On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

Tian-Xiao He,Peter J.-S Shiue

doi:10.1155/2009/709386

Abstract

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

Highlights

Many number and polynomial sequences can be defined, characterized, evaluated, and classified by linear recurrence relations with certain orders
The key idea of our method is to reduce the relation 1.1 of order 2 to a linear recurrence relation of order 1: an can−1 d, n ≥ 1, 1.2 for some constants c / 0 and d and initial condition a0 via geometric sequence
We show some examples of the applications of our method including the presentation of much easier proofs of some well-known formulas of the sequences of order 2

Summary

Introduction

Many number and polynomial sequences can be defined, characterized, evaluated, and classified by linear recurrence relations with certain orders. A number sequence {an} is called sequence of order 2 if it satisfies the linear recurrence relation of order 2: an pan−1 qan−2, n ≥ 2, 1.1 for some nonzero constants p and q and initial conditions a0 and a1. The key idea of our method is to reduce the relation 1.1 of order 2 to a linear recurrence relation of order 1: an can−1 d, n ≥ 1, 1.2 for some constants c / 0 and d and initial condition a0 via geometric sequence.

Main Results and Examples

Identities Constructed from Recurrence Relations

Solutions of Algebraic Equations and Differential Equations