Factorization of Difference Equations by Semiconjugacy with Application to Non-autonomous Linear Equations

F actorization of Difference Equations b y Sem iconjugacy with Application to Non-autonomous Linear E quations H. SED A GHA T* Abstract. The existence o f a semiconjuga te relation p ermits the trans formation of a higher order difference equation on a gr oup in to a n equiv alen t tria ngular system of tw o difference equations o f low er o rders. Int ro ducing time-dep endent form symmetries in this pap er enables us to iden tify the semiconjugate prop erty in a lar g er set of non-autonomo us difference equations than previously considered. W e show that there is a substantial class o f equa tions having this feature that includes the gener al (non-autonomous, non-homogeneo us) linear equatio n with v ariable co efficients in an ar bitrary algebraic field. 1 In tro duction Difference equations of ord er greater than one that are of the follo w ing type x n +1 = f n ( x n , x n − 1 , . . . , x n − k ) (1) determine the forward ev olution o f a v ariable x n in discrete time s ince the time index or th e indep end en t v ariable n is in teger-v alued with n ≥ 0. In pr evious studies of semiconjugate factorizatio ns of d ifference equations of typ e (1), e.g., [3], [4], [5] or [6], the form symmetry lin k in g the higher dimens ional u nfolding map of th e original equation to th at of the lo we r dimens ional factor w as assumed to b e indep enden t of n . While this assu mption did not substanti ally cu r tail th e applicabilit y of the metho d, it did rule out certain non-autonomous equations. F or example, the metho d w ork ed for non-h omogeneous linear equations with constan t co efficien ts but did not apply to linear equations with v ariable coefficients. The m ain goal of this article is to extend the aforemen tioned factorization metho d to allo w time- dep endent form sy m metries where the form sym metry ma y d ep end explicitly on the indep enden t v ariable n . T his extension is significant as it cov ers all non-autonomous equ ations of t yp e (1). In particular, the extended metho d may b e applied to general (non-autonomous, non-homogeneous) linear equations o ver arbitrary algebraic fields to sho w that suc h equations admit semiconjugate factorizat ions v ia eigensequences (i.e., the solutions of an asso ciated discrete Riccati difference equation of low er order). F or ease of reference w e state some of the basic concepts and notation here; add itional bac kground material for this article is a v ailable in [5]. 0 Key wo rds: Time-dep endent form symmetry , semiconjugate f actorization, general linear equation, Riccati differ- ence eq uation, eigensequence 0 *Department of Mathematics, Virginia Commonw ealth Universit y , R ic hmond, Virginia 23284-20 14 USA, Email: hsedagha@vcu.edu 1 As usual, th e num b er k in (1) is a fixed p ositiv e integ er and k + 1 repr esents the or der of the difference equation (1). The u nderlying sp ace of v ariables x n is a group G and f n : G k +1 → G is a give n function for eac h n ≥ 1. If f n = f do es not explicitly dep end on n then (1) is said to b e autonomo us ; it is non-autonomous otherwise. A (forward) solution of Eq.(1) is a s equ ence { x n } ∞ n = − k that is recursiv ely ge nerated b y (1) from a set of k + 1 initia l v alues x 0 , x − 1 , . . . , x − k ∈ G. F orward solutions hav e traditionally b een of greater interest in discrete mo dels that are based on Eq.(1) although other types of solutions (e.g., those having d omain Z , the set of all inte gers) can also b e r eadily defined. Eac h f n is “un folded” by the asso ciated v ecto r map F n : G k +1 → G k +1 that are defi n ed as F n ( u 0 , . . . , u k ) = [ f n ( u 0 , . . . , u k ) , u 0 , . . . , u k − 1 ] , u j ∈ G for j = 0 , 1 , . . . , k . (2) The unfoldings F n determine the equation ( y 0 ,n +1 , y 1 ,n +1 , . . . , y k ,n +1 ) = F n ( y 0 ,n , y 1 ,n , . . . , y k ,n ) in G k +1 . Eac h v ector ( y 0 ,n +1 , . . . , y k ,n +1 ) represen ts a state of the system, or of Eq.(1); G k +1 is the state sp ac e , in analogy to the phase space in differen tial equations. 2 Semiconjugate relation and factoriza tion Let F n b e the u nfolding on G k +1 of f n for eac h n. Th en (1) is equiv alent to X n +1 = F n ( X n ) , X n = ( x n , . . . , x n − k ) . (3) W e are inte rested in derivin g a lo w er dimen s ional equation Y n +1 = Φ n ( Y n ) , Y n = ( y n , . . . , y n − m +1 ) , m ≤ k (4) for (3). If there exists a sequence of maps H n : G k +1 → G m suc h that for every s olution { X n } of (3) Y n = H n ( X n ) , n = 0 , 1 , 2 , . . . (5) is a solution of (4) then Φ n ( H n ( X n )) = Φ n ( Y n ) = Y n +1 = H n +1 ( X n +1 ) = H n +1 ( F n ( X n )) . Therefore, (5) is satisfied for all solutions of (3) and (4) if and only if the sequence { H n } of maps satisfies the f ollo wing equalit y for all n H n +1 ◦ F n = Φ n ◦ H n . (6) If the mappings H n are indep end en t of n, i.e., H n = H for all n then Eq.(6) r educes to H ◦ F n = Φ n ◦ H (7) namely , the time-indep endent semiconjugate relation as d efined in p rior s tudies. W e can no w give the follo win g more general defint ion. 2 Definition 1 L et k ≥ 1 , 1 ≤ m ≤ k . If ther e is a se quenc e of surje ctive maps H n : G k +1 → G m such that Eq.(6) is satisfie d for a given p air of function se qu enc es { F n } and { Φ n } then we say that F n is semic onjugate to Φ n for e ach n and r efer to the se quenc e { H n } as a (time-dep endent) form symm etry of Eq.(3) or e quiv alently, of Eq.(1). Si nc e m < k + 1 , the form symmetry { H n } is or der-r e ducing . T echnicall y , a time-dep en den t form symm etry can also b e defin ed as a single map H : N × G k +1 → G m , H ( n ; u 0 , . . . , u k ) = H n ( u 0 , . . . , u k ) W e choose the sequ ence defin ition d u e to its more intuitiv e con tent. The f ollo wing result extends its time-indep end en t analog in [5 ] and make s pr ecise the concept of semiconjugate f actoriza tion f or the recursiv e difference equation (1). Lemma 2 L et k ≥ 1 , 1 ≤ m ≤ k , let h n : G k − m +1 → G for n ≥ − m + 1 b e a se quenc e of functions on a giv e n non-trivial gr oup G and define the functions H n : G k +1 → G m by H n ( u 0 , . . . , u k ) = [ u 0 ∗ h n ( u 1 , . . . , u k +1 − m ) , . . . , u m − 1 ∗ h n − m +1 ( u m , . . . , u k )] . (8) Then the fol lowing statements ar e true: (a) The fu nc tion H n define d by (8) is su rje ctive for e ach fixe d n ≥ 0 . (b) If { H n } is an or der-r e ducing form symmetry then the differ enc e e quation (1) is e quivalent to the system of e quations t n +1 = φ n ( t n , . . . , t n − m +1 ) , (9) x n +1 = t n +1 ∗ h n +1 ( x n , . . . , x n − k + m ) − 1 (10) whose or ders m and k + 1 − m r e sp e ctively, add up to the or der of (1). (c) The map Φ n : G m → G m in (6) is the unfolding of Eq.(9) for e ach n ≥ 0 ; i.e., e ach Φ n is of sc alar typ e. Pro of. (a) Let n b e a fixed non-negativ e in teger and for j = 0 , . . . , m − 1 denote the j -th co ordin ate function of H n b y η j +1 ( u 0 , . . . , u k ) = u j ∗ h n − j ( u j +1 , . . . , u j + k +1 − m ) (11) No w c h o ose an arbitrary p oint ( v 1 , . . . , v m ) ∈ G m and defi ne u m − 1 = v m ∗ h n − m +1 ( u m , u m +1 . . . , u k ) − 1 , (12) u m = u m +1 = . . . u k = ¯ u 3 where ¯ u is a fixed element of G, e.g., the identit y . Th en v m = u m − 1 ∗ h n − m +1 ( ¯ u, ¯ u . . . , ¯ u ) = u m − 1 ∗ h n − m +1 ( u m , u m +1 . . . , u k ) = η m ( u 0 , . . . , u k ) = η m ( u 0 , . . . , u m − 2 , v m ∗ h n − m +1 ( ¯ u, ¯ u . . . , ¯ u ) − 1 | {z } u m − 1 , ¯ u . . . , ¯ u ) . for an y selection of elemen ts u 0 , . . . , u m − 2 ∈ G. Using the same idea, define u m − 2 = v m − 1 ∗ h n − m +2 ( u m − 1 , ¯ u . . . , ¯ u ) − 1 with u m − 1 defined b y (12) so as to get v m − 1 = u m − 2 ∗ h n − m +2 ( u m − 1 , ¯ u . . . , ¯ u ) = u m − 2 ∗ h n − m +2 ( u m − 1 , u m . . . , u k − 1 ) = η m − 1 ( u 0 , . . . , u k ) = η m − 1 ( u 0 , . . . , u m − 3 , v m − 1 ∗ h n − m +2 ( u m − 1 , ¯ u . . . , ¯ u ) − 1 | {z } u m − 2 , u m − 1 , ¯ u . . . , ¯ u ) for an y c hoice of u 0 , . . . , u m − 3 ∈ G. Con tin uing in this wa y , by in duction w e obtain elemen ts u m − 1 , . . . , u 0 ∈ G suc h that v i = η i ( u 0 , . . . , u m − 1 , ¯ u . . . , ¯ u ) , i = 1 , . . . , m . Therefore, H n ( u 0 , . . . , u m − 1 , ¯ u . . . , ¯ u ) = ( v 1 , . . . , v m ) and it follo ws that H n is onto G m . (b) T o sho w that the SC factorization sys tem consisting of equations (9) and (10) is equiv alen t to Eq.(1) we sho w that: (i) eac h solution { x n } of (1) u niquely generates a solutio n of the s ystem (9) and (10) and conv ersely (ii) eac h solution { ( t n , y n ) } of the sys tem (9) and (10) corresep onds uniqu ely to a solution { x n } of (1). T o establish (i) let { x n } b e the un ique solution of (1) corresp ond ing to a giv en set of initial v alues x 0 , . . . x − k ∈ G. Define the sequence t n = x n ∗ h n ( x n − 1 , . . . , x n − k + m − 1 ) (13) for n ≥ − m + 1 . T h en for eac h n ≥ 0 if H n is defi n ed by (8) it follo ws from th e semiconjugate relation (6) that x n +1 = f n ( x n , . . . , x n − k ) = φ n ( x n ∗ h n ( x n − 1 , . . . , x n − k + m − 1 ) , . . . , x n − m +1 ∗ h n − m +1 ( x n − m , . . . , x n − k )) ∗ [ h n +1 ( x n , . . . , x n − k + m )] − 1 = φ n ( t n , . . . , t n − m +1 ) ∗ [ h n +1 ( x n , . . . , x n − k + m )] − 1 4 Therefore, φ n ( t n , . . . , t n − m +1 ) = x n +1 ∗ h n +1 ( x n , . . . , x n − k + m ) = t n +1 so that { t n } is the u nique solution of the f actor equation (9) with initial v alues t − j = x − j ∗ h − j ( x − j − 1 , . . . , x − j − k + m − 1 ) , j = 0 , . . . , m − 1 . F ur th er, since x n +1 = t n +1 ∗ [ h n +1 ( x n , . . . , x n − k + m )] − 1 for n ≥ 0 by (13), { x n } is the uniqu e solution of the cofactor equation (10) with in itial v alues y − i = x − i for i = 0 , 1 , . . . , k − m and w ith the v alues t n obtained ab o v e. T o establish (ii) let { ( t n , y n ) } b e a solution of the f actor-cofactor system w ith in itial v alues t 0 , . . . , t − m +1 , y − m , . . . y − k ∈ G. Note that these n um b ers determine y − m +1 , . . . , y 0 through the cofactor equation y − j = t − j ∗ [ h − j ( y − j − 1 , . . . , y − j − 1 − k + m )] − 1 , j = 0 , . . . , m − 1 . (14) No w for n ≥ 0 we obtain y n +1 = t n +1 ∗ [ h n +1 ( y n , . . . , y n − k + m )] − 1 = φ n ( t n , . . . , t n − m +1 ) ∗ [ h n +1 ( y n , . . . , y n − k + m )] − 1 = φ n ( y n ∗ h n ( y n − 1 , . . . , y n − k + m − 1 ) , . . . , y n − m +1 ∗ h n − m +1 ( y n − m , . . . , y n − k )) ∗ h n +1 ( y n , . . . , y n − k + m ) − 1 = f n ( y n , . . . , y n − k ) Th us { y n } is the un iqu e solution of Eq.(1) that is generated by the initial v alues (14) and y − m , . . . y − k . Th is completes the p ro of of (b). (c) W e sho w that eac h coord inate function φ j,n is the p r o jection in to co ordinate j − 1 for j > 1 . F rom the defin ition of H n in (8) and the semiconju gate relation (6) w e in fer that H n +1 ( F n ( u 0 , . . . , u k )) = H n +1 ( f n ( u 0 , . . . , u k ) , u 0 , . . . , u k − 1 ) = ( f n ( u 0 , . . . , u k ) ∗ h n +1 ( u 0 , . . . , u k − m ) , u 0 ∗ h n ( u 1 , . . . , u k − m +1 ) , . . . , u m − 2 ∗ h n − m +2 ( u m − 1 , . . . , u k − 1 )) . Matc hing the corresp ond ing comp onent fu nctions in the ab ov e equalit y for j ≥ 2 yields φ j,n ( u 0 ∗ h n ( u 1 , . . . , u k +1 − m ) , . . . , u m − 1 ∗ h n − m +1 ( u m , . . . , u k )) = u j − 2 ∗ h ( u j − 1 , u j . . . , u j + k − m − 1 ) whic h shows that φ j,n maps its j -th co ordin ate to its ( j − 1)-st. Therefore, for eac h n and ev ery ( t 1 , . . . , t m ) ∈ H n ( G k +1 ) we hav e Φ n ( t 1 , . . . , t m ) = [ φ n ( t 1 , . . . , t m ) , t 1 , . . . , t m − 1 ] 5 i.e., Φ n | H n ( G k +1 ) is of scalar typ e. Since by P art (a) H n ( G k +1 ) = G m for eve ry n, it follo ws that Φ n is of s calar typ e. The pair of equations (9) and (10) in T heorem 2 is u n coupled in the sense that (9) is indep en den t of (10). Suc h a p air forms a triangular sy s tem as d efined in [1 ] an d [7 ]. In the next defi nition we use con venien t and suggestiv e terminology to describ e these equations. Definition 3 Eq.(9) is a factor of E q .(1) sinc e it is derive d fr om the semic onjugate factor Φ n . Eq.(10) that links the factor to the original e qu ation is a c ofactor of Eq.(1). We r efer to the system of e q u ations (9) and (10) as a semic onjugate (SC) factorization of Eq .(1). Note that or ders m and k + 1 − m of (9 ) and (10) r esp e ctively, add up to the or der of (1). We r efer to the system of e quations (9) and (10) as a typ e- ( m, k + 1 − m ) or der r e duction of Eq.(1). 3 In v ertible-map criterion In [4] and [5] a u s eful n ecessary and sufficien t condition is obtained by which to d etermine whether the difference equation (1) h as order-reducing form symmetries (not time-dep en den t). In this section we show that th e same useful idea extends to the time-dep endent con text. Applications and examples are d iscu ssed in th e next s ection. Consider the follo wing sp ecial case of (8) w ith m = k H n ( u 0 , u 1 , . . . , u k ) = [ u 0 ∗ h n ( u 1 ) , u 1 ∗ h n − 1 ( u 2 ) , . . . , u k − 1 ∗ h n − k +1 ( u k )] (15) with h n : G → G b eing a sequence of sur jectiv e self-maps of the underlyin g group G for n ≥ − k + 1. If (1) has th e form symm etry (15) then it adm its a type-( k , 1) ord er-reduction and its SC factorizat ion is t n +1 = φ n ( t n , . . . , t n − k +1 ) , (16) x n +1 = t n +1 ∗ h n +1 ( x n ) − 1 . (17) The initial v alues of the factor equation (16) are t − j = x − j ∗ h − j ( x − j +1 ) , j = 0 , 1 , . . . , k − 1 Theorem 4 (Time-dep endent inv e rtible map criterion) Assume that h n : G → G is a se quenc e of bije ctions of G for n ≥ − k + 1 . F o r arbitr ary elements u 0 , v 1 , . . . , v k ∈ G and e v ery n ≥ 0 define ζ 0 ,n ( u 0 ) ≡ u 0 and for j = 1 , . . . , k , ζ j,n ( u 0 , v 1 , . . . , v j ) = h − 1 n − j +1 ( ζ j − 1 ,n ( u 0 , v 1 , . . . , v j − 1 ) − 1 ∗ v j ) . (18) with the usual distinction observe d b etwe en map inversion and gr oup inversion. Then Eq.(1) has the f orm symmetry { H n } define d by (15) if and only if the quantity f n ( ζ 0 ,n , ζ 1 ,n ( u 0 , v 1 ) , . . . , ζ k ,n ( u 0 , v 1 , . . . , v k )) ∗ h n +1 ( u 0 ) (19) 6 is indep endent of u 0 for ev ery n ≥ 0 . In this c ase Eq.(1) has a SC factorization whose factor functions in (16) ar e given by φ n ( v 1 , . . . , v k ) = f n ( ζ 0 ,n , ζ 1 ,n ( u 0 , v 1 ) , . . . , ζ k ,n ( u 0 , v 1 , . . . , v k )) ∗ h n +1 ( u 0 ) . (20) Pro of. Assume first that (19) is indep end ent of u 0 for all v 1 , . . . , v k so that the fu nctions φ n ( v 1 , . . . , v k ) = f n ( ζ 0 ,n , ζ 1 ,n , . . . , ζ k ,n ) ∗ h n +1 ( u 0 ) (21) are w ell d efined. Next, if H n is giv en by (15) then for all u 0 , u 1 , . . . , u k φ n ( H n ( u 0 , u 1 , . . . , u k )) = φ n ( u 0 ∗ h n ( u 1 ) , u 1 ∗ h n − 1 ( u 2 ) , . . . , u k − 1 ∗ h n − k +1 ( u k )) . No w, b y (18) for eac h n and all u 0 , u 1 ζ 1 ,n ( u 0 , u 0 ∗ h n ( u 1 )) = h − 1 n ( u − 1 0 ∗ u 0 ∗ h n ( u 1 )) = u 1 . Similarly , for eac h n and all u 0 , u 1 , u 2 ζ 2 ,n ( u 0 , u 0 ∗ h n ( u 1 ) , u 1 ∗ h n − 1 ( u 2 )) = h − 1 n − 1 ( ζ 1 ,n ( u 0 , u 0 ∗ h n ( u 1 )) − 1 ∗ u 1 ∗ h n − 1 ( u 2 )) = u 2 . Supp ose b y wa y of ind uction th at ζ l,n ( u 0 ∗ h n ( u 1 ) , . . . , u l − 1 ∗ h n − l +1 ( u k )) = u l for 1 ≤ l < j . Th en ζ j,n ( u 0 ∗ h n ( u 1 ) , . . . , u j − 1 ∗ h n − j +1 ( u j )) = h − 1 n − j +1 ( u − 1 j − 1 ∗ u j − 1 ∗ h n − j +1 ( u j )) = u j . Th us by (21) φ n ( H n ( u 0 , u 1 , . . . , u k )) = f n ( u 0 , . . . , u k ) ∗ h n +1 ( u 0 ) No w if F n and Φ n are the u nfoldings of f n and φ n resp ectiv ely , then H n +1 ( F n ( u 0 , . . . , u k )) = [ f n ( u 0 , . . . , u k ) ∗ h n +1 ( u 0 ) , u 0 ∗ h n ( u 1 ) , . . . , u k − 2 ∗ h n − k +2 ( u k − 1 )] = [ φ n ( H n ( u 0 , u 1 , . . . , u k )) , u 0 ∗ h n ( u 1 ) , . . . , u k − 2 ∗ h n − k +2 ( u k − 1 )] = Φ n ( H n ( u 0 , . . . , u k )) 7 and it follo ws that { H n } is a semiconjugate f orm sy m metry for Eq.(1). The existence of a SC factorizat ion with factor f unctions defined by (20) no w follo ws fr om Lemma 2. Con v ers ely , if { H n } as giv en by (15) is a time-dep endent form symmetry of Eq.(1) then the semiconjugate relation implies that for arbitrary u 0 , . . . , u k in G there are fu nctions φ n suc h th at f n ( u 0 , . . . , u k ) ∗ h n +1 ( u 0 ) = φ n ( u 0 ∗ h n ( u 1 ) , . . . , u k − 1 ∗ h n − k +1 ( u k )) . (22) F or eve ry u 0 , v 1 , . . . , v k in G and with f unctions ζ j,n as d efi ned ab o ve , note that ζ j − 1 ,n ( u 0 , v 1 , . . . , v j − 1 ) ∗ h n − j +1 ( ζ j,n ( u 0 , v 1 , . . . , v j )) = v j , j = 1 , 2 , . . . , k . Therefore, abbreviating ζ j,n ( u 0 , v 1 , . . . , v j ) by ζ j,n w e h a ve f n ( ζ 0 ,n , ζ 1 ,n , . . . , ζ k ,n ) ∗ h n +1 ( u 0 ) = φ n ( ζ 0 ,n ∗ h n ( ζ 1 ,n ) , ζ 1 ,n ∗ h n − 1 ( ζ 2 ,n ) , . . . , ζ k − 1 ,n ∗ h n − k +1 ( ζ k ,n )) = φ n ( v 1 , . . . , v k ) whic h is indep en d en t of u 0 . Recall that an algebraic field F =( F , + , · ) is, in particular, a comm utativ e group w ith resp ect to addition. F urther, its set of nonzero elemen ts F \{ 0 } is a commutat iv e group under m ultiplication. A simp le y et imp ortan t typ e of form sym metry m a y b e defined on a fi eld. Definition 5 L et F b e a non-trivial field and { α n } a se quenc e of elements of F such that α n ∈ F \{ 0 } for al l n ≥ − k + 1 . A (time-dep endent) line ar form symmetr y is define d as the fol lowing sp e cial c ase of (15) with h n ( u ) = − α n − 1 u [ u 0 − α n − 1 u 1 , u 1 − α n − 2 u 2 , . . . , u k − 1 − α n − k u k ] . (23) The se quenc e { α n } of nonzer o elements in F may b e c al le d the eigense que nc e of the line ar form symmetry. If E q.(1) has a line ar form symmetry then { α n } i s an ei g ense quenc e of (1 ). The use of the term “eigen” wh ic h is b orro w ed from the theory of linear equations is apt here for tw o reasons. First, the sequ en ce { α n } c haracterizes th e linear f orm sym metry (23) completely and secondly , we fin d b elo w that linear difference equations ind eed h av e linear form symmetries. The existence of a linear form symmetry implies a t yp e-( k, 1) order red u ction for E q.(1) and a SC factorization w here the cofacto r equ ation (17 ) is d etermined more sp ecifically as x n +1 = t n +1 + α n x n . (24) The follo win g necessary and su fficien t condition for the existence of a time-dep endent linear form symmetry is an app lication of Theorem 4. W e drop further menti on of “type-( k , 1)” as we do not discuss an y other order r eduction t yp es in the remainder of th is pap er. 8 Corollary 6 Equation (1) has a time-dep endent line ar form symmetry of typ e (23) with an eigense- quenc e { α n } i n a non-trivial field F i f and only if the quantity f n ( u 0 , ζ 1 ,n ( u 0 , v 1 ) , . . . , ζ k ,n ( u 0 , v 1 , . . . , v k )) − α n u 0 (25) is indep endent of u 0 for al l n ≥ 0 with the functions ζ j,n for j = 1 , . . . , k given by ζ j,n ( u 0 , v 1 , . . . , v j ) = ζ j − 1 ,n ( u 0 , v 1 , . . . , v j − 1 ) − v j α n − j = 1 Q j i =1 α n − i   u 0 − j X i =1 v i i Y p =1 α n − p   . Pro of. The conclusions follo w immediately fr om Theorem 4 usin g h n ( u ) = − α n − 1 u. The last equalit y ab ov e is established from the equalit y pr eceding it by routine calculation. Remark 7 If Eq.(1) has a line ar form symmetry then by Cor ol lary 6 an eigense quenc e of (1) c an b e define d e quivalently as a se quenc e { α n } in F \{ 0 } for which the quantity in (25) i s indep endent of u 0 for ev ery n ≥ 0 . W e close this section with an example of a nonlinear equati on that has a linear form symmetry . F or add itional results and examples, w e refer to [2]. Example 8 Consider the fol lowing thir d-or der nonline ar diffe r enc e e quation x n +1 = ( − 1) n +1 x n − 2 x n − 1 + g n ( x n + x n − 2 ) (26) wher e g n : R → R is a gi v en f u nction for e ach n ≥ − 2 . By Cor ol lary 6 a line ar form symmetry for (26) exists if and only if the quantity ( − 1) n +1 u 0 − 2 ζ 1 ,n + g n ( u 0 + ζ 2 ,n ) − α n u 0 (27) is indep endent of u 0 for al l n . Su b stituting ζ 1 ,n = u 0 − v 1 α n − 1 , ζ 2 ,n = ζ 1 ,n − v 2 α n − 2 = u 0 − v 1 − α n − 1 v 2 α n − 1 α n − 2 . in (27) and r e arr a nging terms give s  ( − 1) n +1 − 2 α n − 1 − α n  u 0 + 2 α n − 1 v 1 + g n  1 + 1 α n − 1 α n − 2  u 0 − 1 α n − 1 α n − 2 v 1 − 1 α n − 2 v 2  9 which is indep endent of u 0 for al l n if the c o efficie nts of the u 0 terms ar e zer os; i.e., for al l n , the numb ers α n satisfy b oth of the fol lowing e quations α n = ( − 1) n +1 − 2 α n − 1 (28) α n − 1 = − 1 α n − 2 . (29) Every solution of E q.(29) is a se qu enc e of p erio d 2  q , − 1 q , q , − 1 q , . . .  (30) wher e q = α − 2 ∈ R . Now (29) yields α − 1 = − 1 /q , which we substitute as an initial value in Eq.(28) to ge t α 0 = − 1 + 2 q . Now to make the p erio d-two se quenc e in (30) also a solution of (28), we r e quir e the ab ove value of α 0 to b e e q ual to q ; thus α 0 = q ⇒ 2 q − 1 = q ⇒ q = 1 . We che c k that if α 0 = q = 1 in (28) then α 1 = 1 − 2 q = − 1 = − 1 q , α 2 = − 1 − 2 − 1 = 1 = q , etc so that b oth of the e quations (28) and (29) gener ate the same se quenc e { α n } wher e α n = ( − 1) n for n ≥ − 2 . It fol lows that { ( − 1) n } i s an eigense qu e nc e for (26). 4 F actorizati on of li near equations W e exp ect that linear difference equ ations are among difference equations that hav e the linear form symmetry and this is in deed the case. The follo wing app lication of Corollary 6 and Th eorem 4 giv es th e semiconju gate factorization for non-autonomous and non-homogeneous linear d ifference equations. Corollary 9 (The gener al line ar e quation) L e t { a i,n } , i = 1 , . . . , k and { b n } b e given se quenc es in a non-trivial field F such that a k ,n 6 = 0 for al l n ≥ 0 . The non-homo gene ous line ar e q u ation of or der k + 1 x n +1 = a 0 ,n x n + a 1 ,n x n − 1 + · · · + a k ,n x n − k + b n (31) has a line ar form symmetry with eigense quenc e { α n } for ev ery solution { α n } in F of the fol- lowing Ric c ati e quation of or der k α n = a 0 ,n + a 1 ,n α n − 1 + a 2 ,n α n − 1 α n − 2 + · · · + a k ,n α n − 1 · · · α n − k (32) 10 The c orr esp onding SC f actorizatio n of (31) is t n +1 = b n − k X i =1 k X j = i a j,n α n − i · · · α n − j t n − i +1 (33) x n +1 = α n x n + t n +1 (34) Pro of. By Corollary 6 it is only necessary to d etermine a sequence { α n } of nonzero elemen ts of F suc h th at f or eac h n the quantit y (25) is indep enden t of u 0 for the follo wing fun ction f n ( u 0 , . . . , u k ) = a 1 ,n u 0 + a 2 ,n u 1 + · · · + a k ,n u k + b n . F or arbitrary u 0 , v 1 , . . . , v k ∈ F and j = 0 , 1 , . . . , k define ζ j,n ( u 0 , v 1 , . . . , v j ) as in Corollary 6. Then th e expression (25) is − α n u 0 + b n + a 1 ,n u 0 + a 2 ,n ζ 1 ,n ( u 0 , v 1 ) + · · · + a k ,n ζ k ,n ( u 0 , v 1 , . . . , v k ) = b n +   k X j =1 a j,n Q j i =1 α n − i − α n   u 0 − k X j =1 a j,n j X i =1 v i Q j p = i α n − p The ab o v e qu an tit y is indep end en t of u 0 if and only if the co efficient of u 0 is zero f or all n ; i.e., if { α n } is a solution of the Riccati difference equation α n = k X j =1 a j,n Q j i =1 α n − i whic h is Eq.(32). It f ollo ws that Eq.(31) has a lin ear form sym metry of type (23 ) with eigensequence { α n } for eac h solution { α n } of the Riccati equation. F or th e corresp ondin g S C f actorization of (31), th e cofactor equation is simply (24) wh ile the factor equation is obtained using the ab ov e calculatio ns and Eq .(20 ) of Theorem 4 as follo ws t n +1 = b n − k X j =1 a j,n j X i =1 t n − i +1 Q j p = i α n − p = b n − k X i =1 k X j = i a j,n α n − i · · · α n − j t n − i +1 . This completes the pro of. Corollary 9 states that any solution of the Riccati equation (32) giv es a form symm etry and a SC factorization of (31) as sp ecified ab o ve. T he n ext example illustrates Corollary 9. 11 Example 10 Consider the se c ond-or der differ enc e e quation x n +1 = ( − 1) n +1 x n + x n − 1 + b n (35) wher e b n , x 0 , x − 1 ar e in a field F which we may take to b e any one of the familiar fields Q , R or C . The asso ciate d R ic c ati e qu ation of (35) is α n = ( − 1) n +1 + 1 α n − 1 . (36) Str aightforwa r d c alculation shows that if α 0 6 = 0 , − 1 then α 1 = α 0 + 1 α 0 , α 2 = − 1 α 0 + 1 , α 3 = − α 0 , α 4 = − α 0 + 1 α 0 , α 5 = 1 α 0 + 1 , α 6 = α 0 . It fol lows that al l solutions of the Ric c ati e q uation (36) with initial value outside the singularity set { 0 , − 1 } ar e eigense quenc e s in F of p erio d 6:  α 0 , α 0 + 1 α 0 , − 1 α 0 + 1 , − α 0 , − α 0 + 1 α 0 , 1 α 0 + 1 , α 0 , . . .  The SC factorization of the line ar e quation (35) is now obtaine d b y Cor ol lary 9 as t n +1 = − 1 α n − 1 t n + b n , x n +1 = α n x n + t n +1 . The next result is concerned with the case of constan t coefficients. The straight forw ard p r o of is omitted. Corollary 11 L et { b n } b e a gi ven se q u enc e in a non-trivial field F a nd let { a i } , i = 1 , . . . , k b e c onstants in F such that a k 6 = 0 . (a) The non-homo gene ous line ar e quation of or der k + 1 x n +1 = a 0 x n + a 1 x n − 1 + · · · + a k x n − k + b n (37) has a line ar form symmetry with eigense quenc e { α n } for every solution { α n } in F of the fol lowing autonomo us R ic c ati e quation of or der k α n = a 0 + a 1 α n − 1 + a 2 α n − 1 α n − 2 + · · · + a k α n − 1 · · · α n − k . (38) (b) Every fixe d p oint of (38) in F is a nonzer o r o ot of the char acteristic p olynomia l of (37), i.e., λ k +1 − a 0 λ k − a 1 λ k − 1 − · · · − a k − 1 λ − a k (39) and thus, an ei g envalue of the homo gene ous p art of (37) in F . As c onstant solutions of (38) such eigenvalues ar e c onstant eigense quenc e s of (37). 12 Example 12 Consider the autonomous se c ond-or der line ar diffe r enc e e quation x n +1 = x n + x n − 1 (40) Eq.(40) has two r e al ei g envalues α ± = 1 ± √ 5 2 as r o ots of the char acteristic p olynomial λ 2 − λ − 1 or e q u ivalently, as fixe d p oints of the Ric c ati e quation α n = 1 + 1 α n − 1 (41) in the field F = R . Thus e ach of α + and α − is a c onstan t eigense qu e nc e of (40) in R and the fol lowing SC factorization is obtaine d i n R : t n +1 = − 1 α + t n = α − t n , x n +1 = α + x n + t n +1 . Note that the SC factorizat ion ab ove has c onstant c o efficie nts also. We note further that sinc e α ± ar e irr ational the ab ove SC factorizat ion i s not v alid if F = Q the field of r ational numb ers. In fact, sinc e the char acteristic p olynomial has no r ational r o ots, it fol lows that ther e ar e no c onstant eigense quenc e s for (40) in Q . H owever, Ric c ati e qu ation (41) is a r ational e quation and thus with a r ational initial value α 0 the c orr esp onding solution of (41) is a solution (non-c onstant) i n Q . F or instanc e, if α 0 = 1 then the c orr esp onding solution of (41) is α n = ϕ n +1 /ϕ n wher e { ϕ n } is the Fib onac ci se quenc e 1,1,2,3,5,8,. . . This r ational eigense quenc e yields the fol lowing SC factorization of (40) that is valid in Q : t n +1 = − ϕ n − 1 ϕ n t n , x n +1 = ϕ n +1 ϕ n x n + t n +1 . We note that lim n →∞ ϕ n +1 /ϕ n = α + in the ab ove f actorization ; in this way the factorization over r ationals is r elate d to the e arlier factorization over the r e als. In a similar fashion, the e qu ation x n +1 = x n − x n − 1 (42) has two c omplex ei genvalues α ± = 1 ± i √ 3 2 13 that ar e r o ots of λ 2 − λ + 1 . Thus, (42) has no c onsta nt eigense que nc es in R but it do es have non-c onstant r e al e i gense quenc es sinc e the Ric c ati e quation α n = 1 − 1 α n − 1 with the initial value α 0 = 2 has a solution  2 , 1 2 , − 1 , 2 , 1 2 , − 1 , . . .  of p erio d thr e e i n R with a c orr esp onding r e al SC f actorization t n +1 = − 1 α n − 1 t n , x n +1 = α n x n + t n +1 . In c ontr ast to the factorization of Eq.(40) ther e i s no simple r elationship b etwe en the factorization of (42) over the r e al numb e rs and its factorizatio n with c onstant eigense quenc es over the c omplex numb ers. Remark 13 Is it p ossible tha t a line ar differ enc e e quation has no eigense quenc es, c onstant or otherwise in a given field F b e c ause the asso ciate d R ic c ati e quation has no solutions at al l in F ? We know the answer to this question in some c ases. If we have a line ar e qu ation (homo gene ous or not) with c onsta nt c o efficients in an algebr aic al ly close d field F (e.g., the field C of c omplex numb ers) then F always c ontains c onstant ei gense quenc es, namely, the r o ots of the char acteristic p olynomial (39). On the other hand, for the finite field Z 3 = { 0 , 1 , 2 } with addition and multiplic a- tion define d mo dulo 3, the line ar e quation (40) has no e igense quenc es. This c an b e shown by testing e ach of the two p ossible nonzer o initial values 1,2 in the R ic c ati e qu ation (41) to verify that b oth le ad to the sing u larity at 0: α 0 = 2 ⇒ α 1 = 1 + 1 2 = 1 + 2 = 0 , α 0 = 1 ⇒ α 1 = 1 + 1 = 2 ⇒ α 2 = 0 . The answer to the q u estion of existenc e of eigense quenc e s in the gener al c ase is not known at this time; in fact, it is not known if a line ar e q uation with r e al c o efficients exists that has no r e al eigense quenc e s. F or “lar ge” fields such as R or C it se ems likely that the gener al line ar e q u ation (31) has an eigense quenc e in the field. The o ccurrence of Riccati difference equation in Corollary 9 m a y seem le ss surpr ising if we recall some basic f acts from [5 ]. In particular, the homogeneous part of (31) is a homog eneous equation of degree one relativ e to the multi plicativ e group F \{ 0 } . Therefore, it has an in v ers ion form symm etry and the factor equation of its S C factorization is non e other than th e Riccati equation (32). Using this fact it is p ossible to restate Corollary 9 without explicit reference to the Riccati equation as follo ws. 14 Corollary 14 Assume that the homo gene ous p art of Eq.(31) has a solution { y n } in the field F such that y n 6 = 0 for al l n. Then { y n +1 /y n } is an eigense qu enc e of (31) whose SC factorization is given b y the p air of e qu ations (33) and (34). Pro of. It is giv en that { y n } satisfies the h omogeneous p art of (31), i.e., y n +1 = a 0 ,n y n + a 1 ,n y n − 1 + a 2 ,n y n − 2 + · · · + a k ,n y n − k . Since y n 6 = 0 for all n, we ma y divide the ab o v e equation by y n to obtain y n +1 y n = a 0 ,n + a 1 ,n y n − 1 y n + a 2 ,n y n − 2 y n + · · · + a k ,n y n − k y n = a 0 ,n + a 1 ,n y n − 1 y n + a 2 ,n y n − 2 y n − 1 y n − 1 y n + · · · + a k ,n y n − k y n − k +1 · · · y n − 1 y n . No w defining α n = y n +1 /y n for all n and sub stituting these terms in the last equation ab o ve yields the Ricca ti equation (32). Thus { y n +1 /y n } is an e igensequence of (31) in F \{ 0 } , as claimed. The SC factorization is obtained as in the p ro of of C orollary 9. Corollary 15 In Eq.(31) let { a i,n } , i = 1 , . . . , k and { b n } b e se q uenc es of r e al numb ers with a i,n ≥ 0 for a l l i, n and a k ,n > 0 for al l n. Then (31) has an eigense q uenc e { y n +1 /y n } an d a SC factorization in R g iven by the p air of e quations (33) and (34). Pro of. If we choose y − j = 1 for j = 0 , . . . , k then the corresp ond ing solution { y n } of the h omo- geneous part of (31 ) is a s equ ence of p ositiv e r eal num b ers. No w an app lication of Corollary 14 completes the pro of. References [1] Alseda, L. and Llibre, J., P erio ds for triangular maps, Bul l. Austr al. M ath. So c. , 47 (1993) 41-53 . [2] Sedaghat, H., F orm Symmetries and R e duction of Or der in Di ffer enc e Equations (forthcoming) CR C Pr ess, Bo ca Raton, 2010. [3] Sedaghat, H., Ev ery homogeneous difference equation of degree one admits a red u ction in ord er, J. Differ enc e Eqs. and A ppl. , 15 (2009) 621-624. [4] Sedaghat, H., Reduction of order in d ifference equations b y semiconjugate factorizat ions, Int. J. Pur e and Appl. Math ., 53 (2009 ) 377-384. [5] Sedaghat, H., Order-Redu cing F orm Symmetries and Semiconjugate F actorizatio ns of Difference Equations (2008) ht tp://arxiv.org/a bs/0804.3 579 15 [6] Sedaghat, H., Reduction of order of separable second ord er difference equations with form symmetries, Int. J. Pur e and Appl. Math ., 27 (2008) 155-163. [7] Smital, J., Why it is imp ortant to und erstand th e d ynamics of triangular maps , J. D iffer enc e Eqs. and Appl. , 14 (2008) 597-606. 16