chapter  10
51 Pages

Chapter 10

Miscellaneous Properties Solutions of Higher Order Linear Difference Equations

Disc:·onjugacy property of a linear homogeneous differential equation allows the possibility of interpolation by its solutions. While this property has been investigated thoroughly for the differential equations, its disc::rete analogs arc not fully developed. In this chapter we shall introduce disconjugacy, right disconjugacy, left disc::onjugacy, right disfocality, eventual disconjuga<:y and eventual right disfocality for the linear homogeneous difference equations, and for each such concept state several results which provide necessary and sufficient conditions. This includes Polya's factorization, and interrelationship between D Markov, D Fekete and D Descartes systems. This is followed by the statement of the discrete analog of a result due to Elias, which bounds the number of certain types of zeros of solutions of linear homogeneous difference equations on a discrete interval. A classification of solutions of these equations based on their behavior in a neighborhood of infinity is also included. Then, we provide explicit representations of polynomials passing through the given boundary conditions which also include (1.5.9) (1.5.14). Such polynomials are called discrete interpolating polynomials. This is followed by the explicit representations of Green's functions for several higher order boundary value problems. For these Green's functions we state several equalities and inequalities whose continuous analogs have proved to be very useful in providing disconjugacy tests and distance between consecutive zeros of the solutions of higher order differential equations. The explicit forms of interpolating polynomials and those of Green's functions help in establishing maximum principles for functions satisfying higher order inequalities. We state some such maximum principles. Finally, in this chapter we have included several results which provide error estimates in polynomial interpolation. Some of these results will be used in the next chapter to study higher order boundary value problems. To limit the size of this volume the proofs of the theorems in this chapter have not been given. However, we observe that almost

all the proofs require special devices and no unified approach seems to be available.