Associated Legendre Functions

The associated Legendre functions, $P_l^{\,m}(x)$ , are the well-behaved solutions of the differential equation

$$\frac{d}{dx}\!\left[(1-x^{\,2})\,\frac{dP_l^{\,m}}{dx}\right]+\left[l\,(l+1)-\frac{m^{\,2}}{1-x^{\,2}}\right]P_l^{\,m} = 0,$$ (292)

for $x$ in the range $-1\leq x\leq +1$ . Here, $l$ is a non-negative integer (known as the degree), and $m$ is an integer (known as the order) lying in the range $-l\leq m\leq l$ . The functions themselves take the form

$$P_l^{\,m}(x)= \frac{(-1)^{\,l+m}}{2^{\,l}\,l!}\,(1-x^{\,2})^{\,m/2}\,\frac{d^{\,l+m}}{dx^{\,l+m}}\,(1-x^{\,2})^{\,l},$$ (293)

which implies that

$$P_l^{\,-m}(x)= (-1)^{\,m}\,\frac{(l-m)!}{(l+m)!}\,P_l^{\,m}(x).$$ (294)

Assuming that $0\leq m\leq l$ , the $P_l^{\,m}(x)$ satisfy the orthogonality condition

$$\int_{-1}^1P_l^{\,m}(x)\,P_k^{\,m}(x)\,dx = \frac{2\,(l+m)!}{(2\,l+1)\,(l-m)!}\,\delta_{lk},$$ (295)

where $\delta_{lk}$ is a Kronecker delta symbol.

The associated Legendre functions of order 0 (i.e., $m=0$ ) are called Legendre polynomials, and are denoted the $P_l(x)$ : that is, $P_l^0(x)=P_l(x)$ . It follows that

$$\int_{-1}^1P_l(x)\,P_k(x)\,dx = \frac{2}{(2\,l+1)}\,\delta_{lk}.$$ (296)

It can also be shown that

$$\frac{1}{(1-2\,x\,t+t^{\,2})^{1/2}} = \sum_{l=0,\infty} P_l(x)\,t^{\,l},$$ (297)

provided $\vert t\vert< 1$ and $\vert x\vert\leq 1$ .

All of the associated Legendre functions of degree less than 3 are listed below:

$$P_0^{\,0}(x) = 1,$$ (298)

$$P_{1}^{\,-1}(x) = (1/2)\,(1-x^{\,2})^{1/2},$$ (299)

$$\ P_1^{\,0}(x) = x,$$ (300)

$$P_1^{\,+1}(x) = -(1-x^{\,2})^{1/2},$$ (301)

$$P_2^{\,-2}(x) =(1/8)\,(1-x^{\,2}),$$ (302)

$$P_2^{\,-1}(x) =(1/2)\,x\,(1-x^{\,2})^{1/2},$$ (303)

$$P_2^{\,0}(x) =(1/2)\,(3\,x^{\,2}-1),$$ (304)

$$P_2^{\,+1}(x) =-3\,x\,(1-x^{\,2})^{1/2},$$ (305)

$$P_2^{\,+2}(x) =3\,(1-x^{\,2}).$$ (306)