A general theory of linear finite-difference equations with a few applications

Abstract

Recently a new formalism has been developed giving the solutions in explicit form of multiterm linear homogeneous recursion relations with nonconstant coefficients. The basic idea was to relate this problem to the one of partitioning an interval into parts of given lengths. This idea is extensively used here to obtain the solutions of linear inhomogeneous difference equations. The resulting method has the advantage of being general: It does not rely on any special device and does not assume any special form for the recursion relation. Applications of these techniques to physical problems are presented elsewhere. Here we show how the method works for first and second order equations. The three-term Legendre polynomial recursion relation with an arbitrary inhomogeneous term is discussed in detail. The Legendre polynomials are then viewed as special cases of combinatorics functions, P (j,m;z), based on the partitions of an interval (j,m), 0⩽j⩽m, into parts of lengths 1 and 2. P (j,m;z) reduces to the Legendre polynomial, Pm(z), for j=0. It is also interesting to note that P (j,m;z) are not orthogonal in general. However, a new set of orthogonal functions can be constructed as the solutions of the inhomogeneous Legendre recursion relation with a suitable choice of the inhomogeneous term.