Associated Legendre function

In mathematics and physics, an associated Legendre function P_l^m is related to a Legendre polynomial P_l by the following equation

$P_{\ell }^{m}(x)=(1-x^{2})^{m/2}{\frac {d^{m}P_{\ell }(x)}{dx^{m}}},\qquad 0\leq m\leq \ell .$

For even m the associated Legendre function is a polynomial, for odd m the function contains the factor (1-x ² )^½ and hence is not a polynomial.

The associated Legendre functions are important in quantum mechanics and potential theory.

Differential equation

Define

$\Pi _{\ell }^{m}(x)\equiv {\frac {d^{m}P_{\ell }(x)}{dx^{m}}},$

where P_l(x) is a Legendre polynomial. Differentiating the Legendre differential equation:

$(1-x^{2}){\frac {d^{2}\Pi _{\ell }^{0}(x)}{dx^{2}}}-2x{\frac {d\Pi _{\ell }^{0}(x)}{dx}}+\ell (\ell +1)\Pi _{\ell }^{0}(x)=0,$

m times gives an equation for Π^m_l

$(1-x^{2}){\frac {d^{2}\Pi _{\ell }^{m}(x)}{dx^{2}}}-2(m+1)x{\frac {d\Pi _{\ell }^{m}(x)}{dx}}+\left[\ell (\ell +1)-m(m+1)\right]\Pi _{\ell }^{m}(x)=0.$

After substitution of

$\Pi _{\ell }^{m}(x)=(1-x^{2})^{-m/2}P_{\ell }^{m}(x),$

and after multiplying through with $(1-x^{2})^{m/2}$, we find the associated Legendre differential equation:

$(1-x^{2}){\frac {d^{2}P_{\ell }^{m}(x)}{dx^{2}}}-2x{\frac {dP_{\ell }^{m}(x)}{dx}}+\left[\ell (\ell +1)-{\frac {m^{2}}{1-x^{2}}}\right]P_{\ell }^{m}(x)=0.$

In physical applications it is usually the case that x = cosθ, then the associated Legendre differential equation takes the form

${\frac {1}{\sin \theta }}{\frac {d}{d\theta }}\sin \theta {\frac {d}{d\theta }}P_{\ell }^{m}+\left[\ell (\ell +1)-{\frac {m^{2}}{\sin ^{2}\theta }}\right]P_{\ell }^{m}=0.$

Extension to negative m

By the Rodrigues formula, one obtains

$P_{\ell }^{m}(x)={\frac {1}{2^{\ell }\ell !}}(1-x^{2})^{m/2}\ {\frac {d^{\ell +m}}{dx^{\ell +m}}}(x^{2}-1)^{\ell }.$

This equation allows extension of the range of m to: $-m\leq \ell \leq m$.

Since the associated Legendre equation is invariant under the substitution m → −m, the equations for P_l ^±m, resulting from this expression, are proportional.

To obtain the proportionality constant we consider

$(1-x^{2})^{-m/2}{\frac {d^{\ell -m}}{dx^{\ell -m}}}(x^{2}-1)^{\ell }=c_{lm}(1-x^{2})^{m/2}{\frac {d^{\ell +m}}{dx^{\ell +m}}}(x^{2}-1)^{\ell },\qquad 0\leq m\leq \ell ,$

and we bring the factor (1−x²)^−m/2 to the other side. Equate the coefficient of the highest power of x on the left and right hand side of

${\frac {d^{\ell -m}}{dx^{\ell -m}}}(x^{2}-1)^{\ell }=c_{lm}(1-x^{2})^{m}{\frac {d^{\ell +m}}{dx^{\ell +m}}}(x^{2}-1)^{\ell },\qquad 0\leq m\leq \ell ,$

and it follows that the proportionality constant is

$c_{lm}=(-1)^{m}{\frac {(\ell -m)!}{(\ell +m)!}},\qquad 0\leq m\leq \ell ,$

so that the associated Legendre functions of same |m| are related to each other by

$P_{\ell }^{-|m|}(x)=(-1)^{m}{\frac {(\ell -|m|)!}{(\ell +|m|)!}}P_{\ell }^{|m|}(x).$

Note that the phase factor (−1)^m arising in this expression is not due to some arbitrary phase convention, but arises from expansion of (1−x²)^m.

Orthogonality relations

Important integral relations are

$\int _{-1}^{1}P_{\ell }^{m}(x)P_{\ell '}^{m}(x)dx={\frac {2\delta _{\ell \ell '}(\ell +m)!}{(2\ell +1)(\ell -m)!}}$     [Proof]

$\int _{-1}^{1}P_{\ell }^{m}(x)P_{\ell }^{n}(x){\frac {dx}{1-x^{2}}}={\frac {\delta _{mn}(\ell +m)!}{m(\ell -m)!}}$

Recurrence relations

The functions satisfy the following difference equations, which are taken from Edmonds.

$(\ell -m+1)P_{\ell +1}^{m}(x)-(2\ell +1)xP_{\ell }^{m}(x)+(\ell +m)P_{\ell -1}^{m}(x)=0$

$xP_{\ell }^{m}(x)-(\ell -m+1)(1-x^{2})^{1/2}P_{\ell }^{m-1}(x)-P_{\ell -1}^{m}(x)=0$

$P_{\ell +1}^{m}(x)-xP_{\ell }^{m}(x)-(\ell +m)(1-x^{2})^{1/2}P_{\ell }^{m-1}(x)=0$

$(\ell -m+1)P_{\ell +1}^{m}(x)+(1-x^{2})^{1/2}P_{\ell }^{m+1}(x)-(\ell +m+1)xP_{\ell }^{m}(x)=0$

$(1-x^{2})^{1/2}P_{\ell }^{m+1}(x)-2mxP_{\ell }^{m}(x)+(\ell +m)(\ell -m+1)(1-x^{2})^{1/2}P_{\ell }^{m-1}(x)=0$

$(1-x^{2}){\frac {dP_{\ell }^{m}}{dx}}(x)=(\ell +1)xP_{\ell }^{m}(x)-(\ell -m+1)P_{\ell +1}^{m}(x)$

$=(\ell +m)P_{\ell -1}^{m}(x)-\ell xP_{\ell }^{m}(x)$