Solvability of a class of hyperbolic-cosine-type difference equations

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, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence, and indica e i changes we e made. The images o o he hi d pa y ma e ial in his a icle a e included in he a icle’s C ea i e Commons licence, unless indica ed o he wise in a c edi line o he ma e ial. I ma e ial is no included in he a icle’s C ea i e Commons licence and you in ended use is no pe mi ed by s a u o y egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission di ec ly om he copy igh holde . To iew a copy o his licence, isi h p://c ea i ecommons.o g/licenses/by/4.0/. S e i´ ce al.Ad ances in Diffe ence Equa ions (2020) 2020:564 Page 2 o 12 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 S e i´ ce al.Ad ances in Diffe ence Equa ions (2020) 2020:564 Page 3 o 12 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, S e i´ ce al.Ad ances in Diffe ence Equa ions (2020) 2020:564 Page 4 o 12 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}. S e i´ ce al.Ad ances in Diffe ence Equa ions (2020) 2020:564 Page 5 o 12 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. S e i´ ce al.Ad ances in Diffe ence Equa ions (2020) 2020:564 Page 6 o 12 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, S e i´ ce al.Ad ances in Diffe ence Equa ions (2020) 2020:564 Page 7 o 12 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. S e i´ ce al.Ad ances in Diffe ence Equa ions (2020) 2020:564 Page 8 o 12 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, S e i´ ce al.Ad ances in Diffe ence Equa ions (2020) 2020:564 Page 9 o 12 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)