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Legendre polynomials, denoted as P_n(x), are essential in scientific fields for solving boundary value problems and are characterized by their orthogonality and parity. They exhibit even or odd symmetry, are normalized at x=1, and have zeros within (-1,1). Recurrence relations facilitate their computation, and their asymptotic behavior is approximated with Bessel functions for large degrees. Shifted Legendre polynomials and Legendre rational functions extend their applicability.

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## Properties of Legendre Polynomials

### Parity of Legendre Polynomials

Legendre polynomials have a distinct parity, either even or odd, depending on their degree

### Normalization and Derivative Properties

Normalization

Legendre polynomials are normalized so that their value at x=1 is unity

Derivative Properties

The derivative of a Legendre polynomial at x=1 is given by P_n'(1) = n(n+1)/2

### Recurrence Relations and Integration Formulas

Legendre polynomials have useful recurrence relations and integration formulas that simplify their computation and reveal their recursive nature

## Asymptotic Behavior and Zeros of Legendre Polynomials

### Asymptotic Behavior

For large degrees, Legendre polynomials can be approximated using other functions to understand their behavior

### Zeros of Legendre Polynomials

The zeros of Legendre polynomials are real, distinct, and symmetric about the origin, with an interlacing property with neighboring polynomials

## Pointwise Evaluations and Shifted Legendre Polynomials

### Pointwise Evaluations

The evaluation of Legendre polynomials at specific points can be determined by their parity and normalization

### Shifted Legendre Polynomials

Shifted Legendre polynomials are adapted to be orthogonal over the interval [0, 1] and are useful in certain types of approximation and computational methods

## Legendre Rational Functions and Their Applications

### Legendre Rational Functions

Legendre rational functions are an extension of Legendre polynomials, retaining orthogonality and serving as eigenfunctions in specific problems

### Applications

Legendre rational functions provide solutions for problems in unbounded domains, demonstrating the versatility and broad applicability of Legendre polynomials

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