View source for Legendre polynomials

''spherical polynomials''

Polynomials orthogonal on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l0580501.png" /> with unit weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l0580502.png" />. The standardized Legendre polynomials are defined by the [[Rodrigues formula|Rodrigues formula]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l0580503.png" /></td> </tr></table>

and have the representation

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The formulas most commonly used are:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l0580505.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l0580506.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l0580507.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l0580508.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l0580509.png" /></td> </tr></table>

where the series on the right-hand side converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805010.png" />.

The first few standardized Legendre polynomials have the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805011.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805012.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805013.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805014.png" /></td> </tr></table>

The Legendre polynomial of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805015.png" /> satisfies the differential equation (Legendre equation)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805016.png" /></td> </tr></table>

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

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and satisfy the uniform and weighted estimates

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805018.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805019.png" /></td> </tr></table>

Fourier series in the Legendre polynomials inside the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805020.png" /> are analogous to trigonometric [[Fourier series|Fourier series]] (cf. also [[Fourier series in orthogonal polynomials|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805021.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805022.png" /> converges if and only if the trigonometric Fourier series of the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805023.png" /></td> </tr></table>

converges at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805024.png" />. In a neighbourhood of the end points the situation is different, since the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805025.png" /> increases with speed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805027.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805028.png" /> and satisfies a [[Lipschitz condition|Lipschitz condition]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805029.png" />, then the Fourier–Legendre series converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805030.png" /> uniformly on the whole interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805031.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805032.png" />, then this series generally diverges at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805033.png" />.

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

See also the references to [[Orthogonal polynomials|Orthogonal polynomials]].

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



====Comments====
Legendre polynomials belong to the families of [[Gegenbauer polynomials|Gegenbauer polynomials]]; [[Jacobi polynomials|Jacobi polynomials]] and [[Classical orthogonal polynomials|classical orthogonal polynomials]]. They can be written as hypergeometric functions (cf. [[Hypergeometric function|Hypergeometric function]]). Their group-theoretic interpretation as [[Zonal spherical functions|zonal spherical functions]] on the two-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805034.png" /> 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.