DISCRETE TRIGONOMETRIC AND HYPERBOLIC SYSTEMS: AN OVERVIEW

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1 DISCRETE TRIGONOMETRIC AND HYPERBOLIC SYSTEMS: AN OVERVIEW Petr Zemáne Deprtment of Mthemtics nd Sttistics, Fculty of Science, Msry University, Kotlářsá, CZ Brno, Czech Republic E-mil: Abstrct. In this pper we present n overview of results for discrete trigonometric nd hyperbolic systems. These systems re discrete nlogues of trigonometric nd hyperbolic liner Hmiltonin systems. We show results which cn be viewed s discrete n-dimensionl extensions of sclr continuous trigonometric nd hyperbolic formules. Dte (corrected finl version): November 6, 013 How to cite: Ulmer Seminre über Funtionlnlysis und Differentilgleichungen 14, pp , University of Ulm, Ulm, Mthemtics Subject Clssifiction. 39A1; 39-06; 6D05; 33B10. Key words nd phrses. Discrete symplectic system; Trigonometric system; Hyperbolic system; Trigonometric function; Hyperbolic function.

2 Discrete trigonometric nd hyperbolic systems 1 1. Introduction In this pper we study the discrete trigonometric nd hyperbolic systems nd certin properties of their solutions. A motivtive fctor for this reserch were results, especilly from [16], which re nown for continuous trigonometric nd hyperbolic systems but they do not hve discrete nlogues yet. We present compct theory for discrete trigonometric nd hyperbolic systems. Hence we recll some results which re nown in the literture. In prticulr, we quote the results for trigonometric systems from [3], where specil coefficients were considered (but results remin true for more generl mtrices s well), nd the results for hyperbolic systems from [7]. All sttements re presented without proofs, becuse the discrete trigonometric nd hyperbolic systems re specil cses of their time scles extensions, which were studied in [33], or they cn be found in some recent ppers. Continuous trigonometric nd hyperbolic systems re specil cses of liner Hmiltonin system, which ws pid interest in [18 1, 34, 36]. On the other hnd, discrete trigonometric nd hyperbolic systems re specil cses of discrete symplectic system, which ws estblished nd investigted in [, 4, 8, 1,, 4, 5, 31, 3, 37, 38]. The system of the form X = Q(t)U nd U = Q(t)X, (CTS) where t [, b], X(t), U(t), nd Q(t) re n n mtrices nd dditionlly the mtrix Q(t) is symmetric for ll t [, b], is clled continuous trigonometric system (CTS). The origin of the study of sclr nd mtrix trigonometric functions cn be found in [7]. Other results were published in [5,14,15,17,8,9,35]. Discrete sclr nd mtrix trigonometric functions were studied in [3, 9 11, 38] nd more recently in [13, 6]. Properties of continuous hyperbolic systems, i.e., the systems in the form X = Q(t)U nd U = Q(t)X, (CHS) where t [, b], X(t), U(t), nd Q(t) re n n mtrices nd dditionlly the mtrix Q(t) is symmetric for ll t [, b], cn be found in the wor [30]. Discrete hyperbolic systems were studied in [7, 38]. This pper is orgnized s follows. In the next section we recll the definition nd bsic properties of the discrete symplectic systems. In Section 3 we show ll (nown nd new) results for discrete trigonometric systems. Similr results for discrete hyperbolic systems re presented in Section 4. In Section 5 we show some identities for sclr continuous time trigonometric nd hyperbolic functions which we cnnot generlize into the n- dimensionl discrete cse nd we explin the reson why it is not possible. Note, tht we consider only rel cse sitution (i.e., for the rel-vlued coefficients nd solutions), but the results hold in the complex domin s well (we only need to replce the trnspose of mtrix by the conjugte trnspose, the term symmetric by Hermitin, nd orthogonl by unitry ).

3 Petr Zemáne. Preliminries on symplectic system The discrete symplectic system is the first order liner difference system of the form z +1 = S z, (1) where {0, 1,,..., N} : = [0, N] Z, N N, z = ( x u ) R n, x, u R n, S R n n, nd dditionlly the mtrix S is symplectic for ll [0, N] Z, i.e., S T J S = J for ll [0, N] Z, where J : = ( 0 I I 0). Since ny symplectic mtrix is invertible, it is obvious, tht symplectic system (1) supplemented with the initil condition z 0 = z (0), where 0 {0, 1,..., N+1}: = [0, N+1] Z nd z (0) R n, hs unique solution. Hence we cn consider symplectic system (1) s mtrix system ( ) ( ) X+1 X = S U, +1 U where X, U R n n. Moreover, when we use the bloc nottion S = ( A B ) C D, where A, B, C, D R n n, we cn rewrite symplectic system (1) in the form X +1 = A X + B U, nd U +1 = C X + D U. (S) The condition for the symplecticity of the mtrix S tes the following (using the bloc nottion) equivlent form A T D C T B = I = D T A B T C, A D T B C T = I = D A T C B T, () A T C C T A = 0 = B T D D T C, A B T B A T = 0 = C D T D C T. (3) If Z = ( X U ) nd Z ( ) = X re ny mtrix solutions of (S), then their Wronsin mtrix Ũ is defined on [0, N + 1] Z s W (Z, Z): = Z T J Z = X T Ũ U T X. It is nown, see [8, Remr 1 (ii)], tht the Wronsin mtrix is constnt on [0, N +1] Z. A solution Z = ( X U ) of (S) is sid to be conjoined solution of (S) if W (Z, Z) 0, i.e., the mtrix X T U is symmetric t one (nd hence for ll) [0, N + 1] Z. Two solutions Z nd Z re clled normlized if W (Z, Z) I. A solution Z is sid to be bsis if rn Z = n for ll [0, N + 1] Z. It is nown tht for ny conjoined bsis Z there lwys exists nother bsis Z such tht Z nd Z re normlized, see [8, Remr 1 (ii)]. If the mtrix solutions Z = ( X U ) nd Z ( ) = X form normlized conjoined bses the Ũ following identities X T Ũ U T X = I = X Ũ T X U T, (4) X XT X X T = 0 = U Ũ T ŨU T (5)

4 Discrete trigonometric nd hyperbolic systems 3 re true for ll [0, N + 1] Z. Moreover, for ll [0, N] Z the following identities hold X +1 Ũ T X +1 U T = A, (6) X +1 X T X +1 XT = B, (7) U +1 Ũ T Ũ+1U T = C, (8) Ũ +1 X T U +1 XT = D. (9) ( ) The mtrix solution Ẑ = XÛ of the initil vlue problem (S), X 0 = 0 nd U 0 = I, with 0 [0, N + 1] Z, is clled the principl solution t Discrete trigonometric system In this section we consider the symplectic system (S) with the mtrix ( ) P Q S =, Q P where P, Q R n n for [0, N] Z. From () nd (3) we get tht the mtrices P nd Q stisfy P T P + Q T Q = I = P P T + Q Q T, (10) P T Q Q T P = 0 = P Q T Q P T. (11) The following definition is in ccordnce with definition of discrete trigonometric system from [3]. Definition 3.1 (Discrete trigonometric system). The system X +1 = P X + Q U, U +1 = Q X + P U, (DTS) where the mtrices P nd Q stisfy identities (10), (11) for ll [0, N] Z, is clled discrete trigonometric system. Remr 3.. By strightforwrd clcultion we my show tht the mtrix S for the trigonometric system stisfies J T S J = S for ll [0, N] Z. Therefore, trigonometric systems re self-reciprocl, see e.g. [3]. Remr 3.3. We justify the terminology trigonometric system. Let us consider sclr discrete trigonometric system x +1 = p x + q u, nd u +1 = q x + p u (1) with the initil conditions x 0 = 0 nd u 0 = 1. The coefficients must stisfy (in order to be trigonometric) p + q = 1 for ll [0, N] Z. Hence the motivtion is the fct tht there exist ϕ (0, π] such tht cos ϕ = p nd sin ϕ = q. Thus, the pir ( 1 ) ( 1 ) x = sin ϕ j nd u = cos ϕ j is the solution of (1). j=0 j=0

5 4 Petr Zemáne By n esy clcultion we cn verify the following lemm. Lemm 3.4. The pir ( X U ) solves system (DTS) if nd only if the pir ( U X) solves the sme trigonometric system. Equivlently ( X U) solves (DTS) if nd only if ( ) X U solves (DTS). Definition 3.5 (Discrete mtrix-vlued trigonometric functions). Let 0 [0, N + 1] Z be fixed. We define the discrete n n mtrix-vlued functions sine (denote Sin ) nd cosine (denote Cos ) by Sin ;0 : = X, Cos ;0 : = U, where the pir ( X U ) is the principl solution t 0 of (DTS). We suppress the index 0 when 0 = 0, i.e., we denote Sin : = Sin ;0 nd Cos : = Cos ;0. Remr 3.6. By using Lemm 3.5, we my define the mtrix-vlued trigonometric functions lterntively s Cos ;0 : = X ) nd Sin ;0 : = Ũ, where the pir is the solution of system (DTS) with the initil conditions X 0 = I nd Ũ 0 = 0. Remr 3.7. The solutions of system (CTS) with n = 1 te the form Sin(t) = sin t Q(τ) dτ nd Cos(t) = cosh t Q(τ) dτ. Hence, we cn see the discrete mtrix functions Sin ;0 nd Cos ;0 s n-dimensionl discrete nlogs of the sclr continuous trigonometric functions for sin(t s) nd cos(t s), which re solution of sclr system (CTS) with Q(t) 1. From the initil conditions nd from the constncy of the wronsin mtrix follows, tht ( Cos Sin ) nd ( Sin Cos) form normlized conjoined bses of system (DTS) nd the mtrix ( ) Cos Sin Φ = Sin Cos is fundmentl mtrix of (DTS). Hence, we cn express every solution ( X U ) of (DTS) in the form X = Cos X 0 Sin U 0, nd U = Sin X 0 + Cos U 0 for ll [0, N + 1] Z. This sttement corresponds to [3, Lemm 1]. The following identities re consequences of formules (4) (9) nd the next two corollries cn be found in [3, Theorem 5 nd Lemm 7]. Corollry 3.8. For ll [0, N + 1] Z hold, while for ll [0, N] Z the identities Cos T Cos + Sin T Sin = I = Cos Cos T + Sin Sin T, (13) Cos T Sin Sin T Cos = 0 = Cos Sin T Sin Cos T (14) we hve the identities Cos +1 Cos T + Sin +1 Sin T = P, (15) Cos +1 Sin T Sin +1 Cos T = Q. (16) From formul (13) we get the following identity, which is discrete mtrix nlogue of the fundmentl formul cos (t) + sin (t) = 1 for sclr continuous time trigonometric functions. Here is the usul Frobenius norm, i.e., V F = ( n ) 1 i,j=1 v ij, see [6, pg. 346]. ( X Ũ

6 Discrete trigonometric nd hyperbolic systems 5 Corollry 3.9. For ll [0, N + 1] Z we hve the identity Cos F + Sin F = n. (17) Remr If the mtrix Cos is invertible for some [0, N + 1] Z, then we cn clculte from (13) nd (14) tht Similrly, Cos 1 Sin 1 = Cos T + Sin T Cos T 1 Sin T. = Sin T + Cos T Sin T 1 Cos T holds if the mtrix Sin is invertible for some [0, N + 1] Z. The following theorem is discrete nlogue of [9, Theorem 1.1] nd is contined in [3, Theorem 9 nd Corollry 10]. Theorem For, l [0, N + 1] Z we hve Sin ;l = Sin Cos T l Cos Sin T l, (18) Cos ;l = Cos Cos T l + Sin Sin T l, (19) Sin = Sin ;l Cos l + Cos ;l Sin l, (0) Cos = Cos ;l Cos l Sin ;l Sin l. (1) Remr 3.1. With respect to Remr 3.7, identities (18) nd (19) re discrete mtrix extensions of the formule sin(t s) = sin(t) cos(s) cos(t) sin(s), cos(t s) = cos(t) cos(s) + sin(t) sin(s). Interchnging the prmeters nd l in (18) nd (19) yields to the following identities, which re n-dimensionl discrete nlogues of the sttement bout the prity for the sclr goniometric functions. This sttement cn be found in [3, Corollry 11]. Corollry Let, l [0, N + 1] Z. Then Sin ;l = Sin T l; nd Cos ;l = Cos T l;. () In Theorem 3.11 we generlized the sum nd difference formules for solutions of one trigonometric system with different initil conditions. Now, we shll del with two trigonometric systems nd their solutions with the sme initil conditions. In the continuous cse it ws done in [16, Theorem 1]. Consider the following two discrete trigonometric systems X +1 = P (i) X + Q (i) U nd U +1 = Q (i) X + P (i) U, (3) with initil conditions X (i) 0 = 0 nd U (i) 0 = I, where i = 1,. We denote by Sin (i) nd Cos (i) the corresponding solutions of system (3) by Definition 3.5. Note, tht for simplicity we use the following nottion (Q (i) )T = Q (i)t, (P (i) )T = P (i)t, (Sin (i) )T =

7 6 Petr Zemáne Sin (i)t, (Cos (i) )T (Cos )T = Cos T = Cos (i)t, (Sin + )T = Sin +T, (Cos+ )T = Cos +T, (Sin )T = Sin T. Now, we put Sin + : = Sin(1) Cos ()T Cos + : = Cos(1) Cos ()T Sin : = Sin(1) Cos ()T Cos : = Cos(1) Cos ()T Theorem Let the mtrices P (i) Sin + nd Cos+ solves the system X +1 = P (1) (X P ()T + U Q ()T U +1 = Q (1) (X P ()T + U Q ()T + Cos (1) Sin (1) Cos (1) + Sin (1), nd Sin ()T, (4) Sin ()T, (5) Sin ()T, (6) Sin ()T. (7) nd Q (i), i = 1,, stisfy (10) nd (11). The pir ) + Q (1) ) + P (1) ( X Q ()T ( X Q ()T with the initil conditions X 0 = 0 nd U 0 = I. The pir Sin nd Cos X +1 = P (1) (X P ()T U Q ()T U +1 = Q (1) (X P ()T U Q ()T ) + Q (1) ) + P (1) (X Q ()T (X Q ()T + U P ()T ), + U P ()T ), + U P ()T ), + U P ()T ), with the initil conditions X 0 = 0 nd U 0 = I. Moreover, for ll [0, N + 1] Z Sin + Sin+T Sin Sin T Sin + Cos+T Sin Cos T solves the system we hve + Cos + Cos+T = I = Sin +T Sin + + Cos+T Cos +, (8) + Cos Cos T = I = Sin T Sin + Cos T Cos, (9) Cos + Sin+T = 0 = Sin +T Cos + Cos+T Sin +, (30) Cos Sin T = 0 = Sin T Cos Cos T Sin. (31) Remr Although the bove two systems re visully of the sme form s the discrete trigonometric system (DTS) nd identities (8) nd (30) for the pir Sin + nd Cos + nd identities (9) nd (31) for Sin nd Cos pper to be lie the properties of normlized conjoined bses of system (S), the solutions ( ) ( Sin + Cos nd Sin ) + Cos re not conjoined bses of their corresponding systems, becuse these systems re not symplectic. When the two systems in (3) re the sme, we obtin the following corollry. It is discrete nlogue of [8, Theorem 1.1]. Corollry Assume tht the mtrices P nd Q stisfy (10) nd (11). Then the system X +1 = P (X P T + U Q T ) + Q ( X Q T + U P T ), U +1 = Q (X P T + U Q T ) + P ( X Q T + U P T ) with initil conditions X 0 = 0 nd U 0 = I possesses the solution X = Sin Cos T, nd U = Cos Cos T Sin Sin T, where the functions Sin nd Cos re the mtrix functions in Definition 3.5. Moreover, the bove mtrices X nd U commute, i.e., X U = U X.

8 Discrete trigonometric nd hyperbolic systems 7 Remr The previous corollry cn be viewed s the discrete n dimensionl nlogy of the double ngle formule for sclr continuous time trigonometric functions sin(t) = sin(t) cos(t) nd cos(t) = cos (t) sin (t). Corollry For ll [0, N + 1] Z the following identities hold Sin (1) Sin ()T = 1 (Cos Cos+ ), (3) Cos (1) Cos ()T = 1 (Cos + Cos+ ), (33) Sin (1) Cos ()T = 1 (Sin + Sin+ ). (34) Remr Identities (3) (34) re discrete n-dimensionl generliztions of sin(t) sin(s) = 1 [cos(t s) cos(t + s)], cos(t) cos(s) = 1 [cos(t s) + cos(t + s)], sin(t) cos(s) = 1 [sin(t s) + sin(t + s)]. According to the definition in [3, pg. 4] we introduce the discrete n-dimensionl nlogues of sclr trigonometric functions tngent nd cotngent. Definition 3.0. Whenever the mtrix Cos is invertible we define the discrete mtrixvlued function tngent (we write Tn) by Tn : = Cos 1 Sin. Whenever the mtrix Sin is invertible we define the discrete mtrix-vlued function cotngent (we write Cotn) by Cotn : = Sin 1 Cos. In the following two theorems we show properties of discrete mtrix-vlued functions tngent nd cotngent, which were published in [3, Corollry 6 nd Lemm 1]. Theorem 3.1. Whenever Tn is defined we hve Tn T = Tn, (35) Cos 1 Cos T 1 Tn = I. (36) Moreover, if the mtrices Cos nd Cos +1 re invertible, then Tn = Cos 1 +1 Q Cos T 1. (37) Remr 3.. In the sclr cse n = 1 identity (35) is trivil. In the sclr continuous time cse (CTS) with Q(t) = q(t) identity (36) tes the form 1 cos (s) tn (s) = 1 with s = t q(τ) dτ π.

9 8 Petr Zemáne Finlly, for the bove s identity (37) tes the form ( t q(t) tn q(τ) dτ) = cos t. q(τ) dτ Theorem 3.3. Whenever Cotn is defined we get Sin 1 Moreover, if the mtrices Sin nd Sin +1 re invertible, then Cotn T = Cotn, (38) Sin T 1 Cotn = I. (39) Cotn = Sin 1 +1 Q Sin T 1. (40) Remr 3.4. In the sclr cse n = 1 identity (38) is trivil. In the sclr continuous time cse (CTS) with Q(t) = q(t) identity (39) reds s 1 sin (s) cotn (s) = 1, where s π is from Remr 3.. And for these vlues of s identity (40) reduces to ( t q(t) cotn q(τ) dτ) = sin t. q(τ) dτ Next, similrly to the definitions of the discrete mtrix functions Sin (i), Cos(i), Sin±, nd Cos ± from (4) (7) we define the following functions Tn (i) : = (Cos(i) ) 1 Sin (i), Cotn(i) : = (Sin(i) ) 1 Cos (i), Tn + : = (Cos+ ) 1 Sin +, Cotn+ : = (Sin+ ) 1 Cos +, Tn : = (Cos ) 1 Sin, Cotn : = (Sin ) 1 Cos. Remr 3.5. Of course, the mtrix-vlued functions Tn ± nd Cotn± hve similr properties s the functions Tn nd Cotn. In prticulr, it follows from (30) nd (31) tht Tn ± nd Cotn± re symmetric. Theorem 3.6. For ll [0, N +1] Z such tht ll involved functions exist, the following identities hold Tn (1) + Tn () = Tn (1) (Cotn (1) Tn (1) Tn () = Tn (1) (Cotn () Tn (1) + Tn () = (Cos () Tn (1) Tn () = (Cos () ) 1 Sin +T + Cotn () Cotn (1) ) Tn(), (41) ) Tn(), (4) (Cos(1) )T 1, (43) ) 1 Sin T (Cos(1) )T 1, (44) Tn + = (Cos() )T 1 (I Tn (1) Tn () ) 1 (Tn (1) Tn = (Cos() )T 1 (I + Tn (1) Tn () ) 1 (Tn (1) Cotn (1) + Cotn () = Cotn (1) (Tn (1) Cotn (1) Cotn () = Cotn (1) (Tn () + Tn () Tn (1) + Tn () Tn () ) Cos()T, (45) ) Cos()T, (46) ) Cotn(), (47) ) Cotn(), (48)

10 Cotn (1) + Cotn () = (Sin () Cotn (1) Cotn () Discrete trigonometric nd hyperbolic systems 9 = (Sin () ) 1 Sin +T (Sin(1) )T 1, (49) ) 1 Sin T (Sin(1) )T 1. (50) Cotn + = (Sin() )T 1 (Cotn (1) + Cotn () ) 1 (Cotn (1) Cotn () Cotn = (Sin() )T 1 (Cotn () Cotn (1) ) 1 (Cotn (1) Cotn () I) Sin ()T, (51) +I) Sin ()T, (5) Remr 3.7. In the sclr continuous time cse (CTS) with Q(t) 1 te the identities (41) nd (4) the following form tn(t) + tn(s) = [cotn(t) + cotn(s)] tn(t) tn(s), tn(t) tn(s) = [cotn(s) cotn(t)] tn(t) tn(s), respectively, identities (43) nd (44) correspond to tn(t) + tn(s) = respectively, identities (45) nd (46) hve the form tn(t + s) = sin(t + s) sin(t s), tn(t) tn(s) = cos(t) cos(s) cos(t) cos(s), tn(t) + tn(s) tn(t) tn(s), tn(t s) = 1 tn(t) tn(s) 1 + tn(t) tn(s), respectively. Although the identities in (47) nd (48) reduce to it is common to write them s cotn(t) + cotn(s) = [tn(t) + tn(s)] cotn(t) cotn(s), cotn(t) cotn(s) = [tn(s) tn(t)] cotn(t) cotn(s), tn(t) tn(s) = tn(t) + tn(s) cotn(t) + cotn(s) = The identities in (49) nd (50) re equivlent to tn(s) tn(t) cotn(t) cotn(s). sin(t + s) cotn(t) + cotn(s) = sin(t) sin(s), sin(t s) cotn(t) cotn(s) = sin(t) sin(s) = sin(s t) sin(t) sin(s), respectively. Finlly, the identities (51) nd (5) re in ccordnce with cotn(t + s) = cotn(t) cotn(s) 1 cotn(t) cotn(s) + 1, cotn(t s) = cotn(t) + cotn(s) cotn(s) cotn(t), respectively.

11 10 Petr Zemáne 4. Discrete hyperbolic system In this section we define discrete hyperbolic mtrix functions nd show nlogous results s for the trigonometric functions in the previous section. Some of these results re nown from [7] but some results nd identities re new. In this section we consider the system (S) with the mtrix S in the form ( ) P Q S =, where the coefficients mtrices P, Q R n n stisfy for ll [0, N] Z the identities see lso [7, identity (10)]. Q P P T P Q T Q = I = P P T Q Q T, (53) P T Q Q T P = 0 = P Q T Q P T, (54) Remr 4.1. It ws shown in [7] tht from identities (53) nd (54) follows tht the mtrix P is necessrily invertible for ll [0, N] Z, nd moreover the identities (P ± Q ) 1 = P T QT nd (P 1 Q ) T = P 1 Q hold, see [7, identities (11) nd (1)]. The following definition ws not stted in [7] explicitly, but it corresponds to the system from [7, identity (19)]. Definition 4. (Discrete hyperbolic system). The system X +1 = P X + Q U, U +1 = Q X + P U, (DHS) where the mtrices P nd Q stisfy identities (53) nd (54) for ll [0, N] Z, is clled discrete hyperbolic system (DHS). Remr 4.3. Similrly to Remr 3.3 we cn show the justifibility of the terminology hyperbolic system. It follows from [7, formules (7) nd (8)], tht the solution of the sclr discrete hyperbolic problem x +1 = p x + q u, nd u +1 = q x + p u, with the initil conditions x 0 = 0 nd u 0 = 1 cn be expressed in the form ( 1 ) ( 1 ) ( 1 ) ( 1 ) x = sgn p i sinh ln p i + q i nd u = sgn p i cosh ln p i + q i. i=0 i=0 By nlogy to Lemm 3.4 we my proove the following lemm. Lemm 4.4. The pir ( X U ) solves the discrete hyperbolic system (DHS) if nd only if the pir ( X U ) solves the sme hyperbolic system. The following definition ws published in [7, Definition 3.1]. Definition 4.5 (Discrete mtrix-vlued hyperbolic functions). Let 0 [0, N + 1] Z be fixed. We define the discrete n n mtrix-vlued functions hyperbolic sine (denoted by Sinh ;0 ) nd hyperbolic cosine (denoted Cosh ;0 ) by i=0 Sinh ;0 : = X, Cosh ;0 : = U, i=0

12 Discrete trigonometric nd hyperbolic systems 11 where the pir ( X U ) is the principl solution of system (DHS) t 0. We suppress the index 0 when 0 = 0, i.e., we denote Sinh : = Sinh ;0 nd Cosh : = Cosh ;0. Remr 4.6. The solutions of the sclr continuous time system (CHS) te the form Sinh(t) = sinh t Q(τ) dτ nd Cosh(t) = cosh t Q(τ) dτ, see [30, pg. 1]. Hence, we cn see discrete mtrix functions Sinh ;0 nd Cosh ;0 s n-dimensionl discrete nlogs of the sclr continuous hyperbolic functions sinh(t s) nd cosh(t s), which re solutions of the sclr system (CHS) with Q(t) 1. Since the solutions ( Cosh Sinh ) nd (Sinh Cosh ) form normlized conjoined bses of (DHS), it follows tht ( ) Cosh Sinh Ψ : = Sinh Cosh is fundmentl mtrix of system (DHS). Therefore, every solution ( X U ) of (DHS) hs the form X = Cosh X 0 + Sinh U 0 nd U = Sinh X 0 + Cosh U 0 for ll [0, N + 1] Z, see lso [7, Theorem 3.1]. From formuls (4) (9) we get the following properties for solutions of discrete hyperbolic systems, which correspond to [7, Corollry 3.1 nd Theorem 3.]. Corollry 4.7. For ll [0, N + 1] Z the identities Cosh T Cosh Sinh T Sinh = I = Cosh Cosh T Sinh Sinh T, (55) Cosh T Sinh Sinh T Cosh = 0 = Cosh Sinh T Sinh Cosh T (56) hold, while for ll t [0, N] Z we hve the identities Cosh +1 Cosh T Sinh +1 Sinh T = P, Sinh +1 Cosh T Cosh +1 Sinh T = Q. Similrly to Corollry 3.9, with using the Frobenius norm, we cn estblish mtrix nlog of the formul cosh (t) sinh (t) = 1, which is [7, Corollry 3.]. Corollry 4.8. For ll [0, N + 1] Z the next identity holds Cosh F Sinh F = n. Remr 4.9. The mtrix-vlued function hyperbolic cosine hs the nturl property of sclr hyperbolic cosine. It follows from (55) tht the mtrix Cosh is invertible for ll [0, N + 1] Z. Moreover, from (55) nd (56) we get Cosh 1 = Cosh T Sinh T Cosh T 1 Sinh T. On the other hnd, the invertibility of the mtrix Sinh is not utomticlly gurnteed. However, if the mtrix Sinh is invertible t some point [0, N + 1] Z, then Sinh 1 = Cosh T Sinh T 1 Cosh T Sinh T. The following sum nd difference formuls were estblished in [7, Theorem 3.3 nd Corollry 3.3].

13 1 Petr Zemáne Theorem For, l [0, N + 1] Z we hve Sinh ;l = Sinh Cosh T l Cosh Sinh T l, (57) Cosh ;l = Cosh Cosh T l Sinh Sinh T l, (58) Sinh = Sinh ;l Cosh l + Cosh ;l Sinh l, (59) Cosh = Cosh ;l Cosh l + Sinh ;l Sinh l. (60) Remr With respect to Remr 4.6 for the sclr continuous time cse with Q(t) 1, identities (57) (58) re mtrix nlogues of sinh(t s) = sinh(t) cosh(s) cosh(t) sinh(s), cosh(t s) = cosh(t) cosh(s) sinh(t) sinh(s). By interchnging the prmeters nd l in (57) nd (58) we get the following corollry, which we cn see s n-dimensionl discrete extensions of the sttement bout the prity for the sclr hyperbolic functions, see lso [7, Corollry 3.4]. Corollry 4.1. Let be, l [0, N + 1] Z. Then the identities re true. Sinh ;l = Sinh T l; nd Cosh ;l = Cosh T l; (61) Now, by using the sme ide nd technique s for the discrete trigonometric functions, we cn derive nlogous results to Theorem 3.14 for discrete hyperbolic systems. For continuous time hyperbolic systems it ws shown in [30, Theorem 4.]. Hence, we consider the following two discrete hyperbolic systems X +1 = P (i) X + Q (i) U, U +1 = Q (i) X + P (i) U (6) with initil conditions X (i) 0 = 0 nd U (i) 0 = I, where i = 1,. Denote by Sinh (i) nd Cosh (i) the corresponding mtrix hyperbolic sine nd hyperbolic cosine functions from Definition 4.5. Note, tht for simplicity we use the following nottion (Q (i) Q (i)t, (P (i) )T = P (i)t (Cosh + )T = Cosh +T, (Sinh (i) )T = Sinh (i)t, (Sinh )T = Sinh T Sinh + : = Sinh(1) Cosh ()T Cosh + : = Cosh(1) Cosh ()T Sinh : = Sinh(1) Cosh ()T Cosh : = Cosh(1) Cosh ()T, (Cosh (i) )T, nd (Cosh )T = Cosh T + Cosh (1) + Sinh (1) Cosh (1) Sinh (1) )T =, = Cosh (i)t, (Sinh + )T = Sinh +T. Now, we put Sinh ()T, (63) Sinh ()T, (64) Sinh ()T, (65) Sinh ()T. (66) For two different discrete hyperbolic systems with the sme initil conditions the following sum nd difference formules hold. For two sme discrete hyperbolic systems with different initil conditions it is shown in Theorem 4.10.

14 Theorem Assume tht P (i) Sinh + nd Cosh+ solves the system Discrete trigonometric nd hyperbolic systems 13 X +1 = P (1) (X P ()T + U Q ()T U +1 = Q (1) (X P ()T + U Q ()T nd Q (i), i = 1,, stisfy (53) nd (54). The pir ) + Q (1) ) + P (1) (X Q ()T (X Q ()T + U P ()T ), + U P ()T ), with the initil conditions X 0 = 0 nd U 0 = I. The pir Sinh nd Cosh system X +1 = P (1) (X P ()T U +1 = Q (1) (X P ()T U Q ()T ) + Q (1) ( X Q ()T + U P ()T ), U Q ()T ) + P (1) ( X Q ()T + U P ()T ). with the initil conditions X 0 = 0 nd U 0 = I. Moreover, for ll [0, N + 1] Z Cosh + Cosh+T Cosh Cosh T Sinh + Cosh+T Sinh Cosh T solves the we hve Sinh + Sinh+T = I = Cosh +T Cosh + Sinh+T Sinh +, (67) Sinh Sinh T = I = Cosh T Cosh Sinh T Sinh, (68) Cosh + Sinh+T = 0 = Sinh +T Cosh + Cosh+T Sinh +, (69) Cosh Sinh T = 0 = Sinh T Cosh Cosh T Sinh. (70) Remr An nlogous sttement s in Remr 3.15 for the pirs ( ) ( Sin + Cos nd Sin ) + Cos now holds for the solutions ( ) ( Sinh + Cosh nd Sinh ) + Cosh. It mens, these two pirs re not conjoined bses of their corresponding systems, becuse these systems re not symplectic. As in Corollry 3.16, when the two systems in (6) re the sme, we obtin from Theorem 4.13 the following. It is discrete nlogue of [30, Corollry 1]. Corollry Assume tht P nd Q stisfy (53) nd (54). Then the system X +1 = P (X P T + U Q T ) + Q (X Q T + U P T ), U +1 = Q (X P T + U Q T ) + P (X Q T + U P T ) with the initil conditions X 0 = 0 nd U 0 = I possesses the solution X = Sinh Cosh T nd U = Cosh Cosh T + Sinh Sinh T, where Sinh nd Cosh re the mtrix functions in Definition 4.5. Moreover, the bove mtrices X nd U commute, i.e. X U = U X. Remr The previous corollry cn be viewed s the n dimensionl discrete nlogy of the double ngle formule for sclr continuous time hyperbolic functions sinh(t) = sinh(t) cosh(t) nd cosh(t) = cosh (t) + sinh (t). Now we cn formulte similr identities s for trigonometric functions in Corollry Corollry For ll [0, N + 1] Z we hve the identities Sinh (1) Sinh ()T = 1 (Cosh+ Cosh ), (71) Cosh (1) Cosh ()T = 1 (Cosh+ + Cosh ), (7) Sinh (1) Cosh ()T = 1 (Sinh+ + Sinh ). (73)

15 14 Petr Zemáne Remr Identities (71) (73) re discrete n-dimensionl nlogues of sinh(t) sinh(s) = 1 [cosh(t + s) cosh(t s)], cosh(t) cosh(s) = 1 [cosh(t + s) + cosh(t s)], sinh(t) cosh(s) = 1 [sinh(t + s) + sinh(t s)]. The next definition of discrete mtrix hyperbolic functions tngent nd cotngent is nown from [7, Definition 3.]. Recll tht the mtrix function Cosh is invertible for ll [0, N + 1] Z, see Remr 4.9. Definition We define the discrete mtrix-vlued function hyperbolic tngent (we write Tnh) by Tnh : = Cosh 1 Sinh. Whenever Sinh is invertible we define the discrete mtrix-vlued function hyperbolic cotngent (we write Cotnh) by Cotnh : = Sinh 1 Cosh. Anlogous results for discrete hyperbolic tngent nd cotngent s in Theorem 3.1 nd Theorem 3.3 for trigonometric functions tngent nd cotngent were published in [7, Theorem 3.4]. Theorem 4.0. For [0, N + 1] Z nd for [0, N] Z we hve Tnh T = Tnh, (74) Cosh 1 Cosh T 1 + Tnh = I, (75) Tnh = Cosh 1 +1 Q Cosh T 1. (76) Remr 4.1. In the sclr cse n = 1 identity (74) is trivil. In the sclr continuous time cse (CHS) identity (75) tes the form 1 cosh (s) + tnh (s) = 1 with s = Finlly, identity (76) tes the form ( t tnh Q(τ) dτ) = Theorem 4.. Whenever Cotnh is defined we get t Q(τ) dτ. Q(t) cosh t. Q(τ) dτ Cotnh T = Cotnh, (77) Cotnh Sinh 1 Moreover, if Sinh nd Sinh +1 re invertible, then Sinh T 1 = I. (78) Cotnh = Sinh 1 +1 Q Sinh T 1. (79)

16 Discrete trigonometric nd hyperbolic systems 15 Remr 4.3. In the sclr cse n = 1 identity (77) is trivil. In the sclr continuous time cse (CHS) identity (78) reds s cotnh (s) 1 sinh (s) = 1, where s 0 is from Remr 4.1. Finlly, for t Q(τ) dτ 0 reduces identity (79) to ( t Q(t) cotnh Q(τ) dτ) = sinh t. Q(τ) dτ Next, similrly to the definitions of the time scle mtrix functions Sinh (i), Cosh(i) Sinh ±, nd Cosh± from (6) (66) we define Tnh (i) : = (Cosh(i) ) 1 Sinh (i), Cotnh(i) : = (Sinh(i) ) 1 Cosh (i), Tnh + : = (Cosh+ ) 1 Sinh +, Cotnh+ : = (Sinh+ ) 1 Cosh +, Tnh : = (Cosh ) 1 Sinh, Cotnh : = (Sinh ) 1 Cosh. Remr 4.4. As in Remr 3.5 we conclude tht the first identities from (69) nd (70) imply the symmetry of the functions Tnh ± nd Cotnh±. Theorem 4.5. For ll [0, N + 1] Z such tht ll involved functions re defined we hve Tnh (1) + Tnh () = Tnh (1) (Cotnh (1) Tnh (1) Tnh () = Tnh (1) (Cotnh () Tnh (1) + Tnh () = (Cosh () Tnh (1) Tnh () = (Cosh () ) 1 Sinh +T + Cotnh () Cotnh (1), ) Tnh(), (80) ) Tnh(), (81) (Cosh(1) )T 1, (8) ) 1 Sinh T (Cosh(1) )T 1, (83) Tnh + = (Cosh() )T 1 (I + Tnh (1) Tnh () ) 1 Tnh = (Cosh() )T 1 (I Tnh (1) Tnh () ) 1 Cotnh (1) + Cotnh () = Cotnh (1) (Tnh (1) + Tnh () (Tnh (1) + Tnh () ) Cosh()T, (84) (Tnh (1) Tnh () ) Cosh()T, (85) ) Cotnh(), (86) Cotnh (1) Cotnh () = Cotnh (1) (Tnh() Tnh (1) ) Cotnh(), (87) Cotnh (1) + Cotnh () = (Sinh () Cotnh (1) Cotnh () = (Sinh () ) 1 Sinh +T (Sinh(1) )T 1, (88) ) 1 Sinh T (Sinh(1) )T 1. (89) Cotnh + = (Sinh() )T 1 (Cotnh (1) + Cotnh () ) 1 (Cotnh (1) Cotnh () Cotnh = (Sinh() )T 1 (Cotnh () Cotnh (1) ) 1 (Cotnh (1) Cotnh () +I) Sinh ()T, (90) I) Sinh ()T, (91)

17 16 Petr Zemáne Remr 4.6. Consider now the sclr continuous time cse (CHS) with Q(t) 1. Then identities (80) nd (81) re equivlent to tnh(t) + tnh(s) = [cotnh(t) + cotnh(s)] tnh(t) tnh(s), tnh(t) tnh(s) = [cotnh(s) cotnh(t)] tnh(t) tnh(s), respectively, identities (8) nd (83) hve the form tnh(t) + tnh(s) = sinh(t + s) sinh(t s), tnh(t) tnh(s) = cosh(t) cosh(s) cosh(t) cosh(s), respectively, while identities (84) nd (85) hve the form tnh(t + s) = tnh(t) + tnh(s) tnh(t) tnh(s), tnh(t s) = 1 + tnh(t) tnh(s) 1 tnh(t) tnh(s). respectively. Moreover, the identities in (86) nd (87) reduce to cotnh(t) + cotnh(s) = [tnh(t) + tnh(s)] cotnh(t) cotnh(s), cotnh(t) cotnh(s) = [tnh(s) tnh(t)] cotnh(t) cotnh(s), respectively, nd the identities in (88) nd (89) correspond in this cse to cotnh(t) + cotnh(s) = sinh(t + s) sinh(s t), cotnh(t) cotnh(s) = sinh(t) sinh(s) sinh(t) sinh(s), respectively. Finlly, the identities (90) nd (91) is in ccordnce with cotnh(t + s) = respectively. cotnh(t) cotnh(s) + 1 cotnh(t) cotnh(s) 1, cotnh(t s) = cotnh(t) + cotnh(s) cotnh(s) cotnh(t), 5. Concluding remrs In this pper we extended to discrete mtrix cse some identities nown for the sclr continuous time trigonometric nd hyperbolic functions (for n overview see e.g. [1, Chpter 4]). On the other hnd, there re still severl trigonometric nd hyperbolic identities which we could not extend to the discrete n-dimensionl cse, e.g. sin x ± sin y = sin x ± y cos x + cos y = cos x + y cos x cos y = sin x + y sinh x ± sinh y = sinh x ± y cosh x + cosh y = cosh x + y cosh x cosh y = sinh x + y cos x y, cos x y, sin y x, cosh x y, cosh x y, sinh x y, (9)

18 Discrete trigonometric nd hyperbolic systems 17 sin (x + y) sin (x y) = sin x sin y, cos (x + y) cos (x y) = cos x sin y, sinh (x + y) sinh (x y) = sinh x sinh y, cosh (x + y) cosh (x y) = sinh x + cosh y. The first identity in (9) reduces to sin x = sin x cos x if we put y = 0. Its right-hnd side seems to be similr to the solution X of the system in Corollry 3.16 but the lefthnd side is not the mtrix-vlued function Sin becuse this system is not trigonometric. Similr resoning cn be used for the remining identities in (9). Nevertheless, we cn clculte the corresponding products for mtrix-vlued functions Sin +, Cos+, Sinh+, nd Cosh+ s on left-hnd side in (93). But we cn not to modify these products in to the forms, which re similr to the right-hnd side in (93), becuse the mtrix multipliction is not commuttive. (93) 6. Acnowledgement The uthor thns to his dvisor Romn Šimon Hilscher for his help nd remrs which led to the present version of the pper. References [1] M. Abrmowitz, I. A. Stegun, editors, Hndboo of Mthemticl Functions with Formuls, Grphs, nd Mthemticl Tbles, New Yor: Dover Publictions, [] C. D. Ahlbrndt, A. C. Peterson, Discrete Hmiltonin Systems: Difference Equtions, Continued Frctions, nd Riccti Equtions, Kluwer Acdemic Publishers Group, Dordrecht, [3] D. R Anderson, Discrete trigonometric mtrix functions. Pnmer. Mth. J. 7 (1997), no. 1, [4] D. R. Anderson, Normlized prepred bses for discrete symplectic mtrix systems, in: Discrete nd Continuous Hmiltonin Systems, R. P. Agrwl nd M. Bohner, editors, Dynm. Systems Appl. 8 (1999), no. 3 4, [5] J. H. Brrett, A Prüfer trnsformtion for mtrix differentil systems, Proc. Amer. Mth. Soc. 8 (1957), [6] D. S. Bernstein, Mtrix Mthemtics. Theory, Fcts, nd Formuls with Appliction to Liner Systems Theory, Princeton University Press, Princeton, 005. [7] P. Bohl, Über eine Differentilgleichung der Störungstheorie, J. Reine Angew. Mth. 131 (1906), [8] M. Bohner, O. Došlý, Disconjugcy nd trnsformtions for symplectic systems, Rocy Mountin J. Mth. 7 (1997), no. 3, [9] M. Bohner, O. Došlý, Trigonometric trnsformtions of symplectic difference systems, J. Differentil Equtions 163 (000), no. 1, [10] M. Bohner, O. Došlý, The discrete Prüfer trnsformtions, Proc. Amer. Mth. Soc. 19 (001), no. 9, [11] M. Bohner, O. Došlý, Trigonometric systems in oscilltion theory of difference equtions, in: Dynmic Systems nd Applictions, Proceedings of the Third Interntionl Conference on Dynmic Systems nd Applictions (Atlnt, GA, 1999), Vol. 3, pp , Dynmic, Atlnt, GA, 001. [1] M. Bohner, O. Došlý, W. Krtz, Sturmin nd spectrl theory for discrete symplectic systems, Trns. Amer. Mth. Soc., to pper. [13] Z. Došlá, D. Šrbáová, Phses of second order liner difference equtions nd symplectic systems, Mth. Bohem. 18 (003), no. 3,

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