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Legendre polynomials are orthogonal polynomials with applications in mathematics and physics, defined on the interval [-1,1]. They are constructed using Bonnet’s recursion formula and are solutions to Legendre's differential equation. These polynomials are essential in solving Laplace's equation in spherical coordinates and contribute to the development of spherical harmonics, playing a significant role in problems with rotational symmetry.

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## Introduction

### Definition

Legendre polynomials are a sequence of orthogonal polynomials named after the French mathematician Adrien-Marie Legendre

### Properties

Orthogonality

Legendre polynomials are orthogonal, meaning the integral of the product of any two distinct Legendre polynomials over the interval [-1,1] is zero

Completeness

Legendre polynomials form a complete set of polynomials, which can represent any function that is square-integrable on the interval [-1,1]

### Construction

Each subsequent Legendre polynomial is constructed to maintain orthogonality with all previous ones

## Recursive Relations

### Bonnet's Recursion Formula

Bonnet's recursion formula expresses P_{n+1}(x) in terms of P_n(x) and P_{n-1}(x), facilitating the computation of Legendre polynomials

### Efficient Computation

The recursive generation of Legendre polynomials allows for efficient computation using Bonnet's recursion formula

## Generating Functions and Differential Equations

### Generating Function

Legendre polynomials can be defined through their generating function, a power series whose coefficients are the polynomials themselves

### Legendre's Differential Equation

Legendre polynomials satisfy Legendre's differential equation, a second-order differential equation with singular points at x = ±1

## Applications and Symmetry Principles

### Applications in Physics

Legendre polynomials have significant applications in physics, particularly in solving Laplace's equation in spherical coordinates

### Symmetry and Group Theory

The properties of Legendre polynomials, such as the addition theorem, are deeply connected to symmetry and group theory, offering insights into their geometric and physical interpretations

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