Solutions of Linear Difference Equations with Variable Coefficients

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 222, 79᎐91Ž. 1998 ARTICLE NO. AY975903

Ranjan K. Mallik*

Department of Electronics and Communication Engineering, Indian Institute of View metadata, citationTechnology, and similar Institution papers of Engineersat core.ac.uk Building, Panbazar, Guwahati 781001, India brought to you by CORE Submitted by Richard A. Duke provided by Elsevier - Publisher Connector

Received May 5, 1997

The explicit solution of a linear difference equation of unbounded order with variable coefficients is presented. As special cases, the solutions of nonhomoge- neous and homogeneous linear difference equations of order N with variable coefficients are obtained. From these solutions, we also get expressions for the product of companion matrices, and the power of a companion matrix. ᮊ 1998 Academic Press Key Words: Linear difference equation; variable coefficients; explicit solution; companion matrices.

1. INTRODUCTION

Linear difference equations or linear recurrences play a significant role in different areas of science and engineering. Pioneering work on the asymptotics of linear difference equations was done by Poincare´ wx 1 . Asymptotics of solutions of linear recurrences with coefficients having series representations have also been studied by Adamswx 2 , Birkhoff wx 3 , Trjitzinskyw 4᎐6 x , Culmer, Harris wx 7 , Sibuya wx 8 , Immink wx 9 , Wimp, and Zeilbergerwx 10 . Further work on convergence properties of linear recurrence sequences has been presented by Kooman and Tijdemanwx 11 . A survey of the literature on explicit solutions of linear recurrences reveals that in the case of linear recurrences with constant coefficients, the explicit solutions in terms of coefficients are well known. That is no longer the case when the coefficients vary with the indexwx 12 . A method of solving linear matrix difference equations with constant coefficients by using

*E-mail: [email protected].

0022-247Xr98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. 80 RANJAN K. MALLIK operator identities has been presented by Verde-Starwx 13 . For equations with variable coefficients, the closed-form solution of the first-order equation is knownwx 14᎐16 . If N y 1 linearly independent solutions of a homogeneous equation of order N are known, then any other linearly independent solution can be obtained explicitly in terms of the known solutions by using the CasoratianŽ. Wronskian in discrete domain wx14, 15, 17, 18 . For a second-order linear homogeneous difference equation with variable coefficients, explicit solutions in terms of coefficients have been presented by Buchbergerwx 19 and in wx 20 through different ap- proaches. Solutions of linear difference equations of unbounded order and of order N with variable coefficients have been represented in terms of determinants of submatrices of a single solution matrix by Kittappawx 21 . Work on the existence and construction of closed-form solutions of linear recurrences with polynomial and rational coefficients has been done by Petkovsekˇ wx 22, 23 . But, in the available open literature, there are no expressions in terms of coefficients for the complete solution of a linear difference equation with varying coefficients when the order is 3 or more, except for cases in which the coefficients have some special properties. This paper presents explicit solutions in terms of coefficients of linear difference equations with variable coefficients, for both the unbounded order case and the Nth-order case.

2. LINEAR DIFFERENCE EQUATION OF UNBOUNDED ORDER

Consider the linear difference equation

ky1 ykks Ýby,iiqx k, kG11Ž. is1 of unbounded order, with integral index k, variable complex coefficients bk,1,...,bk,ky1, and complex forcing term xk. The solution of this equation, which is an expression for yk , k G 1 in terms of only coefficients and forcing terms, is given by the proposition that follows.

PROPOSITION 1. The solution of difference equation Ž.1 is gi¨en by

ky1 ykks Ýcx,iiqx k, kG1,Ž. 2 is1 LINEAR DIFFERENCE EQUATIONS 81 where

kyi j

c b bbmy 1 m 3 k,iks ,ikq ÝÝ,kylk1 ŁyÝn1lnn,kyÝ1lnŽ. m2s s js2Ž.l1,...,lj s l1,...,ljG1 l12ql qиии qljskyi

for i s 1,...,ky1, k G 2. Proof. From the difference equationŽ. 1 , it is clear that its solution is of the formŽ. 2 . Substituting Ž. 2 in Ž. 1 , we obtain, for i s 1,...,ky1, k G 2,

ky1 ky1 iy1 ykks ÝÝÝbx,iiq bcx k,ii,rrqx k is1 is2rs1 ky1 ky2ky1 sÝÝÝbxk,iiq bcx k,ii,rrqx k is1 rs1isrq1 ky2 ky1

sÝÝbk,ikq bc,rr,ii xqbx k,ky1ky1qxk.4Ž. is1 rsiq1

ComparingŽ. 4 with Ž. 2 , we get

ky1 ck,iky b ,iks Ýbc,rr,i for i s 1,...,ky1, k G 2.Ž. 5 rsiq1

If we can show that ck, i given byŽ. 3 satisfies Ž. 5 , then the proposition will be proved. Now usingŽ. 3 , the right-hand side of Ž. 5 can be expressed as

ky1 ky1

ÝÝbck,rr,iksbb,iq1iq1, ikq bb,rr,i rsiq1 rsiq2 kyiy1ky1 bb qÝÝ Ý k,rr,ryl1 js2 rsiqj Ž.l1,...,lj l1,...,ljG1 l12ql qиии qljsryi

= bmy1 m .6Ž. Ł ryÝns1lnn,ryÝs1ln ms2 82 RANJAN K. MALLIK

Substituting l1 s k y r and replacing the index j by j y 1 inŽ. 6 , we get

ky1 Ýbck,rr,i rsiq1 kyiy2 bb bb sk,kyŽkyiy1.kyŽkyiy1.,kyŽkyiy1.y1qÝk,kylk111yl,kylyŽkyl 1yi. l1s1

kyikyiyŽ.jy1 j

bbmy1 m qÝÝ Ýk,kylk1 ŁyÝn1lnn,kyÝ1ln m2s s js3l1s1 Ž.l2,...,lj s l2,...,ljG1 l2qиии qljskyiyl1

kyi j

bbmy 1 m s ÝÝk,kylk1 ŁyÝn1lnn,kyÝ1ln m2s s js2Ž.l1,...,lj s l1,...,ljG1 l12ql qиии qljskyi

sck,ikyb,i Ž.7 for i s 1,...,ky1, k G 2, fromŽ. 3 . Therefore we conclude that the expression for ck, i given byŽ. 3 obeys Ž. 5 , which proves the proposition.

3. LINEAR DIFFERENCE EQUATION OF ORDER N

We consider the linear difference equation

ykqNks Ýay,jkqNyjkqxqN, kG18Ž. js1 of order NNŽ.G2 with variable complex coefficients ak, j, j s 1,...,N, complex forcing term xkqN , and complex initial values y1,..., yN . The solution of this equation, which is an expression for ykqN , k G 1 in terms of only coefficients, initial values, and forcing terms, is given by the following proposition.

PROPOSITION 2. The solution of difference equation Ž.8 with initial ¨alues y1,..., yisgiN ¨en by

N 0

ykqNks ÝÝdy,jNq1yjkq dx,jNq1yjkqxqN, kG1,Ž. 9a js1 js2yk LINEAR DIFFERENCE EQUATIONS 83 where

kqjy1 r d a m 9b k,jks ÝÝŁqlmnyÝ1lnm,lŽ. m1 s rs1 Ž.l1,...,lr s 1Fl1,...,lrFN lrGj l12ql qиии qlrskqjy1

for j s 2 y k,...,N, kG1.

Proof. Difference equationŽ. 8 with initial values y1,..., yN can be treated as a special case ofŽ. 1 in which

ykks x for 1 F k F N,

bk, i s0 for i s 1,...,ky1, 1 F k F N, Ž.10 bk, i s 0 for i s 1,...,kyNy1, k G N q 2,

bk, iksayN,kyifor i s k y N,...,ky1, k G N q 1.

Now the solution ofŽ. 8 with initial values y1,..., yN is given by Proposi- tion 1, with the expression for ck, i given byŽ. 3 having the additional constraints on bk, i inŽ. 10 . The solution can therefore be expressed by usingŽ. 10 as

Nky1 ykksÝÝcy,iiq cx k,iiqx k, kGNq1, or is1 isNq1 N

ykqNksÝcyjqN,Nq1yj Ž.11 js1 0

qÝ ckqN, Nq1yjNxq1yjkqxqN, kG1. js2yk

SinceŽ. 3 can be rewritten as

kyi j c bmm 12 k,iksÝÝ ŁqlmnyÝ1lnn,kyÝ1ln Ž. m1 s s js1Ž.l1,...,lj s l1,...,ljG1 l12ql qиии qljskyi 84 RANJAN K. MALLIK for i s 1,...,ky1, k G 2, it is clear fromŽ. 10 that for each term of the summation on the right-hand side ofŽ. 12 to be nontrivial, the conditions j

1 F l1,...,ljjnjFN, kql y Ý l siql GNq1 ns1 for j s 1,...,kyi Ž.13 need to be satisfied. Therefore we get

kyi j c bmm 14 k,iksÝÝ ŁqlmnyÝ1lnn,kyÝ1ln Ž. m1 s s js1Ž.l1,...,lj s 1Fl1,...,ljFN ljGNq1yi l12ql qиии qljskyi for i s 1,...,ky1, k G N q 1. Substituting

dk,jks c qN,Nq1yjkand a ,jks b qN,kqNyj Ž.15 inŽ. 11 and Ž. 14 respectively, and combining the two results, we obtain the proposition.

Note that difference equationŽ. 8 with initial values y1,..., yN can be expressed in vector form as иии aaak,1 k,2 k,3 aak,Ny1 k,N ykqN y 100иии 00 kqNy1 010иии 00 .s...... y ... . . kq1 000иии 10 y kqNy1 xkqN ykN2 0 =qy ,k1, 16 . q. G Ž. . . yk 0 where the N = N matrix on the right-hand side ofŽ. 16 is the companion matrix for index k. Defining TT ykkJwxwxyqNy1,ykqNy2,..., ykkk, x J x qN,0,...,0 ,Ž. 17a иии aaak,1 k,2 k,3 aak,Ny1 k,N 100иии 00 A010иии 00 Ž.17b kJ...... 000иии 10 LINEAR DIFFERENCE EQUATIONS 85 for k G 1, we can rewriteŽ. 16 as

ykq1s Aykkqx k, kG1,Ž. 18

T where y1 s wxyNN, y y11,..., y is the initial value vector. The solution of Ž.18 with initial value vector y1 can then be given in matrix form by

ky1 иии иии ykq1s AAkky111AyqÝAAkky1 Axlq1lkqx, kG1.Ž. 19 ls1

3.1. Product of Companion Matrices

Consider the homogeneous case of difference equationŽ. 8 with initial values y1,..., yNk, in which x qNs 0 for k G 1. The solution of this homogeneous equation can be expressed, usingŽ. 9a , as

ykqNks Ýdy,jNq1yj, kG1,Ž. 20 js1 where dk, j, j s 1,...,N are given byŽ. 9b . Extending the definition of dk, j to k syŽ.Ny1 , . . . , 0, we can express the solutionŽ. 20 as

ykqNks Ýdy,jNq1yj, kGyŽ.Ny1, Ž. 21 js1 where dk,1,...,dk,N are given byŽ. 9b for k G 1, and by

dd0,1 0,2 иии d0, N иии ddy1,1 y1,2 dy1, N .. .sIN,22Ž. .. . иии ddyŽ Ny1.,1 yŽNy1.,2 dyŽNy1.,N

where I N is the N = N identity matrix, for k syŽ.Ny1 , . . . , 0. How- ever,Ž. 19 implies that the matrix form of the homogeneous solution is

иии ykq1s AAkky111Ay, kG1,Ž. 23 where ykkand A are defined inŽ. 17 . 86 RANJAN K. MALLIK

ComparingŽ.Ž. 17 , 21 , and Ž. 23 , we find that the product of companion matrices can be expressed as

иии AAkky11A иии ddk,1 k,2 ddk,Ny1 k,N иии ddky1,1 ky1,2 ddky1, Ny1 ky1, N s .. . ., .. . . иии ddkyNq1,1 kyNq1,2 ddkyNq1, Ny1 kyNq1, N

kG1,Ž. 24

Ž. Ž. where A kkk G 1 is defined by 17b , and d yiq1, j, the entry in the ith row Ž. иии and the jth column 1 F i, j F N of the product AAkky11A , which is obtained fromŽ. 9b and Ž. 22 , is given by

kyiqj r d a m kyiq1, jks ÝÝŁyiq1qlmnyÝ1lnm,l m1 s rs1 Ž.l1,...,lr s 1Fl1,...,lrFN lrGj l12ql qиии qlrskyiqj if i F minŽ.Ž.k, N , 25a s1ifjsi,i)k,k-N,Ž. 25b s0ifj/i,i)k,k-N.Ž. 25c

It can easily be shown that the characteristic equation of A k is given by

N N nNyj detŽ.Ž.A kNy ␭I sy1 ␭yÝak,j␭ ,26Ž. ½5j1 s which implies

Nq1 detŽ.A kksy Ž1 . a,N.2Ž.7

Based on this result, we have the following proposition for linearly independent solutions of the homogeneous version of difference equationŽ. 8 , that is, the equation

ykqNks Ýay,jkqNyj, kG1.Ž. 28 js1 LINEAR DIFFERENCE EQUATIONS 87

PROPOSITION 3. Difference equation Ž.28 with ak, N / 0, k G 1, has N linearly independent solutions expressed as ykqNks d ,j, j s 1, . . . , N for kGyŽ.Ny1,where dk,1,...,dk,N are gi¨en by Ž.Ž.9b and 22 . Ž. Proof. If is clear from 21 that ykqNks d ,j, j s 1,...,N, are N solutions of difference equationŽ. 28 for k Gy ŽNy1 . . The Casoratian of Ä4 the N sequences dk, jkGyŽNy1., j s 1,...,N is given by the determinant

dk ,1 иии dk,N иии dkq1,1 dkq2, N .. .. иии dkqNy1,1 dkqNy1, N иии dkqNy1,1 dkqNy1, N

N иии ?@dkqNy2,1 dkqNy2, N syŽ.1 2 .. ..

dk ,1 иии dk,N

?@N ¡Ž.y12 for k syŽ.Ny1 kqNy1 s~N ?@2 Nq1 Ž.y1 Ł Ž.y1 ai,N for k GyŽ.Ny2 ¢ is1 /0,Ž. 29 usingŽ.Ž.Ž. 22 , 24 , 27 , and the fact that ai, N / 0, i s 1,...,kqNy1, Ž. Ä4 kGy Ny2 . Therefore dk, jkGyŽNy1., j s 1,...,N, are linearly independent sequences, which implies that ykqNks d ,j, j s 1,...,N, are N linearly independent solutions of the difference equation for k Gy Ž.Ny1.

3.1.1. Power of the Companion Matrix for the Constant Coefficient Case Consider the case of the homogeneous Nth-order difference equation

Ž.28 with initial values y1,..., yNkin which a ,jjs a for all k G 1, j s 1,...,N, that is, the equation

ykqNjks ÝayqNyj, kG130Ž. js1 with initial values y1,..., yN . We can rewriteŽ. 30 as

ykq1s Ayk, k G 1,Ž. 31 88 RANJAN K. MALLIK

where ykkis defined inŽ. 17a , A is defined in Ž. 17b with a ,jjreplaced by a , and y1 is the initial value vector. FromŽ. 23 , the matrix form of the solution toŽ. 31 with initial value vector y1 is given by

k ykq11s A y , k G 1.Ž. 32 Thus the kth power of the N = N companion matrix A gives the solutions for ykq1,..., ykqN of the Nth-order homogeneous linear difference equa- tionŽ. 30 with constant coefficients a1,...,aN and initial values y1,..., yN. Using the result inŽ. 24 and Ž. 25 for the product of companion matrices, we obtain

иии ddk,1 k,2 ddk,Ny1 k,N ddиии dd k ky1,1 ky1,2 ky1, Ny1 ky1, N A s .. . ., .. . . иии ddkyNq1,1 kyNq1,2 ddkyNq1, Ny1 kyNq1, N

kG1,Ž. 33 k where dkyiq1, j, the entry in the ith row and the jth column of A ,is expressed as

dkyiq1, j

kyiqj r a if i min k, N , 34a s ÝÝŁlm F Ž.Ž. m1 rs1 Ž.l1,...,lr s 1Fl1,...,lrFN lrGj l12ql qиии qlrskyiqj s1ifjsi,i)k,k-N,Ž. 34b s0ifj/i,i)k,k-N.Ž. 34c

Our aim is to obtain an alternative expression for dkyiq1, j when i F minŽ.k, N in terms of powers of the coefficients a1,...,aN. NowŽ. 34a can be rewritten as

N ¡kyiqj ¦ dki1, jlls~¥Ž.Ž.aиии aa35 yq ÝÝ Ý 1ry1¨ jr1 Ž. ¨ss l1,...,lry1 ¢§1l,...,l N F1 ry1F иии l12ql q qlry1skyiqjy¨ for i minŽ.k, N . The sum of a иии a over all l ,...,l satisfying F ll1 ry1 1ry1 иии 1Fl1 ,...,lry112FN, l ql q qlry1s k y i q j y ¨ LINEAR DIFFERENCE EQUATIONS 89 where r s 1,...,kyiqj is the same as the sum of

t12q t q иии qtN tt1 N a1 иии aN ž/t1,...,tN over all t1,...,tN satisfying

t1 ,...,tN G0, t12q 2t q иии qNtNs k y i q j y ¨ .

ThereforeŽ. 35 implies, for i F minŽ.k, N ,

dkyiq1, j

NN¡ ¦ t12qtqиии qtN aatm. sÝÝ~¥t,...,t Łm¨ j Ž.t,...,t ž/1 N ms1 ¨s 1N ¢§t,...,t 0 1 NG t12q2t qиии qNtNskyiqjy¨ Ž.36

Ž. Replacing t¨¨by t y 1 in 36 , we get

иии tjjqtq1qqtN dkyiq1, j s Ý t12t иии tN Ž.t1,...,tN q q q t1,...,tNG0 t12q2t qиии qNtNskyiqj

N t12qtqиии qtN = atm Ž.37 t,...,t Łm ž/1 N ms1 for i s 1, . . . , minŽ.k, N , j s 1,...,N, kG1. This is an alternative way of expressingŽ. 34a in terms of the coefficients a1,...,aN of the companion matrix A. The result is consistent with the expression for the combinatorial power of the companion matrix presented by Chen and Louckw 24, Theo- rem 3.1x , obtained by using the concept of a digraph to represent a matrix.

4. CONCLUSION

The explicit solutions of the linear difference equations presented here utilize the combinatorial properties of the indices of the coefficients. The 90 RANJAN K. MALLIK solution of the difference equation of unbounded order results in the solution of the Nth-order equation, which, in turn, provides expressions for the product of companion matrices and the positive integral powers of a companion matrix.

ACKNOWLEDGMENTS

The author thanks his colleague Dr. S. Ponnusamy of the Department of Mathematics, Indian Institute of Technology, Guwahati, for his valuable suggestions. He also expresses his gratitude to the anonymous referee for hisrher constructive comments.

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