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].

See also the references to Orthogonal polynomials.

References

[1]	A.M. Legendre, Mém. Math. Phys. présentés à l'Acad. Sci. par divers savants , 10 (1785) pp. 411–434
[2]	E.W. Hobson, "The theory of spherical and ellipsoidal harmonics" , Cambridge Univ. Press (1931)

Comments

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.