Research Article Developing a Series Solution Method of q-difference Equations

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1 Appied Mahemaics Voume 2013, Arice ID , 4 pages hp://dx.doi.org/ /2013/ Research Arice Deveoping a Series Souion Mehod of q-difference Equaions Hsuan-Ku Liu Deparmen of Mahemaics and Informaion Educaion, Naiona Taipei Universiy of Educaion, Taiwan Correspondence shoud be addressed o Hsuan-Ku Liu; hiu.nccu@gmai.com Received 16 November 2012; Acceped 14 Apri 2013 Academic Edior: Fiomena Cianciaruso Copyrigh 2013 Hsuan-Ku Liu. This is an open access arice disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he origina wor is propery cied. The series souion is widey appied o differenia equaions on R bu is no found in q-differenia equaions. Appying he Tayor and muipicaion rue of wo generaized poynomias, we deveop a series souion of inear homogeneous q-difference equaions. As an exampe, he series souion mehod is used o find a series souion of he second-order q-difference equaion of Hermie s ype. 1. Inroducion Severa wors have been done receny for series souions on cerain ime scaes. One of he difficuies for deveoping a heory of series souions for inear homogeneous equaions on ime scaes is ha he formua for muipicaion by wo generaized poynomias is no easiy found. If he ime scae has consan graininess, Haie and Ha 1 provided an exac formua for he muipiciy of wo generaized poynomias. Using he obained resus, he series souions for inear dynamicequaionsareproposedonheimescaesr and T hz (difference equaions wih sep size h). On generaized ime scaes, Mozyrsa and Pawłuszewicz 2 presened he formua for he muipiciy of he generaized poynomias of degree one and degree n N. Le 0<q<1andusehenoaions q N {q n n N}, q N q N {0}, (1) where N denoes he se of posiive inegers. Liu 3presened a formua for he muipicaion of wo q-poynomias. The obained resus are used o deveop a series souion mehod of he second-order difference equaions on q N.Precisey,he second-order q-difference equaion is described as u ΔΔ () +g() u Δ () +f() u () 0, q N, (2) where f and g are boh q-anayic funcions a 0 in he inerva (c, d). As an exampe, he series souion mehod is appied o consider he q-hermie s equaion of he form u ΔΔ () u Δ () +λu() 0, q N (3) wih iniia condiion u(0) a and u Δ (0) b. Thispaper is organizedasfoows: in Secion 2, basic ideas on q-cacuus are inroduced. The series souion mehod is deveoped in Secion 3 and is appied o consider he q- Hermie s equaion in Secion 4.Finay,a concise concusion is provided in Secion A Basic Inroducion o Time Scaes A ime scae means an arbirary nonempy cosed subse of hereanumbers.thecacuusofimescaeswasiniiaedby Liu 3 in order o creae a heory ha can unify discree and coninuous anaysis. Then, we inroduce he dea derivaive by saring o defineheforwardandbacwardjumpoperaors. Definiion 1. Le T be a ime scae. For T, we define he forward jump operaor σ:t T by σ () : inf {s > s T}, (4) whie he bacward jump operaor ρ:t T by ρ () : sup {s < s T}. (5)

2 2 Appied Mahemaics Definiion 2. The graininess funcion μ : T 0,) is defined by μ () : σ (). (6) According o he basic definiions, we can give some usefu reaionships concerning he dea derivaive. Le a and q be rea numbers such ha 0<q<1.The q-shif facoria 4 is defined by (a; q) 0 1, (a;q) n n 1 (1 aq ), n1,2,...,n. Assume f:t R is a funcion and T. Theqderivaive 5a is defined by (7) f Δ () f(q) f(). (8) (q 1) A q-difference equaion is an equaion ha conains qderivaives of a funcion defined on q N. Definiion 3. OnheimescaeT,heq-poynomias h (, 0 ): T R are defined recursivey as foows: h 0 (, s) 1, h +1 h (τ, s) Δτ. (9) s Hence, for each fixed s, he dea derivaive of h wih respec o saisfies h Δ (, s) h 1 (, s), 1. (10) By compuing he recurrence reaion, he q-poynomias can be represened as 1 sq h (, s) (11) j0 qj on q N 5. Agarwa and Bohner 6 give a Tayor s formua for funcions on a genera ime scae. On q N,heTayor sformua canberewrienashefoowingform. Theorem 4. Le n N.Supposef is n imes differeniabe on q N.Leα, q N.Onehas f () n 1 h f Δ (α) + ρ n 1 () α h n 1 (, σ (τ)) f Δn (τ) Δτ. (12) Before deveoping he series souion mehod, we inroduce he q-anayic funcion on q N. Definiion 5. A rea-vaued funcion f:q N R is said o be q-anayic a 0 if and ony if here is a power series cenered a 0 ha converges o f near 0 ; ha is, here exis coefficiens {a } and poins c, d qn such ha c< 0 <dand for a (c,d) q N. f () a h (, 0 ) (13) The producion rue of wo q-poynomias a 0 which wi be used o derive he series souion in foowing secions 3. Theorem 6. Le h i (, 0) and h j (, 0) be wo q-poynomias a zero. One has Proof. Since we have h i+j (, 0) i 1 ( h i (, 0) h j (, 0) h i+j (, 0) (q; q) j h i+j (, 0). (14) i+ μ0 i+ )( μ0 qμ ) μ0 qμ i h i (, 0) ( μ0 qμ )j ( μ0 qμ h i (, 0) ( )( μ0 qμ h i (, 0) h j (, 0) ( This impies ha μ0 μ0 qμ qμ, (15) i+ i 1 μ0 qμ ) i+ q μ )( i 1 μ0 qμ ) +i μ0 qμ ). (16) h i (, 0) h j (, 0) ( )h μ0 qμ i+j (, 0) +i μ0 qμ (1 q υ+i+1 ) (1 q υ+1 ) h i+j (, 0) (q; q) j h i+j (, 0). (17) Proposiion 7. Le h i () and h j () be any wo q-poynomias. One has h i (, 0) h j (, 0) h j (, 0) h i (, 0). (18)

3 Appied Mahemaics 3 Proof. Wihou oss of generaiy, we suppose i>jand have (q, q) j (q j+1 ;q) i (q, q) i (1 qj+1 ) (1 q i+j ) (1 q i ) (1 qj+1 ) (1 q i+j ) (1 q i ) 0. (1 qi+1 ) (1 q i+j ) (1 q) (1 q j ) (1 qi+1 ) (1 q i+j )(1 q j+1 ) (1 q i ) (1 q j )(1 q j+1 ) (1 q i ) This impies ha (19) we ge g () u Δ () G () h (, 0) U (+1) h (, 0) G () U (+1 )(h (, 0) h (, 0)) f () u () G () U (+1 ) (q+1,q) h (q, q) (, 0), F () h (, 0) U () h (, 0) (q, q) j (q j+1 ;q) i. (20) (q, q) i F () U ( )(h (, 0) h (, 0)) Therefore, we have by Theorem 6. h i (, 0) h j (, 0) h j (, 0) h i (, 0) (21) 3. Deveoping Series Souions Mehod Using he Tayor series on ime scaes, we deveop a series souion mehod for soving q-difference equaions in his secion. Consider a second-order q-differenceequaion u ΔΔ () +g() u Δ () +f() u () 0, q N, (22) where f and g are boh q-anayic funcions a 0 in he inerva (c, d). Hence, here exis wo sequences of coefficiens {F()} and {G()} such ha f () F () h (, 0), g() G () h (, 0) (23) for a (c,d) q N. Onecanfindapowerseriessouionofheform u () by carrying ou he foowing seps. Sep 1. Since U () h (, 0), (24) F () U ( ) (q+1,q) h (q, q) (, 0). Subsiuing (24)and(26)ino(22), we ge he equaion (26) U (+2) + G () U (+1 ) (q+1,q) (q, q) h (, 0) 0. + F () U ( ) (q+1,q) (q, q) (27) Sep 2. Se he coefficiens of he power series equa o zero. Thagivesarecurrencereaionhareaesaercoefficiens in he power series (24) o he earier ones. Tha is, U (+2) + + G () U (+1 ) (q+1,q) (q, q) F () U ( ) (q+1,q) 0. (q, q) (28) Sep 3. Find a coefficiens U() in erms of he firs wo coefficiens U(0) and U(1), hus wriing he q-series in he form u Δ () u ΔΔ () U (+1) h (, 0), U (+2) h (, 0), (25) U () h (, 0) U(0) u 1 () +U(1) u 2 (), (29) where u 1 and u 2 are wo ineary independen q-series souions.

4 4 Appied Mahemaics 4. Appicaions In his secion, he series souion mehod is appied o consider he q-hermie sequaionswihiniiacondiions. Consider he q-hermie s equaion of he form u ΔΔ () u Δ () +λu() 0, q N, (30) wih u(0) a and u Δ (0) b. Le u () U () h (, 0), (31) hen u(0)au(0)and u Δ (0) b U(1). Appying (31) ino (30), we have U (2) λu(0), (q ;q) U (+2) 1 λ U () (1 q ) (q; q) 1 λu(), (32) where 1,2,... This impies ha U (2) (1 q2( 1) ) i1 (1 q2(i 1) ) U (2 + 1) (1 q2 1 ) Hence, we ge i1 u () a(1+ 1 i1 +b(h 1 () + (1 q2i 1 ) (1 q2(i 1) ) 1 au 1 () +bu 2 (). i1 λu(2 ( 1)) λa, λu(2 1) λb. (1 q2i 1 ) λh 2 ()) λh 2+1 ()) By compuing he Wronsian of u 1 and u 2 a 0,wege (33) (34) Exampe 8. Consider he q-hermie s equaion wih q1/2 of he form u ΔΔ u Δ +u0 (36) wih u(0) 1 and u Δ 0.Subsiuing(31)ino(36)yieds U (2) Π i1 1 (1/2)2(i 1) (1/2) and U(2 + 1) 0 which impies ha u () Concusion 1 1Π i1 1 (1 2 ) 2i 3 Π i1 1 (1 2 ) 2i 3h 2 () 1 h 2 () 1 2 h 4 () 3 8 h 6 (). (37) (38) Oneareawhichisheacofdeveopmenisheheoryof series souions on q-difference equaions. In his paper, we presen he formua for he muipiciy of wo q-poynomias a 0. Thepurposeisoprovidehebasicmechanicsforfinding he series souions of inear homogeneous q-difference equaion. As an exampe we consider series souion of he q- Hermie s equaion of he form u ΔΔ u Δ +λu0,wihhe iniia condiions u(0) a and u Δ (0) b.usinghepresened mehod, he series souion of he Hermie s equaion can be obained ieraivey. In fuure sudies, we woud appy he presened mehod o find he series souion of oher qdifference equaions on q N. References 1 B. D. Haie and L. M. Ha, Poynomia and series souions of dynamic equaions on ime scaes, Dynamic Sysems and Appicaions,vo.12,no.1-2,pp , D. Mozyrsa and E. Pawłuszewicz, Hermie s equaions on ime scaes, Appied Mahemaics Leers,vo.22,no.8,pp , H.-K. Liu, Appicaion of a differenia ransformaion mehod o srongy noninear damped q-difference equaions, Compuers & Mahemaics wih Appicaions, vo.61,no.9,pp , T. H. Koornwinder, q-specia funcions: a uoria, hp://arxiv.org/abs/mah/ M. Bohner and A. Peerson, Dynamic Equaions on Time Scaes: An Inroducion wih Appicaions, Birhäuser, Boson, Mass, USA, R.P.AgarwaandM.Bohner, Basiccacuusonimescaesand some of is appicaions, Resus in Mahemaics,vo.35,no.1-2, pp. 3 22, Wu 1,u 2 (0) u 1 (0) u Δ 2 (0) u 2 (0) u Δ 1 (0) 10. (35) This impies ha u 1 and u 2 are wo ineary independen souions.

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