Trigonometric Recurrence Relations and Tridiagonal Trigonometric Matrices

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1 International Journal of Difference Equations. ISSN Volume 1 Number pp c Research India Publications Trigonometric Recurrence Relations and Tridiagonal Trigonometric Matrices Ondřej Došlý Department of Mathematics Masary University Janáčovo nám. 2a CZ Brno Czech Republic dosly@math.muni.cz Šára Pechancová Department of Mathematics Faculty of Civil Engineering Brno University of Technology Žižova 17 CZ Brno Czech Republic pechancova.s@fce.vutbr.cz Abstract We show that every tridiagonal symmetric matrix can be transformed by a special transformation into the so-called tridiagonal trigonometric matrix. The relationship of this transformation to 2 2 trigonometric symplectic system and to three-term trigonometric recurrence relations is discussed as well. AMS subject classification: 39A10. Keywords: Three-term recurrence relation symplectic difference system trigonometric transformation trigonometric system Sturm-Liouville difference equation. 1. Introduction In this paper we consider the three-term symmetric recurrence relation r 1 x 2 β x 1 r x 0 r 0 {0... N 1} 1.1 Research supported by the Grant 201/04/0580 of the Czech Grant Agency and the Research Project MSM of the Czech Ministry of Education Received April ; Accepted April

2 20 Ondřej Došlý and Šára Pechancová which is closely related to the 2 2 symplectic difference system x 1 a x b u u 1 c x d u 1.2 with a 1 b 1 r c r 1 r β and d 1 c r and to the tridiagonal symmetric matrix β 0 r r 1 β 1 r r 2 β 2 r L r N 3 β N 3 r N r N 2 β N 2 r N r N 1 β N 1 More precisely under the closely related we mean that a sequence x {x } N1 0 is a solution of 1.1 if and only if there exists u {u } N 0 such that x u solves 1.2 and this happens if and only if x {x } N1 0 with x 0 0 x N1 solves the linear system Lx 0. The integer N in 1.1 and 1.3 can be taen arbitrarily large and can be understood N when dealing with asymptotic properties of solutions of or with Jacobi matrices which are 1.3 with N. For a general bacground of the qualitative theory of equation 1.1 system 1.2 and the tridiagonal symmetric matrices Jacobi matrices 1.3 we refer to the boos [ ]. Our principal concern are transformations and oscillatory properties of 1.1 and 1.2 and their relationship to positive definiteness of 1.3. Recall that a 2 2 difference a b system 1.2 is said to be symplectic if the 2 2 matrix S is symplectic i.e. c d S T J S J J It is not difficult to verify that the 2 2 matrix S is symplectic if and only if det S 1 i.e. ad bc 1. In the special case a d : p and b c : q i.e. p 2 q 2 1 the system s 1 p s q c c 1 q s p c 1.5 is called trigonometric system. This terminology is motivated by the fact that there exists ϕ [0 2π such that cos ϕ p sin ϕ q and a solution of 1.5 is s sin 1 1 ϕj c cos ϕj. 1.6

3 Trigonometric Difference Relations and Matrices 21 Another important property of 1.5 is that it is self-reciprocal i.e. the so-called reciprocity transformation see e.g. [8] x 0 1 y u 1 0 z transforms 1.5 into itself. This means that the matrix with s c given by 1.6 c s Z s c forms the fundamental matrix of 1.5 which is symplectic and orthogonal. The aim of this paper is to deal with various aspects of the transformation theory of 1.2 in particular with the so-called trigonometric transformation of 1.2. First we establish an alternative proof of the main result of [4] which deals with this transformation. Then we study 1.2 under the assumption b 0. We show that then 1.2 and an associated trigonometric system 1.5 can be written in the form of three-term relation 1.1. We introduce the concepts of the trigonometric three-term symmetric recurrence relation and the tridiagonal trigonometric symmetric matrix and using these concepts we present a geometric proof of the relationship between positivity of the quadratic functional associated with 1.5 and nonexistence of generalized zeros of a certain solution of this system. We also compare our results with their continuous counterparts which are treated in details in the boos [5 11]. 2. Preliminaries Symplectic system 1.2 is closely related to the discrete quadratic functional Fx u; 0 N N { a c x 2 2b c x u b d u} considered over the class of {x u } N1 0 satisfying the boundary condition x 0 0 x N1 and the first equation of 1.2 x 1 a x b u the so-called equation of motion. This class of {x u} will be referred to as the class of admissible sequences for F. It is nown that the functional Fx u; 0 N is positive over the class of admissible x {x u} with x 0 if and only if the solution given by the initial condition x 0 0 u u 0 1 has no generalized zero in 0 N 1] i.e. x 1 0 whenever x 0 and x x 1 b N see e.g. [3]. We will also need some results of the transformation theory of symplectic difference systems see e.g. [8] for details. Let R h 0 1 g h

4 22 Ondřej Došlý and Šára Pechancová where h g are real-valued sequences. The transformation x y R u z 2.2 transforms system 1.2 into a system which is again symplectic y 1 ã y b z z 1 c y d z with the sequences ã b c d given by the formulas ã a h b g h 1 b b h h 1 c g 1 a h b g h 1 c h d g 2.3 d g 1b h 1 d h. 3. Trigonometric Recurrence Relations and Matrices We start with an alternative proof of the statement that every 2 2 symplectic system can be transformed by a transformation preserving oscillatory behavior into a trigonometric system. This result is formulated in [4] for general 2n 2n symplectic systems. The fact that we consider here a scalar 2 2 symplectic system enables to present a proof which is based on the direct computation and which is more transparent than that given in [4]. x [1] x [2] Theorem 3.1. Let is symplectic and let u [1] h 2 u [2] 2 x [1] be solutions of 1.2 such that the matrix x [2] 2 g x[1] u[1] x[2] u[2] u [1] x [2] u [2] h. 3.1 Then transformation 2.2 transforms 1.2 into trigonometric system 1.5. Proof. According to transformation formulas 2.3 we need to prove the identities ã d and b c i.e. the identities a h b g g 1b h 1 d h 1 h b g 1 a h b g h 1 c h d g. h h 1

5 Trigonometric Difference Relations and Matrices 23 We have and ã a h b g h 1 a h 2 b u[1] x[2] u[2] h h 1 d b 1 u[1] 1 b x [2] 1 u[2] 1 h2 1 d h h 1 1 [ b a h h b u [1] 1 b a x [2] b u [2] [a 1 h h 1 c d u [1] c x [2] d u [2] 2 b c a d a b u[1] a d b c b x [2] a h 2 b u[1] x[2] u[2] h h 1 u[2] [ 1 x [2] ] a d b c 2 ] ] 2 x [2] 1 d 2 b c a d so the first identity is proved. To prove the second identity b c we proceed as follows: ch h 1 h h 1 g 1 a h b g h h 2 1c h d g 1 u[1] 1 x[2] 1 u[2] 1 [a h 2 b u[1] h 2 1 [c h 2 d u[1] x[2] u[2] ] x[2] u[2] [ u[1] x[2] u[2] 1 u[1] 1 x[2] 1 u[2] 1 b h 21d ] ] h 2 [a 1 u[1] 1 x[2] 1 u[2] 1 c h 2 1 [ ] h 2 a b c a d u[1] x[2] u[2] [ 2 ] 2 u [1] u [2] h 2 b a d b c u[1] x[2] u[2] h 2 a a d b c } h 2 a u[1] x[2] u[2] u[1] x[2] u[2] ] {[b a d b c 2a b d ] u[1] x[2] u[2] [ 2 u [1] u [2] [ b u[1] x[2] u[2] ] 2 h 2 b h 2a ]

6 24 Ondřej Došlý and Šára Pechancová {[ 2 b x [2] b u[2] x[2] u[1] ] [ 2 2 u [1] u [2] 2 b b h h 1 ] 2 } 2 u[1] x[2] u[2] which proves that b c since h 0. Definition 3.2. A three-term recurrence relation 1.1 is said to be trigonometric if r N 1 and there exists a sequence e {e } N 0 e { 1 1} such that β e 1 sgn r 1 r1 2 1 e sgn r r A tridiagonal matrix L of the form 1.3 is said to be trigonometric if there exist ϕ 0... ϕ N 1 0 2π ϕ j π j 0... N 1 such that β cotg ϕ 1 cotg ϕ and r 1 sin ϕ. 3.3 Now we relate the concepts from the previous definition. Lemma 3.3. The following statements are equivalent: i Recurrence relation 1.1 is trigonometric; ii The associated symplectic difference system 1.2 is trigonometric; iii The related symmetric tridiagonal matrix 1.3 is trigonometric. Proof. i ii: Suppose that 1.1 is a trigonometric recurrence relation and put q 1 r p e r r Then obviously p 2 q 2 1 and p 1 q 1 p q e 1 r 1 r r 1 e r r 2 1 r β i.e. 1.1 can be written in the form 1 p1 x 2 p x 1 1 x q 1 q 1 q q

7 Trigonometric Difference Relations and Matrices 25 Put u 1 q x 1 p x i.e. x 1 p x q u and using 3.5 u 1 1 q 1 x 2 p 1 x 1 p q x 1 1 q x p q p x q u 1 q x p2 1 x p u q q x p u so x u is a solution of 1.5 with p q given by 3.4. ii iii: System 1.5 with q 0 is trigonometric if and only if p 2 q 2 1 i.e. there exists ϕ 0 2π ϕ π such that sin ϕ q cos ϕ p. In the previous part of the proof we have shown that if x u is a solution of 1.5 with q 0 then x {x } N1 0 solves 3.5 and hence if x 0 0 x N1 it solves the linear system Lx 0 where r 1 q 1 sin ϕ β p 1 q 1 p q cotg ϕ 1 cotg ϕ i.e. the matrix L in 1.3 is trigonometric. iii i: This implication is an immediate consequence of the relationship between Jacobi matrices and three-term symmetric recurrence relations see e.g. [10] or [13]. In the next statement we use the previous lemma and Theorem 3.1 to show that any tridiagonal symmetric matrix can be reduced via a diagonal transformation matrix to a tridiagonal trigonometric matrix. Theorem 3.4. Given a tridiagonal symmetric N N matrix L there exists a sequence {f } N 1 0 such that L can be expressed in the form L diag {f 0... f N 1 } L diag {f 0... f N 1 } 3.6 where L is a tridiagonal trigonometric matrix. Proof. Consider a symplectic difference system 1.2 with b 0. Then from the first equation in this system u 1 b x 1 a x and substituting into the second equation we get the recurrence relation x 2 b 1 a1 d x 1 x 0 b 1 b b

8 26 Ondřej Došlý and Šára Pechancová i.e. 1.1 with r 1 b and β a 1 b 1 d b. The same idea applied to the symplectic system which results from 1.2 upon the transformation 2.2 gives the recurrence relation 1.1 with r 1 b h h 1 b β ã1 b1 d b 1 h 2 ah bg 1 b 1 h 1 h 2 h 2 a 1 1 h 2 1 b 1 h 2 1β. d b 1 h g 1 b h 1 d b h h 1 Now by Theorem 3.1 there are sequences h g such that 2.2 transforms 1.2 into 1.5 and this fact coupled with the statement of Lemma 3.3 gives the required result where f : h N 1. In the last statement of this section we give a discrete analogue of the result that the trigonometric quadratic functional with a positive function q Fx; a b b a [ ] 1 qt x 2 qtx 2 dt is positive over the class of nontrivial differentiable functions satisfying xa 0 xb if and only if b a qt dt < π see e.g. [5]. Theorem 3.5. The tridiagonal trigonometric matrix 1.3 where β r are given by N with ϕ 0 π is positive definite if and only if ϕ < π. In particular the m-th principal minor D m of L m 1... N 1 is given by the formula m sin j0 ϕ j D m. 3.7 q 0 q m Proof. We prove the statement by induction. Denote q sin ϕ p cos ϕ. Then we have D 1 p 0 p 1 cotg ϕ 0 cotg ϕ 1 sinϕ 0 ϕ 1. q 0 q 1 q 0 q 1 0

9 Trigonometric Difference Relations and Matrices 27 Consider the determinant of the 1-th order D 1. Expanding D 1 by the 1-th row using the Laplace rule we get D 1 sin p p 1 D 1 D q q 1 q 2 1 sin ϕ cos ϕ 1 cos ϕ sin ϕ 1 q 0 q 1 q 2q 1 1 j0 ϕ j cos ϕ cos sin ϕ cos ϕ 1 sin 1 cos j0 ϕ j q 0 q 1 q 2q 1 1 j0 ϕ j cos ϕ q 0 q 1 q 2 q 1 1 j0 ϕ j sin ϕ sin 1 j0 ϕ j q 2 sin ϕ sin ϕ cos ϕ 1 cos ϕ sin ϕ 1 sin 1 cos ϕ sin ϕ 1 cos j0 ϕ j q 0... q 1 q 2q 1 sin ϕ sin ϕ 1 sin 1 j0 ϕ j 1 j0 ϕ j cos ϕ q 0... q 1 q 2q 1 1 sin j0 ϕ 1 j sin ϕ 1 cos 2 ϕ 1 cos j0 ϕ j sin 2 ϕ cos ϕ 1 1 sin ϕ cos ϕ 1 sin j0 ϕ j q 0... q 1 q 2 q 1 1 sin 2 ϕ [ sin j0 ϕ j sin ϕ 1 cos 1 sin ϕ cos ϕ [sin j0 ϕ j q 0... q 1 q 2 q 1 cos ϕ cos ϕ sin ϕ 1 cos q 0... q 1 q 2q 1 1 j0 ϕ j cos ϕ 1 cos cos ϕ 1 ] 1 j0 ϕ j sin ϕ 1 ] q 0... q 1 q 2q 1 1 sin 2 ϕ cos j0 ϕ 1 j ϕ 1 sin ϕ cos ϕ sin j0 ϕ j ϕ 1 sin ϕ sin 1 j0 ϕ j q 0... q 1 q 2 q 1 q 0... q 1 q 2 q 1 sin 1 j0 ϕ j q 0... q 1 1 j0 ϕ j sin ϕ which proves 3.7 and also the statement of theorem.

10 28 Ondřej Došlý and Šára Pechancová 4. Remars i As we have already mentioned earlier the transformation theory of the second order differential equation rtx ctx was studied in details in the monographs [511]. It was shown there among others that 4.1 can be transformed via the transformation x hty into the equation 1 qt y qty 0 q 1 rh Theorem 3.1 can be regarded as a discrete analogue of this statement. Theorem 3.1 which is not a new result in our paper only its proof is new has been used following the continuous pattern to study oscillatory properties of the second order difference equation r x c x 1 0 r in the recent papers [6712]. However the relationship of 4.3 to three-term recurrence relations 1.1 and Jacobi matrices 1.3 is not considered in those papers. ii It was shown in [4] that the sequence h in Theorem 3.1 can be taen in such a way that if b 0 in 1.2 then q > 0 in 1.5 see [4]. Consequently one can suppose without loss of generality that ϕ 0 π in trigonometric matrix 1.3 see Definition 3.2. iii Assume that the recurrence relation 1.1 is trigonometric i.e. in view of Lemma 3.3 there exist ϕ 0 2π \ {π} such that x 2 cotg ϕ 1 cotg ϕ x 1 x sin ϕ 1 sin ϕ By a direct computation or again using Lemma 3.3 one can verify that then 1 sin 1 ϕj x [2] cos ϕj is the fundamental system of solutions of 4.4. This is a discrete analogue of the fact that t t x 1 t sin qs ds x 2 t cos qs ds is the fundamental system of solution of 4.2. iv Using the previous remar 4.4 with ϕ 0 π is nonoscillatory if and only if ϕ <. An important concept of the theory of three-term symmetric recurrences is the so-called recessive solution which is a solution x with the property x lim 0 x

11 Trigonometric Difference Relations and Matrices 29 for any solution x linearly independent of x. The recessive solution of 4.4 can be computed explicitly it is the solution given by the formula x sin ϕ j. We hope to use this fact in investigating asymptotic properties of 1.1 in a subsequent paper. References [1] R.P. Agarwal Difference Equations and Inequalities. Theory Methods and Applications. Second edition. Monographs and Textboos in Pure and Applied Mathematics 228. Marcel Deer Inc. New Yor [2] R.P. Agarwal M. Bohner S.R. Grace and D. O Regan Discrete Oscillation Theory Hindawi Publishing Corporation New Yor [3] M. Bohner Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions J. Math. Anal. Appl. 199: [4] M. Bohner and O. Došlý Trigonometric transformations of symplectic difference systems J. Differential Equations 1631: [5] O. Borůva Linear differential transformations of the second order The English Universities Press Ltd. London [6] Z. Došlá and Š. Pechancová Conjugacy and phases of second order difference equations submitted. [7] Z. Došlá and D. Šrabáová Phases of linear difference equations and symplectic systems Math. Bohememica 128: [8] O. Došlý R. Hilscher Linear Hamiltonian difference systems: transformations recessive solutions generalized reciprocity Dynam. Systems Appl. 8: [9] S. Elaydi An Introduction to Difference Equations. Third edition. Undergraduate Texts in Mathematics. Springer New Yor [10] W.G. Kelley and A.C. Peterson Difference Equations An Introduction with Applications. Second edition. Harcourt/Academic Press San Diego [11] F. Neuman Global Properties of Linear Ordinary Differential Equations Mathematics and its Applications Kluwer Academic Publishers Group Dordrecht [12] Š. Ryzí On the first and second phases of 2 2 symplectic difference systems Stud. Univ. Žilina Math. Ser. 17: [13] G. Teschl Jacobi Operators and Completely Integrable Nonlinear Lattices American Mathematical Society New Yor j