Closed-form solution of a general three-term recurrence relation

Ivan Gonoskov

doi:10.1186/1687-1847-2014-196

Abstract

We present a closed-form solution for n th term of a general three-term recurrence relation with arbitrary given n-dependent coefficients. The derivation and corresponding proof are based on two approaches, which we develop and describe in detail. First, the recursive-sum theory, which gives the exact solution in a compact finite form using a recursive indexing. Second, the discrete dimensional-convolution procedure, which transforms the solution to the non-recursive expression of n, including a finite number of elementary operations and functions.

Highlights

1 Introduction A general three-term recurrence relation is usually defined by the following expression: Wn+ = AnWn + BnWn, ( )
In that way we develop two approaches for the expressions with a finite number of terms
Possible applications of the developed approaches, namely the R-sum theory and the discrete dimensional-convolution procedure, are not limited by the considered statement. They could be used for solving other recursive problems, in particular many-term recurrence relations

Summary

Introduction

Where n ≥ (n ∈ N), Wn is unknown function of n, An and Bn are arbitrary given functions of n, and W = C , W = C are initial conditions (we assume not to have the trivial case when C = C = and Wn ≡ ) This well-known relation has a large number of applications and plays an important role in many areas of mathematics and physics. The recurrence relation Eq ( ) corresponds to the finite difference equation for the general second order differential equation with unknown function f (x) and arbitrary given U(x), namely: f – U(x) · f = It is widely used for the analytical and numerical analysis (and approximations) in corresponding physical and mathematical applications; see for example [ ]. Three-term recurrence relations appear naturally when one uses the Frobenius method for solving some linear differential equations and studying some special functions; see [ ]

Objectives

Conclusion