Legendre function

The general Legendre equation reads

${\displaystyle (1-x^{2})\,y''-2xy'+\left[\lambda (\lambda +1)-{\frac {\mu ^{2}}{1-x^{2}}}\right]\,y=0,\,}$

where the numbers λ and μ may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when λ is an integer (denoted n), and μ = 0 are the Legendre polynomials P_n; and when λ is an integer (denoted n), and μ = m is also an integer with |m| < n are the associated Legendre polynomials. All other cases of λ and μ can be discussed as one, and the solutions are written P^μ
_λ, Q^μ
_λ. If μ = 0, the superscript is omitted, and one writes just P_λ, Q_λ. However, the solution Q_λ when λ is an integer is often discussed separately as Legendre's function of the second kind, and denoted Q_n.

This is a second order linear equation with three regular singular points (at 1, −1, and ∞). Like all such equations, it can be converted into a hypergeometric differential equation by a change of variable, and its solutions can be expressed using hypergeometric functions.

Since the differential equation is linear and of second order, it has two linearly independent solutions, which can both be expressed in terms of the hypergeometric function, ${\displaystyle _{2}F_{1}}$. With ${\displaystyle \Gamma }$ being the gamma function, the first solution is

${\displaystyle P_{\lambda }^{\mu }(z)={\frac {1}{\Gamma (1-\mu )}}\left[{\frac {1+z}{1-z}}\right]^{\mu /2}\,_{2}F_{1}\left(-\lambda ,\lambda +1;1-\mu$ ;{\frac {1-z}{2}}\right),\qquad {\text{for }}\ |1-z|<2}

and the second is,

${\displaystyle Q_{\lambda }^{\mu }(z)={\frac {{\sqrt {\pi }}\ \Gamma (\lambda +\mu +1)}{2^{\lambda +1}\Gamma (\lambda +3/2)}}{\frac {e^{i\mu \pi }(z^{2}-1)^{\mu /2}}{z^{\lambda +\mu +1}}}\,_{2}F_{1}\left({\frac {\lambda +\mu +1}{2}},{\frac {\lambda +\mu +2}{2}};\lambda +{\frac {3}{2}};{\frac {1}{z^{2}}}\right),\qquad {\text{for}}\ \ |z|>1.}$

These are generally known as Legendre functions of the first and second kind of noninteger degree, with the additional qualifier 'associated' if μ is non-zero. A useful relation between the P and Q solutions is Whipple's formula.

Plot of the first five Legendre functions of the second kind.

The nonpolynomial solution for the special case of integer degree ${\displaystyle \lambda =n\in \mathbb {N} _{0}}$, and ${\displaystyle \mu =0}$, is often discussed separately. It is given by

${\displaystyle Q_{n}(x)={\frac {n!}{1\cdot 3\cdots (2n+1)}}\left(x^{-(n+1)}+{\frac {(n+1)(n+2)}{2(2n+3)}}x^{-(n+3)}+{\frac {(n+1)(n+2)(n+3)(n+4)}{2\cdot 4(2n+3)(2n+5)}}x^{-(n+5)}+\cdots \right)}$

This solution is necessarily singular when ${\displaystyle x=\pm 1}$.

The Legendre functions of the second kind can also be defined recursively via Bonnet's recursion formula

${\displaystyle Q_{n}(x)={\begin{cases}{\frac {1}{2}}\log {\frac {1+x}{1-x}}&n=0\\P_{1}(x)Q_{0}(x)-1&n=1\\{\frac {2n-1}{n}}xQ_{n-1}(x)-{\frac {n-1}{n}}Q_{n-2}(x)&n\geq 2\,.\end{cases}}}$

The nonpolynomial solution for the special case of integer degree ${\displaystyle \lambda =n\in \mathbb {N} _{0}}$, and ${\displaystyle \mu =m\in \mathbb {N} _{0}}$ is given by

${\displaystyle Q_{n}^{m}(x)=(-1)^{m}(1-x^{2})^{\frac {m}{2}}{\frac {\mathrm {d} ^{m}}{\mathrm {d} x^{m}}}Q_{n}(x)\,.}$

The Legendre functions can be written as contour integrals. For example,

${\displaystyle P_{\lambda }(z)=P_{\lambda }^{0}(z)={\frac {1}{2\pi i}}\int _{1,z}{\frac {(t^{2}-1)^{\lambda }}{2^{\lambda }(t-z)^{\lambda +1}}}dt}$

where the contour winds around the points 1 and z in the positive direction and does not wind around −1. For real x, we have

${\displaystyle P_{s}(x)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\left(x+{\sqrt {x^{2}-1}}\cos \theta \right)^{s}d\theta ={\frac {1}{\pi }}\int _{0}^{1}\left(x+{\sqrt {x^{2}-1}}(2t-1)\right)^{s}{\frac {dt}{\sqrt {t(1-t)}}},\qquad s\in \mathbb {C} }$

The real integral representation of ${\displaystyle P_{s}}$ are very useful in the study of harmonic analysis on ${\displaystyle L^{1}(G//K)}$ where ${\displaystyle G//K}$ is the double coset space of ${\displaystyle SL(2,\mathbb {R} )}$ (see Zonal spherical function). Actually the Fourier transform on ${\displaystyle L^{1}(G//K)}$ is given by

${\displaystyle L^{1}(G//K)\ni f\mapsto {\hat {f}}}$

where

${\displaystyle {\hat {f}}(s)=\int _{1}^{\infty }f(x)P_{s}(x)dx,\qquad -1\leq \Re (s)\leq 0}$

• Ferrers function

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• Courant, Richard; Hilbert, David (1953), Methods of Mathematical Physics, Volume 1, New York: Interscience Publisher, Inc.
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• Whittaker, E. T.; Watson, G. N. (1963), A Course in Modern Analysis, Cambridge University Press, ISBN 978-0-521-58807-2

• Legendre function P on the Wolfram functions site.
• Legendre function Q on the Wolfram functions site.
• Associated Legendre function P on the Wolfram functions site.
• Associated Legendre function Q on the Wolfram functions site.