a emen s hold:
(a) I x0∈Cis gi en by (3), hen he solu ion o he equa ion is gi en by (22).
(b) I x0≥2,gene al solu ion o he equa ion is gi en by
xn=x0+x2
0–4
26n
+x0+x2
0–4
2–6n
,n∈N0.
(c) I x0≤–2,gene al solu ion o he equa ion is gi en by he ollowing o mula:
xn=x2
0–2–x0x2
0–4
26n–1
+x2
0–2–x0x2
0–4
2–6n–1
,n∈N.
(d) I |x0|≤2and x0=2cosθ o some θ∈[0,2π), hen he solu ion o he equa ion is
gi en by
xn=2cos6nθ,n∈N0.
Rema k 1Rela ions(10), (11), (14), (17), and (20) a e well known and a e equen ly used
in a ious si ua ions such as in sol ing he polynomial equa ions
akxk+ak–1xk–1 +···+a1x+a0=0,
in he case aj=ak–j,j=0,k(see, e.g., [37]).
2.2 Main equa ion
Byusing hep ocedu ep ecedingP oposi ions1–4,o he sol ablehype bolic-cosine- ype
diffe ence equa ions can be ound. Howe e , he co esponding ela ions become mo e
and mo e complica ed, so he me hod is no so effec i e. No e ha equa ions (2), (12),
(15), (18), and (21)canbew i enin he o m
xn+1 =Pk(xn), n∈N0, (23)
whe e
P2( )= 2–2,
P3( )= 3–3 ,
P4( )= 4–4 2+2,
P5( )= 5–5 3+5 ,
P6( )= 6–6 4+9 2–2.
Hence, i is o some in e es o find a polynomial class (Pk)k∈Ncon aining hem.
Todo his,i shouldbesaid ha a e yuse ul ac ela ed o hesequenceo polynomials
Pk( ), k∈N, is ha hey sa is y a linea ecu si e ela ion o second o de . Namely, le
:=a+1
a,
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hen
Pk( )=ak+1
ak. (24)
Since
ak+1
aka+1
a=ak+1 +1
ak+1 +ak–1 +1
ak–1 ,
we ha e Pk+1( )–P1( )Pk( )+Pk–1( )=0, ha is,
Pk+1( )– Pk( )+Pk–1( )=0 (25)
o k≥2, which is he desi ed ecu si e ela ion.
F om his, since ini ial alues a e
P1( )= and P2( )= 2–2, (26)
all hepolynomialsPk( )canbecalcula ed ecu si ely.Mo eo e ,sincei isahomogeneous
linea diffe ence equa ion, i can be sol ed in a closed o m.
Indeed, hecha ac e is icpolynomialassocia edwi hequa ion(25)is
P2(λ)=λ2– λ+1,
and i s oo s a e
λ1,2 = ±√ 2–4
2. (27)
Hence, gene al solu ion o equa ion (25) has he ollowing o m:
Pk( )=c1 +√ 2–4
2k+c2 –√ 2–4
2k,k∈N. (28)
F om (26)and(28), we ha e
c1 +√ 2–4
2+c2 –√ 2–4
2= ,
c1 +√ 2–4
22+c2 –√ 2–4
22= 2–2.
(29)
The de e minan o sys em (29)is
=
+√ 2–4
2 –√ 2–4
2
( +√ 2–4
2)2( –√ 2–4
2)2=–√ 2–4.
Hence, a e some calcula ions, we ha e
c1=1

–√ 2–4
2
2–2 ( –√ 2–4
2)2= 1 (30)