Order-reducing Form Symmetries and Semiconjugate Factorizations of Difference Equations

Order-Reducing F orm Symmetrie s and Semiconjugate F actorizations of Diffe rence Equations H. SED A GHA T Abstract. The scalar difference equation x n +1 = f n ( x n , x n − 1 , · · · , x n − k ) may exhibit symmetries in its form that allo w for redu ction of ord er through sub- stitution or a c hange of v ariables. Suc h form symmetries can b e defined gen- erally using the semiconjugate relation on a group whic h yields a reduction of order through the semiconjugate factorization of the difference equation of order k + 1 i n to equations of lesser orders. Differen t classes of equations are considered including separable e quations and homogeneous equ ations of de- gree 1. Applications include giving a complete factorization of the linear n on- homogeneous d ifference equation of order k + 1 in to a system of k + 1 first order linear non-homogeneous equati ons in whic h the coefficien ts are the eigen- v alues of the h igher ord er equation. F o rm symmetries are also used to explain the complicated m ultistable b eha vior of a separable, s econd order exp on ential equation. Keyw ords. F orm symmetry , order reduction, semico njugate, groups, difference equations, linear non -h omogeneous, separable, h omogeneous of degree 1, multistabilit y 1 In tro duc t ion Certain difference equations ha v e symmetries in their expressions that allow a reduc- tion of their orders through substitutions of new v ariables. F or instance, consider the second order, scalar difference equation x n +1 = x n + g n ( x n − x n − 1 ) (1) where g n is a real function for eac h integer n . T his equation has a symmetry in its form that is easy to iden tify when (1) is re-written as x n +1 − x n = g n ( x n − x n − 1 ) . (2) 1 No w, setting t n = x n − x n − 1 c hanges Eq.(2) to t he first order equation t n +1 = g n ( t n ) . (3) The expression x n − x n − 1 is an example of what w e ma y call a form s ymm etry . Substituting a new v ariable t n for this form symmetry in (2) ga v e the lo w er order equation (3). The form symmetry also establishes a link b et w ee n the second or der equation and the first order one, in the sense that information ab out eac h solution { t n } o f (3) can then b e translated in to information ab out the corresp onding solution of (1) using the equation x n = x n − 1 + t n = x 0 + n X k =1 t k (4) where x 0 is an initial v alue for (1). Along similar lines, the non-homogeneous linear difference equation x n +1 + px n + q x n − 1 = α n with 1 + p + q = 0 (5) has at least tw o form symmetries. First, setting p = − 1 − q and rearranging terms in (5) rev eals the fo rm symmetry x n − x n − 1 and the corresp onding o rder-reducing substitution: x n +1 − x n = α n + q ( x n − x n − 1 ) ⇒ t n +1 = α n + q t n . (6) F urther, x n +1 − q x n = α n − ( p + q ) x n − q x n − 1 = α n + x n − q x n − 1 ⇒ t n +1 = α n + t n . (7) Th us x n − q x n − 1 is also a form symmetry of (5). Note that the co efficien ts q and 1 of t n in (6) and (7), respectiv ely , a r e b oth eigen v alues o f the homogeneous part of (5), i.e., ro ots of the c haracteristic p olynomial z 2 + pz + q when p + q = − 1. Later in this article we sho w that this r elationship holds for all linear difference equations. These a nd many other types of know n form symmetries of difference equations of t ype x n +1 = f n ( x n , x n − 1 , · · · , x n − k ) . (8) can b e defined in terms of semiconjugacy; in [22] there is a basic disc ussion of this topic for real functions but w e giv e a more general definition in this a rticle. The idea of reducing order via semiconjugacy is basically simple; w e find functions that are 2 semiconjugate to the giv en functions f n but whic h hav e few er v ariables than k + 1 , i.e., the order of (8). Then the semic onjugate relation sim ultaneously determines b oth a form symmetry and a factorization of Eq.(8). This factorization of difference equations, whic h is also a formal r epresen tation of the substitution pro cess discussed ab ov e, yields a pair of lo w er order equations. This pair is made up o f a factor equation suc h as (3) and an asso ciated equation suc h as (4) that is deriv ed f rom the form symmetry and relates the factor equation to the original one. The orders of the factor equation and the asso ciated one alw ays add up to the order of the original scalar equation. The aim of this article is to formalize, w ithin the framew ork of semiconjugacy , the concept o f form symme try and its use in reduction of order by substitutions. In addition to unifying v arious ad ho c t echniq ues, this approac h also giv es rise t o new metho ds fo r analyzing higher order difference equations. Some of these new metho ds are describ ed in this article a long with examples and applications. Symmetries of a different kind ha v e already b een used to study hig her o rder differ- ence equations or syste ms of first order o nes. A w ell-kno wn approach inv olve s ada pta - tion of t he Lie symmetry concept from differential equations t o difference equations; see, e.g., [9] for a general discussion of the discrete case co v ering v a rious topics suc h as reduction of order, in tegrabilit y and finding explicit solutions. Also see [4], [13], [15] for additional ideas and tec hniques. The main difference b etw een t he concepts of form symme try and Lie symm etry ma y be summed up a s follows: F orm symmetries are sough t in the difference equation itself whereas Lie symmetries exist in the solu- tions of the equation. T he existence of either type of symmetry can yield v aluable information ab out the dynamics and the solutions of the difference equation with a v ariet y of applications suc h as reduction of order. 2 Semiconjugate forms The material in this section s ubstan tially extends the notions in [2 2] to the more general group con text. F or related results and bac kground material on difference equations see, e.g., [1], [3], [5 ], [8], [12 ]. In this article, G denotes a non-trivial group. The gr oup structure pro vides a suitable framew or k for our results. How ev er, in most applications G turns out t o b e a substructure of a more complex ob ject suc h as a ve ctor space , a ring or an alge- bra p ossessing a compatible or natur a l metric top ology . In t ypical studies in v olving difference equations and discrete dynamical systems , G is a group of real or complex n um b ers. W e may define each mapping f n on the am bien t sturcture as long as the 3 follo wing in v ariance condition holds f n ( G k +1 ) ⊂ G for all n. (9) If (9) holds then for eac h set o f initial v alues x 0 , x − 1 , . . . , x − k in G Eq.(8 ) recur- siv ely generates a solution or orbit { x n } ∞ n = − 1 in G. Before defing form symmetries fo r (8), it is neces sary to discuss some general defin tions in v olving systems. Let 1 ≤ m ≤ k . W e sa y that a self map F n = [ f 1 ,n , . . . , f k +1 ,n ] of G k +1 is semic on- jugate to a self map Φ n of G m if there is a function H : G k +1 → G m suc h that fo r ev ery n, H ◦ F n = Φ n ◦ H . (10) Eac h mapping Φ n is called a semiconjugate factor of the corresponding F n . Sup- p ose that H ( u 0 , . . . , u k ) = [ h 1 ( u 0 , . . . , u k ) , . . . , h m ( u 0 , . . . , u k )] Φ n ( t 1 , . . . , t m ) = [ φ 1 ,n ( t 1 , . . . , t m ) , . . . , φ m,n ( t 1 , . . . , t m )] where h j : G k +1 → G and φ j,n : G m → G are the cor r esp o nding comp onen t functions. Then iden tit y (10) is equiv alent to t he system h j ( f 1 ,n ( u 0 , . . . , u k ) , . . . , f k +1 ,n ( u 0 , . . . , u k )) = φ j,n ( h 1 ( u 0 , . . . , u k ) , . . . , h m ( u 0 , . . . , u k )) j = 1 , 2 , . . . , m. (11) If the functions f j,n are given then (11) is a system of functional equations whose solutions h j , φ j,n giv e the maps H and Φ n . The functions Φ n on G m define a system with low er dime nsion than that defined b y the functions F n on G k +1 . F or a giv en solution { X n } of the equation X n +1 = F n ( X n ) , X 0 ∈ G k +1 (12) let Y n = H ( X n ) for n = 0 , 1 , 2 , . . . Then Y n +1 = H ( X n +1 ) = H ( F n ( X n )) = Φ n ( H ( X n )) = Φ n ( Y n ) so that { Y n } satisfies the low er order equation Y n +1 = Φ n ( Y n ) , Y 0 = H ( X 0 ) ∈ G m . (13) The relationship b et w een the solutions of (12) and those of (13) is not g enerally straigh tforw ard; ho w ev er, inf o rmation ab out the solutions of (13) can shed ligh t on the dynamics of (12). The case where G = R , m = 1 a nd F n = F is time-indep enden t (or autonomous) is of some in t erest b ecause in this case Eq.(13) is a first order difference equation on the real n umbers and as suc h its dynamics ar e m uch b etter understo o d than that o f the higher dimens ional Eq.(12). This case is discussed in detail in [22]. 4 3 Order-redu cing form symmetries F or the scalar difference equation (8) that is of in terest in this article, eac h F n is the asso ciated v ector ma p (or unfolding) o f the function f n in Eq.(8), i.e., F n ( u 0 , . . . , u k ) = [ f n ( u 0 , . . . , u k ) , u 0 , . . . , u k − 1 ] . Ev en if eac h suc h F n is semiconjugate to an m -dimensional map Φ n as in (10), the preceding discussion only giv es the s ystem (13) in whic h the maps Φ n are not necessarily of scalar type similar to F n . While this may b e unav oidable in some cases , adding a f ew reasonable restrictions can ensure that each Φ n is also of scalar t ype. T o this end, define h 1 ( u 0 , . . . , u k ) = u 0 ∗ h ( u 1 , . . . , u k ) (14) where h : G k → G is a f unction to b e determined and ∗ denotes the group op eration. This restriction on H mak es sense for Eq.(8 ), whic h is of recursiv e t ype; i.e., x n +1 giv en explicitly by functions f n . With these restrictions o n H and F n the first equation in (11) is giv en b y f n ( u 0 , . . . , u k ) ∗ h ( u 0 , . . . , u k − 1 ) = g n ( u 0 ∗ h ( u 1 , . . . , u k ) , . . . , h m ( u 0 , . . . , u k )) (15) where for not a tional con v enien ce w e ha v e set g n . = φ 1 ,n : G m → G. Eq.(15) is a functional equation in whic h the functions h, h j , g n ma y b e determined in terms of the giv en functions f n . Our aim is ultimately t o extract a scalar equation of order m suc h as t n +1 = g n ( t n , . . . , t n − m +1 ) (16) from (1 5) in suc h a w ay that the maps Φ n will b e of scalar t yp e. The basic framew ork is already in place; let { x n } b e a solution of Eq.(8) and define t n = x n ∗ h ( x n − 1 , . . . , x n − k ) . Then the left hand side of (1 5) is x n +1 ∗ h ( x n , . . . , x n − k +1 ) = t n +1 . whic h gives the initial part of the difference eq uation (1 6). In order that the righ t hand side of (15) coincide with that in (16) it is necessary to define h j ( x n , . . . , x n − k ) = t n − j +1 = x n − j +1 ∗ h ( x n − j , . . . , x n − k − j +1 ) , j = 2 , . . . , m. (17) 5 Since the left hand side of (17) do es not dep end o n terms x n − k − 1 , . . . , x n − k − j +1 it follo ws that the function h j m ust b e constan t in its last few co ordinates. Since h deos not depend o n j the num b er of constan t co ordinates is found f r o m t he last function h m . Sp ecifically , we hav e h m ( x n , . . . , x n − k ) = x n − m +1 ∗ h ( x n − m , . . . , x n − k | {z } k − m +1 v ariables , x n − k − 1 , . . . , x n − k − m +1 | {z } m − 1 terms with h const an t ) (1 8) The preceding condition leads to the necess ary restrictions o n h and every h j for a consisten t deriv atio n of (1 6 ) from (1 5), so (18 ) is a consistency condition. No w from (17) and (18) w e o btain for ( u 0 , . . . , u k ) ∈ G k +1 h j ( u 0 , . . . , u k ) = u j − 1 ∗ h ( u j , u j +1 . . . , u j + k − m ) , j = 1 , . . . , m. (19) W e refer to H = [ h 1 , . . . , h m ] as a form symmetry for Eq.(8) if the comp onents h j are defined by (19). Since the range of H has a low er dimension than its domain, w e sa y that H is an or der-r e d ucing form symmetry . Using the forms in (1 9) for h j in (15) for eve ry solution { x n } of Eq.(8) w e obtain the following pa ir of equations from (15), the first of whic h is just (16): t n +1 = g n ( t n , . . . , t n − m +1 ) , (20a) x n +1 = t n +1 ∗ h ( x n , . . . , x n − k + m ) − 1 . (20b) The p o w er − 1 represen ts group inv ersion in G. The first equation (20 a) may b e called a factor of Eq.(8) since it is distilled from the semiconjugate factor Φ n . The second equation (20b) that links the factor to the original equation may b e called a c ofactor o f Eq.(8). W e call the s ystem of equations (20) a semic onjugate (SC) factorization o f Eq.(8). Note that if { t n } is a given solution of (20a) then using this sequence in (20b) pro duces a solution { x n } o f (8). Con v ersely , if { x n } is a solution of (8) then the sequence t n = x n ∗ h ( x n − 1 , . . . , x n − k + m − 1 ) is a solution of (20a) with initial v alues t − j = x − j ∗ h ( x − j − 1 , . . . , x − j − k + m − 1 ) , j = 0 , . . . , m − 1 . Since solutions of the pair o f equations (20a) and (20b) coincide with the solutions of the scalar equation (8), we sa y that the pair (20a) and (20b) is equiv alent to ( 8 ). The following summarizes the preceding discussions. Theorem 1 . L et k ≥ 1 , 1 ≤ m ≤ k and supp ose that ther e ar e functions h : G k − m +1 → G a n d g n : G m → G that satisfy e quations ( 15) and (19 ). Then with the or der-r e d ucin g form symme try H ( u 0 , . . . , u k ) = [ u 0 ∗ h ( u 1 , . . . , u k +1 − m ) , . . . , u m − 1 ∗ h ( u m , . . . , u k )] 6 Eq.(8) is e q uiva lent to the SC factorization c onsisting of the p air of e quations (20a) and (20b) whose or ders m and k + 1 − m r e s p e ctively, add up to the or der o f (8). In this setting w e sa y that the SC facto r izat io n (20) giv es a typ e-( m, k + 1 − m ) or der r e d uction for Eq.(8 ), or that (8) is a type-( m, k + 1 − m ) equation. A second order difference equation ( k = 1) can hav e only the order-reduction ty p e (1,1) into t w o first order equations although the factor and cofactor equations are not uniquely defined. In general, a higher order difference equation may ha v e more than one SC factorization. A third order equation can hav e tw o order-reduction t yp es, namely , (1,2) and (2,1). Of the k p ossible order reduction t yp es for an equation of order k + 1 the tw o extreme ones, namely , (1 , k ) and ( k , 1) ha v e the extra appeal of having an equation of order 1 as either a factor or a cofactor. In the next tw o sec tions w e discuss classes of higher order difference equations havin g one of these order-reduction t yp es. W e not e that the SC factorization of Theorem 1 do es not require the determina- tion of φ j,n for j ≥ 2 . F or completeness, w e close this section b y showin g that eac h co ordinate function φ j,n pro jects in to co ordinate j − 1 for j > 1 , th us sho wing that Φ n is of scalar t yp e, i.e., it is the unfolding o f Eq.(16 ) in the same sense that F n unfolds (8). If the maps h j are give n b y (1 9) then for j ≥ 2 (1 1 ) giv es φ j,n ( h 1 ( u 0 , . . . , u k ) , . . . , h m ( u 0 , . . . , u k )) = h j ( f n ( u 0 , . . . , u k ) , u 0 , . . . , u k − 1 ) = u j − 2 ∗ h ( u j − 1 , u j . . . , u j + k − m − 1 ) = h j − 1 ( u 0 , . . . , u k ) . Therefore, for eac h n and for eve ry ( t 1 , . . . , t m ) ∈ H ( G k +1 ) w e hav e Φ n ( t 1 , . . . , t m ) = [ g n ( t 1 , . . . , t m ) , t 1 , . . . , t m − 1 ] i.e., Φ n | H ( G k +1 ) is of scalar t yp e. F urther, if H is defined comp onen t-wise b y (19) then H ( G k +1 ) = G m ; i.e., H is on to G m so that Φ n is of scalar t yp e. T o prov e the onto claim, w e pic k arbitra ry [ t 1 , . . . , t m ] ∈ G m and set u m − 1 = t m ∗ h ( u m , u m +1 . . . , u k ) − 1 where u m = u m +1 = . . . u k = 1 (t he group iden tit y). Then t m = u m − 1 ∗ h (1 , 1 . . . , 1) = u m − 1 ∗ h ( u m , u m +1 . . . , u k ) = h m ( u 0 , . . . , u k ) = h m ( u 0 , . . . , u m − 2 , t m ∗ h (1 , 1 . . . , 1) − 1 , 1 . . . , 1) . for an y c hoice of u 0 , . . . , u m − 2 ∈ G. Similarly , define u m − 2 = t m − 1 ∗ h ( u m − 1 , u m . . . , u k − 1 ) − 1 7 so as to get t m − 1 = u m − 2 ∗ h ( u m − 1 , u m . . . , u k − 1 ) = h m − 1 ( u 0 , . . . , u k ) = h m − 1 ( u 0 , . . . , u m − 3 , t m − 1 ∗ h ( u m − 1 , 1 . . . , 1) − 1 , u m − 1 , 1 . . . , 1) for any c hoice of u 0 , . . . , u m − 3 ∈ G. Con tin uing in this wa y , induction leads to selection of u m − 1 , . . . , u 0 suc h that t j = h j ( u 0 , . . . , u m − 1 , 1 , . . . , 1) , j = 1 , . . . , m and it is pro ved that H is onto G m . 4 HD1 and oth er t yp e- ( k , 1) factorizations If m = k then the function h : G → G in (19) is of one v ariable and w e obtain a t ype-( k , 1) order reduction with form symmetry H ( u 0 , . . . , u k ) = [ u 0 ∗ h ( u 1 ) , u 1 ∗ h ( u 2 ) . . . , u k − 1 ∗ h ( u k )] (21) and SC fa ctorization t n +1 = g n ( t n , t n − 1 , . . . , t n − k +1 ) x n +1 = t n +1 ∗ h ( x n ) − 1 where t he functions g n : G k → G are determined b y the giv en functions f n in (8) as in the previous section. The simplest example of a non- constan t h in this setting is the iden tity function h ( u ) = u for all u ∈ G. An example of a t ype-( k , 1) difference equation ha ving this t yp e of form symmetry ov er (0 , ∞ ) under ordinary multiplic ation is the ratio nal equation x n +1 = ax n − 1 x n x n − 1 + b , a, b > 0 . (22) The term x n x n − 1 in the denominator suggests m ultiplying (22) b y x n on b oth sides and substituting t n = x n x n − 1 = x n h ( x n − 1 ) 8 to get the SC factorization t n +1 = at n t n + b = g ( t n ) , t 0 = x 0 x − 1 (23) x n +1 = t n +1 x n = t n +1 h ( x n ) . F urther, Eq.(23) can b e made linear by the c ha ng e of v ar iables s n = 1 /t n . F or an exhaustiv e tr eatmen t of (22 ) based on these ideas, see [16]. Another form symmetry of t ype (21) that is defined on C or R is based on h ( u ) = cu where c is a fixed, no nzero complex or real n um ber. This t ype of form sym metry (with real c ) has b een used in e.g., [7] and [17]. In the case where h ( u ) = u − 1 is based on group in v ersion, it is p ossible to identify the class of functions f n that ha v e the form symmetry (21). Equation (8) is said to b e homo gen e ous of de gr e e 1 (HD1) if for ev ery n = 1 , 2 , 3 , . . . the functions f n are homogeneous of degree 1 relativ e to the group G , i.e., f n ( u 0 ∗ t, . . . , u k ∗ t ) = f n ( u 0 , . . . , u k ) ∗ t for all t, u i ∈ G, i = 0 , . . . , k , n ≥ 1 . If G is non-commutativ e then t his definition giv es a “r ig h t v ersion” of the HD1 prop erty ; a “ left v ersion” can b e defin ed analogously . W e note that the tw o equations (1) and (5 ) in the In tro duction are HD1 relativ e t o the additiv e group of real n um bers. F or commen ts on homog eneous functions and their abundance on groups we refer to [20]; though stated for functions of tw o v ariables, the results in [20] easily extend to an y n umber of v ariables. The follo wing result sho ws t ha t the HD1 prop ert y c ha rac- terizes the in v ersion-based fo rm symmetry and yields a t ype-( k , 1) order-reduction in ev ery case. Theorem 2 . Eq.(8 ) has the inversion-b ase d form symmetry H ( u 0 , . . . , u k ) = [ u 0 ∗ u − 1 1 , . . . , u k − 1 ∗ u − 1 k ] , h ( t ) = t − 1 (24) if and only if f n is HD1 r elative to G for al l n . In this c a se, (8) has a typ e-( k , 1 ) or der-r e d uction with the SC factorization t n +1 = f n (1 , t − 1 n , ( t n ∗ t n − 1 ) − 1 , . . . , ( t n ∗ t n − 1 ∗ · · · ∗ t n − k +1 ) − 1 ) (25a) x n +1 = t n +1 ∗ x n . (25b) Note that the factor differ enc e e q uation (25a) has or der k and its c ofac tor (25b) is line ar non-autonomous of or der one in x n . 9 Pro of. First, assume that (8) has the form symmetry (24) that satisfies Eq.(15) for giv en functions g n , i.e., f n ( u 0 , . . . , u k ) ∗ u − 1 0 = g n ( u 0 ∗ u − 1 1 , . . . , u k − 1 ∗ u − 1 k ) . (26) Let t ∈ G b e arbitrary . Then f o r all n (26) implies f n ( u 0 ∗ t, . . . , u k ∗ t ) = g n (( u 0 ∗ t ) ∗ ( u 1 ∗ t ) − 1 , . . . , ( u k − 1 ∗ t ) ∗ ( u k ∗ t ) − 1 ) ∗ ( u 0 ∗ t ) = [ g n ( u 0 ∗ u − 1 1 , . . . , u k − 1 ∗ u − 1 k ) ∗ u 0 ] ∗ t = f n ( u 0 , . . . , u k ) ∗ t. It follo ws that f n is HD1 relativ e to G for all n and the first part of the theorem is prov ed. The con v erse is prov ed in a straightforw ard fashion; see [2 1]. Remarks. 1. Equation (25b) can be solv ed explicitly in terms of a solution { t n } of (2 5a) a s follo ws: x n = Q n − 1 i =0 t n − i ∗ x 0 n = 1 , 2 , 3 , . . . (27) where the m ultiplicativ e no t ation is used f o r it erations of the g roup opreation ∗ . In additiv e (and comm utative) notation, (27) tak es the form x n = x 0 + P n i =1 t i . (28) 2. W e can quick ly construct Eq.(25a) directly fr o m (8) in the HD 1 case by making the substitutions 1 → x n , ( t n t n − 1 · · · t n − i +1 ) − 1 → x n − i for i = 1 , 2 , . . . , k. (29) Recall that 1 represen t s the gr o up iden tit y in m ultiplicativ e notation. In additiv e notation (29) t a k es the form 0 → x n , − t n − t n − 1 · · · − t n − i +1 → x n − i for i = 1 , 2 , . . . , k . (30) Previous studies in v olving HD1 equations implicitly use the idea behind Theorem 2 ab o v e to reduce second order equations to first order ones; see e.g. [6], [10], [16]. Examples 1- 3 next illustrate Theorem 2 a nd some asso ciated concepts . Example 1 . Consider the rational delay difference equation x n +1 = x n  a n x n − k +1 x n − k + b n  , x 0 , x − 1 , . . . , x − k > 0 (31) 10 where { a n } , { b n } are seque nces of p ositiv e real n um b ers. This equation is HD1 relativ e to the group (0 , ∞ ) under ordinar y multiplication. Th us Theorem 2 and (29) giv e the SC factor izat io n of (3 1) as t n +1 = (1)  a n ( t n t n − 1 . . . t n − k +2 ) − 1 ( t n t n − 1 . . . t n − k +1 ) − 1 + b n  = a n t n − k +1 + b n , x n +1 = t n +1 x n . In this case, the factor equation is linear non-homogeneous with a t ime dela y of k − 1 and can b e solv ed to obtain an explicit solution of (31) through (27), if desired. Alternativ ely , w e can quic kly deriv e information ab out the asymptotic b ehav ior of (31). F or instance, if lim n →∞ a n = a > 0 , lim n →∞ b n = b ≥ 0 , a + b 6 = 1 then w e conclude that all positive solutions of (31) conv erge to zero if a + b < 1 a nd to ∞ if a + b > 1 . Example 2 . This example illustrates a situation where (8) and (25a) are b oth HD1, although with respect to differen t groups. Conside r the third order equation x n +1 = x n + a ( x n − x n − 1 ) 2 x n − 1 − x n − 2 , a 6 = 0 . (32) Relativ e to the additiv e g r o up G = R , this eq uation is HD1 with the eviden t fo rm x n − x n − 1 . Sp ecifically , (32) has the f orm symmetry H ( u, v , w ) = [ u − v , v − w ] , h ( t ) = − t. Making the substituion t n = x n − x n − 1 , or using (30 ) w e get the SC f actorization t n +1 = at 2 n t n − 1 , t 0 = x 0 − x − 1 , t − 1 = x − 1 − x − 2 (33) x n +1 = t n +1 + x n . This is a t ype-( 2 ,1) order reduction. Note that t n 6 = 0 for n ≥ − 1 if initial v alues satisfy x 0 , x − 2 6 = x − 1 . (34) Relativ e to the multiplicativ e g roup of all nonzero real num b ers, the second order equation (33) is HD1 with form symm etry H ( u, v ) = u v , h ( s ) = 1 s . 11 Making the substitution (29) g ives the ty p e-(1,1) order reduction s n +1 = as n , s 0 = t 0 t − 1 t n +1 = s n +1 t n No w usin g (27) and (28) w e obtain the follo wing form ula for solutions of ( 3 2) sub ject to (34): x n = x 0 + t 0 n X j =1 s j 0 a j ( j +1) / 2 , s 0 = t 0 t − 1 = x 0 − x − 1 x − 1 − x − 2 . Example 3 . Consider the following v arian t of Eq.(32): x n +1 = x n + a ( x n − x n − 1 ) x n − 1 − x n − 2 , a 6 = 0 (35) sub ject to (34). As in Example 2, the HD1 form x n − x n − 1 giv es the SC factor izat io n t n +1 = at n t n − 1 , t 0 = x 0 − x − 1 , t − 1 = x − 1 − x − 2 (36) x n +1 = t n +1 + x n . (37) Unlik e (33), Eq .(36) is not HD1. But a straightforw ar d calculation sho ws that ev ery solution of (36) has p erio d 6 as follows  t − 1 , t 0 , at 0 t − 1 , a 2 t − 1 , a 2 t 0 , at − 1 t 0  . (38) Th us w e ma y use (37) and (28) to calculate the corresponding s olution of (35) explicitly: F or eac h n, there are in tegers δ n ≥ 0 and 0 ≤ ρ n ≤ 5 suc h that n = 6 δ n + ρ n . If σ is t he sum of the six n um b ers in ( 38) then the explicit solution of Eq.(35 ) ma y b e stated as x n = x 0 + P n i =1 t i = x 0 + σ 6 ( n − ρ n ) + P n i = n − ρ n +1 t i where ev ery t i is in the set (38) and the last term is zero if ρ n = 0 . 12 Remark . ( The ful l triangular factoriz a tion pr op erty ) The equation in Example 2 has an in teresting extra feature: it can b e fully SC factored as a system of first order difference equations s n +1 = as n t n +1 = s n +1 t n = as n t n x n +1 = t n +1 + x n = as n t n + x n This sy stem is triangular in the sense t ha t eac h equ ation is indep enden t of the v ariables in the equations b elow it. F o r a general discussion of the p erio dic solutions of systems of this t ype see [2], [11]. If a difference equation of order k + 1 has the prop ert y that it can be factored completely into a triangular system of first order difference equations then w e sa y that the difference equation has the ful l triang ular factorization pr op erty or tha t it is FTF. Clearly , ev ery HD 1 equation of order 2 is FTF but it is b y no means clear if all HD1 equations of order 3 or greater are FTF. F or instance, it is no t ob vious that Eq.(35) in Example 3 do es in fact ha v e the F TF prop erty . The difficult y there is due to the non-HD1 nature of (36) whic h leads t o a form sy mmetry that inv olv es complex functions (see Example 5 b elow). F or non-HD1 equations, the FTF prop erty is not clear ev en for equations of order 2. But in Corollary 1 in the next section w e sho w that ev ery line ar non-homog eneous equation of order k + 1 is FTF with a complete factorization in to a triangular system of linear non- homogeneous first order equations. 5 Separability and t yp e - ( 1 , k ) factorization The class o f HD1 functions do es not include certain familiar functions. F or example, the linear non-homog eneous function φ ( u, v ) = au + bv + c is HD1 relativ e to the group of all real num b ers under addition o nly when a + b = 1; it is HD1 relativ e to the group (0 , ∞ ) under ordinary multiplication only when c = 0 and a, b ≥ 0. These restrictions suggest tha t a prop er study of order reducible form symmetries fo r linear difference equations do es not b elong in the con text of HD1 equations. In this section w e define a class of equations that prop erly includes all linear non- homogeneous difference eq uations with constan t coefficien ts as w ell as some other in teresting non-HD1 equations. Before discus sing this class, recall that a t ype-(1 , k ) equation has a SC facto r izat io n with factor of order m = 1 and cofactor of or der k . Therefore, the form symmetry is a scalar function that may b e written a s H ( u 0 , . . . , u k ) = u 0 ∗ h ( u 1 , . . . , u k ) . 13 No w w e define a function φ : G k +1 → G to b e sep ar a ble (or algebr aic al ly factor able) relativ e to G if t here are k + 1 functions φ j : G → G , j = 0 , 1 , . . . , k suc h that for a ll u 0 , . . . , u k ∈ G, φ ( u 0 , . . . , u k ) = φ 0 ( u 0 ) ∗ · · · ∗ φ k ( u k ) . Note that ev ery linear non-homogeneous function is trivially separable relativ e to ev ery additiv e subgroup of the complex num b ers C . T he rational function φ ( u, v ) = au p /v is separable relativ e to the group of nonzero real n um b ers under m ultiplication for ev ery in teger p but it is HD1 relative to the same group if and o nly if p = 2 . The exp o nen tial function φ ( u, v ) = v e a − bu − cv is separable relativ e to the group o f non-zero real num b ers under m ultiplication but it is not HD 1 relativ e to that group. 5.1 Additiv e forms W e define Eq.(8) to b e s eparable if ev ery function f n is separable relative to the underlying group G . In this section w e consider the fo llowing separable v ersion of (8 ) o v er the gro up o f complex num b ers C under addition: x n +1 = α n + φ 0 ( x n ) + φ 1 ( x n − 1 ) + · · · φ k ( x n − k ) . (39) with x − j , α n ∈ C , φ j : C → C , j = 0 , 1 , . . . , k . (40) It is not strictly neces sary for the sak e of a pplications tha t the maps φ j b e defined on all of C (indeed, in most applicatio ns they a r e defined on the set R of real n um bers) but w e mak e a strong assumption to reduce the amount of tec hnical details in this article. The use o f complex n umbers is necessary because form symmetries of (39) ma y b e complex ev en if all quan tities in (40) a re r eal (this happ ens in particular fo r linear equations). The next result fro m [19] sho ws that Eq.(39) has an order-reducing form symme- try if one of the k + 1 functions φ 0 , . . . , φ k can b e expressed as a particular linear com bination of the remaining k functions. The fo r m symmetry in this case gives a t ype-( 1 , k ) order reduction of (39). Theorem 3 . Assume that ther e is a c onstant c ∈ C such that the functions φ 0 , . . . , φ k in Eq.(39) satisfy c k +1 z − c k φ 0 ( z ) − c k − 1 φ 1 ( z ) − · · · − cφ k − 1 ( z ) − φ k ( z ) = 0 for all z . (41) Then (39) has the fol lowing form symmetry 14 H ( z 0 , z 1 , . . . , z k ) = z 0 + h 1 ( z 1 ) + · · · + h k ( z k ) (42) wher e h j ( z ) = c j z − c j − 1 φ 0 ( z ) − · · · − φ j − 1 ( z ) , j = 1 , . . . k (43) The form symmetry in (42) and (4 3 ) yields the typ e- (1 , k ) or der r e duction z n +1 = α n + cz n , z 0 = x 0 + h 1 ( x − 1 ) + · · · + h k ( x − k ) (44) x n +1 = z n +1 − h 1 ( x n ) − · · · − h k ( x n − k +1 ) . (45) Remark. No te that the factor equation (44) has order 1 and t he cofactor (45) has order k in this case. F or r eference, w e note that ( 4 1) and (43) imply the following ch k ( z ) = φ k ( z ) . (46) A significan t feature of Eq.(45) is that it has the same fo rm as (3 9 ). Th us if the functions h 1 , . . . , h k satisfy the analog of (4 1 ) for some constan t c ′ ∈ C then Theorem 3 can b e applied t o (45). T he next result exploits this feature b y applying Theorem 3 to a linear non-homogeneous equation repeatedly un til w e are left with a triangular system of fir st order linear equations. Corollary 1 . The line ar non-homo g ene ous diffe r enc e e quation of or d e r k + 1 with c onstant c o efficients x n +1 + b 0 x n + b 1 x n − 1 + · · · + b k x n − k = α n (47) wher e b 0 , . . . , b k , α n ∈ C has the FTF pr o p erty a nd is e quiva l e nt to the fol lowin g triangular system of k + 1 first or der l i n e ar non-homo gene ous e quations z 0 ,n +1 = α n + c 0 z 0 ,n , z 1 ,n +1 = z 0 ,n +1 + c 1 z 1 ,n . . . z k ,n +1 = z k − 1 ,n +1 + c k z k ,n in which z k ,n = x n is the solution of Eq.(47) and the c onstants c 0 , c 1 , . . . , c k ar e the eigenvalues of the homo gene ous p art of (47), i.e., r o ots o f the cha r acteristic p olyno- mial P ( z ) = z k +1 + b 0 z k + b 1 z k − 1 + · · · + b k − 1 z + b k . (48) 15 Pro of . Defining φ j ( z ) = − b j z for j = 1 , . . . k and applying Theorem 3 abov e yields the SC factorization z 0 ,n +1 = α n + c 0 z 0 ,n x n +1 = z 0 ,n +1 − β 1 , 0 x n − · · · − β 1 ,k − 1 x n − k +1 (49) where c 0 satisfies (41) c k +1 0 z + c k 0 b 0 z + c k − 1 0 b 1 z + · · · + c 0 b k − 1 z + b k z = 0 for all z ∈ C , i.e. c 0 is a ro ot of the ch aracteristic p olynomial P in ( 48). F urther, the n um b ers β 1 ,j are give n via the function h j in (43) a nd (46) as h j ( z ) = β 1 ,j − 1 z , β 1 ,j − 1 = c j 0 + c j − 1 0 b 0 + · · · + b j − 1 , c 0 β 1 ,k − 1 = − b k − 1 . Alternativ ely , the num b ers β 1 ,j ma y b e calculated from the recursion β 1 ,j = c 0 β 1 ,j − 1 + b j , j = 1 , . . . k − 1 , β 1 , 0 = c 0 + b 0 , c 0 β 1 ,k − 1 = − b k − 1 . (50) Next, since Eq.(49), i.e., x n +1 + β 1 , 0 x n + · · · + β 1 ,k − 1 x n − k +1 = z 0 ,n +1 is of the same t ype as (47), Theorem 3 can b e applied to it to yield the SC factorization z 1 ,n +1 = z 0 ,n +1 + c 1 z 1 ,n x n +1 = z 1 ,n +1 − β 2 , 0 x n − · · · − β 2 ,k − 2 x n − k +2 in whic h c 1 satisfies (41) fo r (49), i.e., the p o w er is reduced b y 1 and co efficien ts adjusted appropriately as in c k 1 + β 1 , 0 c k − 1 1 + β 1 , 1 c k − 2 1 + · · · + β 1 ,k − 2 c 0 + β 1 ,k − 1 = 0 . No w w e sho w that c 1 is also a ro ot of P in (48). Define P 1 ( z ) = z k + β 1 , 0 z k − 1 + · · · + β 1 ,k − 2 z + β 1 ,k − 1 so that c 1 is a ro ot of P 1 . If it is shown that ( z − c 0 ) P 1 ( z ) = P ( z ) then P 1 divides P so c 1 is a ro ot of P . Direct calculation using (50) sho ws ( z − c 0 ) P 1 ( z ) = z k +1 + β 1 , 0 z k + β 1 , 1 z k + · · · + β 1 ,k − 1 z − c 0 z k − c 0 β 1 , 0 z k − 1 − · · · − c 0 β 1 ,k − 2 z − c 0 β 1 ,k − 1 = z k +1 + ( c 0 + b 0 ) z k + ( c 0 β 1 , 0 + b 1 ) z k − 1 · · · + ( c 0 β 1 ,k − 2 + b k − 2 ) z − c 0 z k − c 0 β 1 , 0 z k − 1 − · · · − c 0 β 1 ,k − 2 z − c 0 β 1 ,k − 1 = P ( z ) . 16 Therefore, the ab o v e pro cess inductiv ely generates t he system in the statemen t of this corollary . Remarks . (Op e r ator factorization, c omple m entary and p articular solutions) 1. The triangular SC factorizatio n of Coro llary 1 is essen tially what is obtained through op erator factorization. If E x n = x n +1 represen ts the forward shift op erator then as is w ell-know n, the eigen v alues factor the op erator P ( E ) with P defined b y (48); i.e., (4 7) can b e written as ( E − c 0 )( E − c 1 ) · · · ( E − c k ) x n − k = α n . (51) No w if we define ( E − c 1 ) · · · ( E − c k ) x n − k = y 0 ,n (52) then (51) can b e written as y 0 ,n +1 − c 0 y 0 ,n = α n whic h is the first equation in the triang ula r system of Corollar y 1 with y 0 ,n = z 0 ,n . W e ma y con tin ue in this fashion b y applying the same idea to ( 52); w e set ( E − c 2 ) · · · ( E − c k ) x n − k = y 1 ,n and write (52) a s y 1 ,n +1 − c 1 y 1 ,n = z 0 ,n whic h is the second equation in the triangular system if y 1 ,n = z 1 ,n − 1 . The reduction in the time index n here is due to the remov al of one o ccurrence of E . Pro ceeding in this fashion, setting y j,n = z j,n − j at eac h step, w e ev en tually arrive at ( E − c k ) x n − k = y k − 1 ,n ⇒ x n +1 − k = y k − 1 ,n + c k x n − k . Th us, with y k − 1 ,n = z k − 1 ,n − k +1 the preceding equation is the same as the last equation in the system of Corollar y 1. 2. With Corollary 1 we may obtain the eigenv alues and b oth the particular solu- tion and the solution of the homogeneous part of (47) sim ultaneously without needing to guess linearly indep enden t solutions, namely , t he complex exp onentials. W e indi- cate how this is do ne in the second order case k = 1 whic h is also represen tativ e of the higher order cases . First, for a giv en sequence s = { s n } of complex num b ers and for eac h c ∈ C , let us define the quan tity σ n ( s ; c ) = n X j =1 c j − 1 s n − j 17 and note that for sequences s, t and n umbers a, b ∈ C , σ n ( as + bt ; c ) = aσ n ( s ; c ) + bσ n ( t ; c ) , i.e., σ n ( · , c ) is a linear op erator on the space of complex sequences for eac h n ≥ 1 and each c ∈ C . F urt her, if s n = ab n then it is easy to see that σ n ( s ; c ) =  a ( b n − c n ) / ( b − c ) , c 6 = b anb n − 1 , c = b . (53) No w, if k = 1 then the semiconjugate factorization of (4 7 ) in to first o rder equations is z n +1 = α n + c 0 z n , z 0 = x 0 + ( c 0 − b 0 ) x − 1 (54) x n +1 = z n +1 + c 1 x n . (55) A straightforw ard inductiv e argument gives the solution o f (54) as z n = z 0 c n 0 + σ n ( α ; c 0 ) . (56) Next, insert (56) in to (55), set γ n = z n +1 and rep eat the ab o v e argumen t to obtain the general solution of (47) for k = 1, i.e., x n = x 0 c n 1 + σ n ( γ ; c 1 ) . If c 1 6 = c 0 then from (53) w e obta in after combining some terms and noting that γ 0 = z 1 = α 0 + c 0 z 0 , x n =  α 0 + c 0 z 0 c 0 − c 1  c n 0 +  x 0 − α 0 + c 0 z 0 c 0 − c 1  c n 1 + σ n ( σ ′ ( α ; c 0 ); c 1 ) . where σ ′ = { σ n +1 ( α ; c 0 ) } . W e recognize the first tw o terms of the abov e sum as giving the solution of the homog eneous part of (47) and the last term a s g iving the particular solution. In t he case of rep eat eigenv alues, i.e., c 1 = c 0 again from (53 ) w e get x n = [ x 0 c 0 + ( α 0 + c 0 z 0 ) n ] c n − 1 0 + σ n ( σ ′ ( α ; c 0 ); c 0 ) . 5.2 Multipli cativ e forms As another applicatio n of Theorem 3 w e consider the follo wing difference equation on the p ositive real line y n +1 = β n ψ 0 ( y n ) ψ 1 ( y n − 1 ) · · · ψ k ( y n − k ) , (57) β n , y − j ∈ (0 , ∞ ) , ψ j : (0 , ∞ ) → (0 , ∞ ) , j = 0 , . . . k . 18 T aking the logarthim of Eq.(57) changes it s m ultiplicative form to an additiv e one. Specifically b y defining x n = ln y n , y n = e x n , ln β n = α n , φ j ( r ) = ln ψ j ( e r ) , j = 0 , . . . k , r ∈ R w e can tr a nsform (57) into (39). Then Theorem 3 implies the f o llo wing generalization of the main result of [18]. Corollary 2 . Eq.( 57) has a fo rm symmetry H ( t 0 , t 1 , . . . , t k ) = t 0 h 1 ( t 1 ) · · · h k ( t k ) (58) if ther e is c ∈ C such that the fol l o wing is true for al l t > 0 , ψ 0 ( t ) c k ψ 1 ( t ) c k − 1 · · · ψ k ( t ) = t c k +1 . (59) The functions h j in (58) ar e given as h j ( t ) = t c j ψ 0 ( t ) − c j − 1 · · · ψ j − 1 ( t ) , j = 1 , . . . k (60) and the form symmetry in (58 ) and (60) yields the typ e- (1 , k ) or der r e duction r n +1 = β n r c n , r 0 = y 0 h 1 ( y − 1 ) · · · h k ( y − k ) (61a) y n +1 = r n +1 h 1 ( y n ) · · · h k ( y n − k +1 ) . (6 1b) Example 4 ( A simple equation with complicated m ultistable solutions). Equations of t ype (39) or (57) are capable of ex hibiting complex behavior, including the generation of co existing stable solutio ns of many differen t t yp es that r ange from p erio dic to c haotic. As a sp ecific example of suc h m ultistable equations consider the following second-order equation x n +1 = x n − 1 e a − x n − x n − 1 , x − 1 , x 0 > 0 . (62) Note that Eq.(62) has up to t wo isolat ed fixed p oints. One is the origin whic h is rep elling if a > 0 (eigen v alues of linearization are ± e a/ 2 ) and the other fixe d point is ¯ x = a/ 2. If a > 4 then ¯ x is unstable and no n- h yperb olic b ecause the eigen v alues of the linearization of (62) are − 1 a nd 1 − a/ 2 . The computer-generated diagram in Figure 1 shows the v ariety o f stable p erio dic and non-p erio dic solutions that o ccur with a = 4 . 6 and one initial v alue x − 1 = 2 . 3 fixed and the other initial v alue x 0 19 Figure 1: Bifur cations of solutions of Eq.(62) with a c h anging initial v alue; a = 4 . 6 is fixed. c hanging from 2.3 to 4.8; i.e., approa ching (or mo ving a w ay from) the fixed p o in t ¯ x on a straig ht line segmen t in the plane. In Fig ur e 1, f o r ev ery gr id v alue of x 0 in the range 2.3-4.8 , the last 200 (of 300) p oin ts of the solution { x n } a r e plotted v ertically . In this figure, stable solutions with p erio ds 2, 4, 8, 12 and 16 can b e easily identified. All of the solutions that a pp ear in Figure 1 represen t c o existing stable orbits of Eq.(62). There are also p erio dic and non- p erio dic solutions whic h do not a pp ear in Fig ure 1 b ecause they are unstable (e.g., the fixed p oint ¯ x = 2 . 3). Additional bifurcations of b oth p erio dic and no n- p erio dic solutions o ccur outside the range 2.3-4.8 whic h ar e not sho wn in Fig ure 1. Understanding the b eha vior for solutions of Eq.(62) is made easier when w e lo ok at its SC f a ctorization given by (61a) and (6 1b). Here k = 1 and ψ 0 ( t ) = e − t , ψ 1 ( t ) = te − t , β n = e a for all n. Th us (59) takes the form ψ 0 ( t ) c ψ 1 ( t ) = t c 2 for all t > 0 e − ct te − t = t c 2 for all t > 0 20 The last equalit y is true if c = − 1 , which leads to the form symmetry h 1 ( t ) = t − 1 ψ 0 ( t ) − 1 = 1 te − t ⇒ H ( u 0 , u 1 ) = u 0 u 1 e − u 1 and SC fa ctorization r n +1 = e a r n , r 0 = x 0 h 1 ( x − 1 ) = x 0 x − 1 e − x − 1 (63) x n +1 = r n +1 h 1 ( x n ) = r n +1 x n e − x n . ( 6 4) All solutions of (63) with r 0 6 = e a/ 2 are p erio dic with p erio d 2:  r 0 , e a r 0  =  x 0 x − 1 e − x − 1 , x − 1 e a − x − 1 x 0  . Hence the orbit of eac h nontriv ial solution { x n } of (62) in the plane is restricte d to the pair of curv es ξ 1 ( t ) = e a r 0 te − t and ξ 2 ( t ) = r 0 te − t . (65) No w, if x − 1 is fixed a nd x 0 c hanges, then r 0 c hanges prop ortiona t ely t o x 0 . These c hanges in initial v alues are reflected as c hanges in p ar am e ters in (64). The orbits of the one dimensional map bte − t where b = r 0 or e a /r 0 exhibit a v a riet y of b ehaviors as the par ameter b changes according to well-kno wn rules such as the f undamen tal ordering of cycles and the o ccurrence of chaotic b eha vior with the a pp earnce of p erio d- 3 orbits when b is large enough; see, e .g., [3], [5], [14], [22 ]. Eq.(64) splits these b eha viors ev enly o v er the pair of curv es (65) as the initial v alue x 0 c hanges; see F ig ure 2 whic h sho ws t he orbits of (62) fo r tw o differen t initial v alues x 0 with a = 4 . 6 ; the first 100 p oin ts of e ac h orbit ar e discarded in thes e images s o as to highlight the asymptotic b eha vior of each or bit . The splitting ov er the pair of curv es ξ 1 , ξ 2 also explains wh y o dd p erio ds do not a pp ear in Figure 1. Example 5 . W e no w com bine differen t types of form symmetry to sho w that Eq.(35) in Example 3 has the FTF prop ert y . If w e assume that a > 0 , x 0 > x − 1 > x − 2 , x 0 ≥ 0 (66) then the multiplic ativ e group (0 , ∞ ) of p ositiv e real num b ers is the in v ar ian t set of Eq.(36). Since ( 36) is obvious ly s eparable ov er (0 , ∞ ), w e c hec k equalit y (59) in Corollary 2 with ψ 0 ( t ) = t and ψ 1 ( t ) = 1 /t : t c − 1 = t c 2 for all t > 0 . 21 Figure 2 : Two of the orbits in Figure 1 sho wn here on their lo ci of t w o curv es ξ 1 , ξ 2 in the state sp ace. This condition holds if c 2 − c + 1 = 0 . The quadratic has complex ro ots c ± = 1 ± i √ 3 2 so b y Corollary 2, Eq.(36 ) has a form symmetry H ( u 0 , u 1 ) = u 0 h 1 ( u 1 ) , where h 1 ( t ) = t c + t − 1 = t − c − and an SC factorization r n +1 = ar c + n , r 0 = t 0 h 1 ( t − 1 ) (67) t n +1 = r n +1 t c − n . (68) The three equations (67), (68) and (37 ) establish that Eq.(35) has t he FTF prop- ert y with a f actorization r n +1 = ar c + n , t n +1 = ar c + n t c − n , x n +1 = ar c + n t c − n + x n . It is notew orth y that in spite of the o ccurrence of complex exp onen ts, this system generates p ositiv e s olutions from p ositiv e initial v alues. This fa ct may seem less 22 surprising if Eq.(36) is tra nsformed in to a linear equation (with complex eigen v alues) b y taking logar it hms as in the b eginning of this section. Remark . Under the added restrictions (66), Coro llary 2 ma y also b e applied to Eq.(33) of Example 2 to obtain an SC factor izatio n using the separable type of form symmetry . Is this SC factorization differen t from that in Ex ample 2? T o see that they are in fact the same, let ψ 0 ( t ) = t 2 and ψ 1 ( t ) = 1 /t in the equalit y (59) and require that t 2 c − 1 = t c 2 for all t > 0 . This holds if c is a ro ot of the quadratic c 2 − 2 c + 1 = 0 , i.e., c = 1 . Thus using (60) w e calculate the form symmetry as H ( u 0 , u 1 ) = u 0 h 1 ( u 1 ) , where h 1 ( t ) = t 1 t − 2 = t − 1 . 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[19] Sedag ha t , H., Difference equations with order-reducing fo rm symmetries, to ap- p ear. [20] Sedag ha t , H., All homogeneous second order difference equations of degree one ha v e semiconjugate factorizations, J. D i ff er enc e Eqs. and Appl. , 13 (200 7) 453 - 456. 24 [21] Sedag ha t , H., Ev ery homogeneous difference equation of degree 1 admits a re- duction in o rder, J. Differ enc e Eqs. and Appl., to appear . [22] Sedag ha t , H., Nonline ar Differ e nc e Equations: T he ory with Applic ations to So- cial Scienc e Mo dels , Kluw er, Dordrec h t, 2003. Departmen t of Mathematics, Virginia Common w ealth Univ ersit y , Ric hmond, Virginia 23284-2 014, USA Email: hs edagha@v cu.edu 25