# Legendre Polynomials, P_{n}(x)

synonyms: zonal harmonics

Note the parity of Legendre polynomials: P_{n}(-x)=(-1)^{n}P_{n}(x);
polynomials with even n are **even** and those with odd n are **odd**.

## Recursion Formula: (n+1)P_{n+1}(x)
= (2n+1)xP_{n}(x) - nP_{n-1}(x)

Note the recursion formula is useful to determine successive Legendre polynomials once
P_{0}=1 and P_{1}=x are known. Also, this formula confirms that even
parity applies to even n and that odd parity applies to odd n.

A further recurrence relation: .

# Associated Legendre Functions.

P_{l}^{m}(x) (l=0,1,2,...;
m=0,1,2,...l)

## Rodrigues Formula:

## Generating Function:

## Examples:

## Recursion Formula:

## Differential Equation:

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Created or up-dated 07/20/99
by R.D. Poshusta