# Legendre polynomials

spherical polynomials

Polynomials orthogonal on the interval $[ - 1 , 1 ]$ with unit weight $\phi ( x) = 1$. The standardized Legendre polynomials are defined by the Rodrigues formula

$$P _ {n} ( x) = \ \frac{1}{n ! 2 ^ {n} } \frac{d ^ {n} }{d x ^ {n} } ( x ^ {2} - 1 ) ^ {n} ,\ \ n = 0 , 1 \dots$$

and have the representation

$$P _ {n} ( x) = \ \frac{1}{2 ^ {n} } \sum _ { k= } 0 ^ { [ } n/2] \frac{( - 1 ) ^ {k} ( 2 n - 2 k ) ! }{k ! ( n - k ) ! ( n - 2 k ) ! } x ^ {n-} 2k .$$

The formulas most commonly used are:

$$( n + 1 ) P _ {n+} 1 ( x) = \ ( 2 n + 1 ) x P _ {n} ( x) - n P _ {n-} 1 ( x) ,$$

$$P _ {n} ( - x ) = ( - 1 ) ^ {n} P _ {n} ( x) ; \ P _ {n} ( 1) = 1 ,\ P _ {n} ( - 1 ) = ( - 1 ) ^ {n} ,$$

$$( 1 - x ^ {2} ) P _ {n} ^ { \prime } ( x) = n P _ {n-} 1 ( x) - x n P _ {n} ( x) ,$$

$$P _ {n+} 1 ^ { \prime } ( x) - P _ {n-} 1 ^ { \prime } ( x) = ( 2 n + 1 ) P _ {n} ( x) .$$

The Legendre polynomials can be defined as the coefficients in the expansion of the generating function

$$\frac{1}{\sqrt {1 - 2 x t + t ^ {2} }} = \ \sum _ { n= } 0 ^ \infty P _ {n} ( x) t ^ {n} ,$$

where the series on the right-hand side converges for $x \in [ - 1 , 1 ]$.

The first few standardized Legendre polynomials have the form

$$P _ {0} ( x) = 1 ,\ P _ {1} ( x) = x ,\ \ P _ {2} ( x) = \frac{3 x ^ {2} - 1 }{2} ,$$

$$P _ {3} ( x) = \frac{5 x ^ {3} - 3 x }{2} ,\ P _ {4} ( x) = \frac{35 x ^ {4} - 30 x ^ {2} + 3 }{8} ,$$

$$P _ {5} ( x) = \frac{63 x ^ {5} - 70 x ^ {3} + 15 x }{8} ,$$

$$P _ {6} ( x) = \frac{231 x ^ {6} - 315 x ^ {4} + 105 x ^ {2} - 5 }{16} .$$

The Legendre polynomial of order $n$ satisfies the differential equation (Legendre equation)

$$( 1 - x ^ {2} ) \frac{d ^ {2} y }{d x ^ {2} } - 2 x \frac{dy}{dx} + n ( n + 1 ) y = 0 ,$$

which occurs in the solution of the Laplace equation in spherical coordinates by the method of separation of variables. The orthogonal Legendre polynomials have the form

$$\widehat{P} _ {n} ( x) = \ \sqrt { \frac{2 n + 1 }{2} } P _ {n} ( x) ,\ n = 0 , 1 \dots$$

and satisfy the uniform and weighted estimates

$$| \widehat{P} _ {n} ( x) | \leq \ \sqrt { \frac{2 n + 1 }{2} } ,\ \ x \in [ - 1 , 1 ] ,$$

$$( 1 - x ^ {2} ) ^ {1/4} | \widehat{P} _ {n} ( x) | \leq \sqrt { \frac{2 n + 1 }{\pi n } } ,\ x \in [ - 1 , 1 ] .$$

Fourier series in the Legendre polynomials inside the interval $( - 1 , 1 )$ are analogous to trigonometric Fourier series (cf. also Fourier series in orthogonal polynomials); there is a theorem about the equiconvergence of these two series, which implies that the Fourier–Legendre series of a function $f$ at a point $x \in ( - 1 , 1 )$ converges if and only if the trigonometric Fourier series of the function

$$F ( \theta ) = \ ( \sin \theta ) ^ {1/2} f ( \cos \theta )$$

converges at the point $\theta = { \mathop{\rm arc} \cos } x$. In a neighbourhood of the end points the situation is different, since the sequence $\{ \widehat{P} _ {n} ( \pm 1 ) \}$ increases with speed $\sqrt n$. If $f$ is continuous on $[ - 1 , 1 ]$ and satisfies a Lipschitz condition of order $\alpha > 1 / 2$, then the Fourier–Legendre series converges to $f$ uniformly on the whole interval $[ - 1 , 1 ]$. If $\alpha = 1 / 2$, then this series generally diverges at the points $x = \pm 1$.

These polynomials were introduced by A.M. Legendre [1].

Legendre polynomials belong to the families of Gegenbauer polynomials; Jacobi polynomials and classical orthogonal polynomials. They can be written as hypergeometric functions (cf. Hypergeometric function). Their group-theoretic interpretation as zonal spherical functions on the two-dimensional sphere $S ^ {2} = \mathop{\rm SO} ( 3) / \mathop{\rm SO} ( 2)$ serves as a prototype, both from the historical and the didactical point of view. A noteworthy consequence of this interpretation is the addition formula for Legendre polynomials.