Analysis of linear difference equations with variable coefficients

In this paper, we studies the explicit solution of a linear difference equation of unbounded order with variable coefficients is presented. The solutions of nonhomogeneous 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. 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.

Introduction

Asymptotics of solutions of linear recurrences with coefficients having series representations have also been studied by many authors [1-7] . Further work on convergence properties of linear recurrence sequences has been presented by Kooman and Tijdeman [8] . 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 index [5] . A method of solving linear matrix difference equations with constant coefficients by using operator identities has been presented by Verde-Star [9] . Work on the existence and construction of closed-form solutions of linear recurrences with polynomial and rational coefficients has been done by Petkovsek [10-11] . 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.

Formulations

Consider the linear difference equation (1) of unbounded order, with integral index k, variable complex coefficients b k, 1 ,..., b k, k-1 , and complex forcing term x k . The solution of this equation, which is an expression for y k , k1 in terms of only coefficients and forcing terms, is given by the proposition that follows. Proposition 1. The solution of difference equation ( 1) is given by (2) http://www.mathsjournal.com Where (3) for i=1,..., k-1, k2.

Proof. From the difference equation ( 1), it is clear that its solution is of the form (2). Substituting ( 2) in (1), we obtain, for i =1,..., k-1, k2, ( 4) Comparing ( 4) with ( 2), we get (5) If we can show that c k, i given by ( 3) satisfies ( 5), then the proposition will be proved. Now using (3), the right-hand side of ( 5) can be expressed as (6) Substituting l 1 =k-r and replacing the index j by j-1 in (6), we get (7) for i=1,..., k-1, k2, from (3). Therefore we conclude that the expression for c k,i given by ( 3) obeys ( 5), which proves the proposition.

Linear difference equation of order N

We know that the linear difference equation (8) of order N (N2) with variable complex coefficients a k, j , j=1,..., N, complex forcing term x k+N , and complex initial values y 1 ,..., y N . The solution of this equation, which is an expression for y k+N , k1 in terms of only coefficients, initial values, and forcing terms, is given by the following proposition. Proposition 2. The solution of difference equation ( 8) with initial values y 1 ,..., y N is given by (9a) Where (9b) for j=2-k,..., N, k1.

Proof. Difference equation (8). With initial values y 1 ,..., y N can be treated as a special case of (1). In which (10) Now the solution of ( 8) with initial values y 1 ,..., y N is given by Proposition 1, with the expression for c k, i given by (3) having the additional constraints on b k,i in (10). The solution can therefore be expressed by using (10) as (11) Since (3) can be rewritten as (12) for i=1,..., k-1, k2, it is clear from (10) that for each term of the summation on the right-hand side of (12) to be nontrivial, the conditions In ( 11) and ( 14) respectively, and combining the two results, we obtain the proposition. Note that difference equation ( 8) with initial values y 1 ,..., y N can be expressed in vector form as (16) where the N=N matrix on the right-hand side of ( 16) is the companion matrix for index k. Defining (17a) (17b) for k1, we can rewrite (16) as (18) where y 1 = [y N , y N-1 ,..., y 1 ] T is the initial value vector. The solution of (18) with initial value vector y 1 can then be given in matrix form by ( 19)

Product of Companion Matrices

Consider the homogeneous case of difference equation ( 8) with initial values y 1 ,..., y N , in which x k+N =0 for k1. The solution of this homogeneous equation can be expressed, using (9a), as 22), ( 24), ( 27), and the fact that a i, N 0, i=1,..., k+N-1, k-(N-2). Therefore {d k, j } k-(N-1) , j-1,..., N, are linearly independent sequences, which implies that y k+N =d k, j , j=1,..., N, are N linearly independent solutions of the difference equation for ky -(N-1).

Power of the companion matrix for the constant coefficient case

Consider the case of the homogeneous Nth-order difference equation ( 28) with initial values y 1 ,..., y N in which a k, j =a j for all k1, j= 1,..., N, that is, the equation (

With initial values y 1 ,..., y N . We can rewrite (30). As (31) where y k is defined in (17a), A is defined in (17b) with a k, j replaced by a j , and y 1 is the initial value vector. From (23), the matrix form of the solution to (31) with initial value vector y 1 is given by (32) Thus the kth power of the N×N companion matrix A gives the solutions for y k+1 ,..., y k+N of the Nth-order homogeneous linear difference equation (30) with constant coefficients a 1 ,..., a N and initial values y 1 ,..., y N . Using the result in (24) and (25) for the product of companion matrices, we obtain (33) where d k-i+1, j , the entry in the ith row and the jth column of A k , is expressed as

Our aim is to obtain an alternative expression for d k-i+1, j when i  min k, N. in terms of powers of the coefficients a 1 ,..., a N . Now (34a). Can be rewritten as (35) for imin (k, N). The sum of a l1 .... a lr-1 over all l 1 ,..., l r-1 satisfying where r=1,..., k-i+j is the same as the sum of Over all t 1 ,..., t N satisfying Therefore (35) implies, for imin (k, N), (36) Replacing t v by t v -1 in (36), we get (37) for i=1,..., min (k, N), j=1,..., N, k1. This is an alternative way of expressing (34a) in terms of the coefficients a 1 ,..., a N of the companion matrix A.

Conclusion

The explicit solutions of the linear difference equations presented here utilize the combinatorial properties of the indices of the coefficients. The 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.

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