Придружени Лежандрови полиноми
P
ℓ
m
(
x
)
{\displaystyle P_{\ell }^{m}(x)}
(
1
−
x
2
)
y
″
−
2
x
y
′
+
(
ℓ
[
ℓ
+
1
]
−
m
2
1
−
x
2
)
y
=
0
,
{\displaystyle (1-x^{2})\,y''-2xy'+\left(\ell [\ell +1]-{\frac {m^{2}}{1-x^{2}}}\right)\,y=0,\,}
m
уреди
Придружени Лежандрови полиноми
P
ℓ
m
(
x
)
{\displaystyle P_{\ell }^{m}(x)}
Лежандровим полиномима (m ≥ 0)
P
ℓ
m
(
x
)
=
(
−
1
)
m
(
1
−
x
2
)
m
/
2
d
m
d
x
m
(
P
ℓ
(
x
)
)
{\displaystyle P_{\ell }^{m}(x)=(-1)^{m}\ (1-x^{2})^{m/2}\ {\frac {d^{m}}{dx^{m}}}\left(P_{\ell }(x)\right)\,}
За обичне Лежандрове полиноме вреди:
(
1
−
x
2
)
d
2
d
x
2
P
ℓ
(
x
)
−
2
x
d
d
x
P
ℓ
(
x
)
+
ℓ
(
ℓ
+
1
)
P
ℓ
(
x
)
=
0.
{\displaystyle (1-x^{2}){\frac {d^{2}}{dx^{2}}}P_{\ell }(x)-2x{\frac {d}{dx}}P_{\ell }(x)+\ell (\ell +1)P_{\ell }(x)=0.}
Члан (−1)m
P
ℓ
(
x
)
=
1
2
ℓ
ℓ
!
d
ℓ
d
x
ℓ
[
(
x
2
−
1
)
ℓ
]
,
{\displaystyle P_{\ell }(x)={\frac {1}{2^{\ell }\,\ell !}}\ {\frac {d^{\ell }}{dx^{\ell }}}\left[(x^{2}-1)^{\ell }\right],}
па се онда придружени Лежандров полином може приказати као:
P
ℓ
m
(
x
)
=
(
−
1
)
m
2
ℓ
ℓ
!
(
1
−
x
2
)
m
/
2
d
ℓ
+
m
d
x
ℓ
+
m
(
x
2
−
1
)
ℓ
.
{\displaystyle P_{\ell }^{m}(x)={\frac {(-1)^{m}}{2^{\ell }\ell !}}(1-x^{2})^{m/2}\ {\frac {d^{\ell +m}}{dx^{\ell +m}}}(x^{2}-1)^{\ell }.}
Лежандрови полиноми могу да се прикажу и као специјални случај хипергеометријске функције :
P
λ
μ
(
z
)
=
1
Γ
(
1
−
μ
)
[
1
+
z
1
−
z
]
μ
/
2
2
F
1
(
−
λ
,
λ
+
1
;
1
−
μ
;
1
−
z
2
)
{\displaystyle P_{\lambda }^{\mu }(z)={\frac {1}{\Gamma (1-\mu )}}\left[{\frac {1+z}{1-z}}\right]^{\mu /2}\,_{2}F_{1}(-\lambda ,\lambda +1;1-\mu ;{\frac {1-z}{2}})}
Претпостављајући
0
≤
m
≤
ℓ
{\displaystyle 0\leq m\leq \ell }
m :
∫
−
1
1
P
k
m
P
ℓ
m
d
x
=
2
(
ℓ
+
m
)
!
(
2
ℓ
+
1
)
(
ℓ
−
m
)
!
δ
k
,
ℓ
{\displaystyle \int _{-1}^{1}P_{k}^{m}P_{\ell }^{m}dx={\frac {2(\ell +m)!}{(2\ell +1)(\ell -m)!}}\ \delta _{k,\ell }}
При томе је
δ
k
,
ℓ
{\displaystyle \delta _{k,\ell }}
Кронекерова делта функција .
Осим тога они задовољавају релацију ортогоналности за фиксни ℓ:
∫
−
1
1
P
ℓ
m
P
ℓ
n
1
−
x
2
d
x
=
{
0
if
m
≠
n
(
ℓ
+
m
)
!
m
(
ℓ
−
m
)
!
if
m
=
n
≠
0
∞
if
m
=
n
=
0
{\displaystyle \int _{-1}^{1}{\frac {P_{\ell }^{m}P_{\ell }^{n}}{1-x^{2}}}dx={\begin{cases}0&{\mbox{if }}m\neq n\\{\frac {(\ell +m)!}{m(\ell -m)!}}&{\mbox{if }}m=n\neq 0\\\infty &{\mbox{if }}m=n=0\end{cases}}}
уреди
P
0
0
(
x
)
=
1
{\displaystyle P_{0}^{0}(x)=1}
P
1
−
1
(
x
)
=
−
1
2
P
1
1
(
x
)
{\displaystyle P_{1}^{-1}(x)=-{\begin{matrix}{\frac {1}{2}}\end{matrix}}P_{1}^{1}(x)}
P
1
0
(
x
)
=
x
{\displaystyle P_{1}^{0}(x)=x}
P
1
1
(
x
)
=
−
(
1
−
x
2
)
1
/
2
{\displaystyle P_{1}^{1}(x)=-(1-x^{2})^{1/2}}
P
2
−
2
(
x
)
=
1
24
P
2
2
(
x
)
{\displaystyle P_{2}^{-2}(x)={\begin{matrix}{\frac {1}{24}}\end{matrix}}P_{2}^{2}(x)}
P
2
−
1
(
x
)
=
−
1
6
P
2
1
(
x
)
{\displaystyle P_{2}^{-1}(x)=-{\begin{matrix}{\frac {1}{6}}\end{matrix}}P_{2}^{1}(x)}
P
2
0
(
x
)
=
1
2
(
3
x
2
−
1
)
{\displaystyle P_{2}^{0}(x)={\begin{matrix}{\frac {1}{2}}\end{matrix}}(3x^{2}-1)}
P
2
1
(
x
)
=
−
3
x
(
1
−
x
2
)
1
/
2
{\displaystyle P_{2}^{1}(x)=-3x(1-x^{2})^{1/2}}
P
2
2
(
x
)
=
3
(
1
−
x
2
)
{\displaystyle P_{2}^{2}(x)=3(1-x^{2})}
P
3
−
3
(
x
)
=
−
1
720
P
3
3
(
x
)
{\displaystyle P_{3}^{-3}(x)=-{\begin{matrix}{\frac {1}{720}}\end{matrix}}P_{3}^{3}(x)}
P
3
−
2
(
x
)
=
1
120
P
3
2
(
x
)
{\displaystyle P_{3}^{-2}(x)={\begin{matrix}{\frac {1}{120}}\end{matrix}}P_{3}^{2}(x)}
P
3
−
1
(
x
)
=
−
1
12
P
3
1
(
x
)
{\displaystyle P_{3}^{-1}(x)=-{\begin{matrix}{\frac {1}{12}}\end{matrix}}P_{3}^{1}(x)}
P
3
0
(
x
)
=
1
2
(
5
x
3
−
3
x
)
{\displaystyle P_{3}^{0}(x)={\begin{matrix}{\frac {1}{2}}\end{matrix}}(5x^{3}-3x)}
P
3
1
(
x
)
=
−
3
2
(
5
x
2
−
1
)
(
1
−
x
2
)
1
/
2
{\displaystyle P_{3}^{1}(x)=-{\begin{matrix}{\frac {3}{2}}\end{matrix}}(5x^{2}-1)(1-x^{2})^{1/2}}
P
3
2
(
x
)
=
15
x
(
1
−
x
2
)
{\displaystyle P_{3}^{2}(x)=15x(1-x^{2})}
P
3
3
(
x
)
=
−
15
(
1
−
x
2
)
3
/
2
{\displaystyle P_{3}^{3}(x)=-15(1-x^{2})^{3/2}}
уреди
(
ℓ
−
m
+
1
)
P
ℓ
+
1
m
(
x
)
=
(
2
ℓ
+
1
)
x
P
ℓ
m
(
x
)
−
(
ℓ
+
m
)
P
ℓ
−
1
m
(
x
)
{\displaystyle (\ell -m+1)P_{\ell +1}^{m}(x)=(2\ell +1)xP_{\ell }^{m}(x)-(\ell +m)P_{\ell -1}^{m}(x)}
2
m
x
P
ℓ
m
(
x
)
=
−
1
−
x
2
[
P
ℓ
m
+
1
(
x
)
+
(
ℓ
+
m
)
(
ℓ
−
m
+
1
)
P
ℓ
m
−
1
(
x
)
]
{\displaystyle 2mxP_{\ell }^{m}(x)=-{\sqrt {1-x^{2}}}\left[P_{\ell }^{m+1}(x)+(\ell +m)(\ell -m+1)P_{\ell }^{m-1}(x)\right]}
1
−
x
2
P
ℓ
m
(
x
)
=
1
2
ℓ
+
1
[
P
ℓ
−
1
m
+
1
(
x
)
−
P
ℓ
+
1
m
+
1
(
x
)
]
{\displaystyle {\sqrt {1-x^{2}}}P_{\ell }^{m}(x)={\frac {1}{2\ell +1}}\left[P_{\ell -1}^{m+1}(x)-P_{\ell +1}^{m+1}(x)\right]}
1
−
x
2
P
ℓ
m
(
x
)
=
1
2
ℓ
+
1
[
(
ℓ
−
m
+
1
)
(
ℓ
−
m
+
2
)
P
ℓ
+
1
m
−
1
(
x
)
−
(
ℓ
+
m
−
1
)
(
ℓ
+
m
)
P
ℓ
−
1
m
−
1
(
x
)
]
{\displaystyle {\sqrt {1-x^{2}}}P_{\ell }^{m}(x)={\frac {1}{2\ell +1}}\left[(\ell -m+1)(\ell -m+2)P_{\ell +1}^{m-1}(x)-(\ell +m-1)(\ell +m)P_{\ell -1}^{m-1}(x)\right]}
1
−
x
2
P
ℓ
m
+
1
(
x
)
=
(
ℓ
−
m
)
x
P
ℓ
m
(
x
)
−
(
ℓ
+
m
)
P
ℓ
−
1
m
(
x
)
{\displaystyle {\sqrt {1-x^{2}}}P_{\ell }^{m+1}(x)=(\ell -m)xP_{\ell }^{m}(x)-(\ell +m)P_{\ell -1}^{m}(x)}
1
−
x
2
P
ℓ
m
′
(
x
)
=
1
2
[
(
ℓ
+
m
)
(
ℓ
−
m
+
1
)
P
ℓ
m
−
1
(
x
)
−
P
ℓ
m
+
1
(
x
)
]
{\displaystyle {\sqrt {1-x^{2}}}{P_{\ell }^{m}}'(x)={\frac {1}{2}}\left[(\ell +m)(\ell -m+1)P_{\ell }^{m-1}(x)-P_{\ell }^{m+1}(x)\right]}
(
1
−
x
2
)
P
ℓ
m
′
(
x
)
=
1
2
ℓ
+
1
[
(
ℓ
+
1
)
(
ℓ
+
m
)
P
l
−
1
m
(
x
)
−
l
(
l
−
m
+
1
)
P
l
+
1
m
(
x
)
]
{\displaystyle (1-x^{2}){P_{\ell }^{m}}'(x)={\frac {1}{2\ell +1}}\left[(\ell +1)(\ell +m)P_{l-1}^{m}(x)-l(l-m+1)P_{l+1}^{m}(x)\right]}
(
x
2
−
1
)
P
ℓ
m
′
(
x
)
=
ℓ
x
P
ℓ
m
(
x
)
−
(
ℓ
+
m
)
P
ℓ
−
1
m
(
x
)
{\displaystyle (x^{2}-1){P_{\ell }^{m}}'(x)={\ell }xP_{\ell }^{m}(x)-(\ell +m)P_{\ell -1}^{m}(x)}
(
x
2
−
1
)
P
ℓ
m
′
(
x
)
=
1
−
x
2
P
ℓ
m
+
1
(
x
)
+
m
x
P
ℓ
m
(
x
)
{\displaystyle (x^{2}-1){P_{\ell }^{m}}'(x)={\sqrt {1-x^{2}}}P_{\ell }^{m+1}(x)+mxP_{\ell }^{m}(x)}
(
x
2
−
1
)
P
ℓ
m
′
(
x
)
=
−
(
ℓ
+
m
)
(
ℓ
−
m
+
1
)
1
−
x
2
P
ℓ
m
−
1
(
x
)
−
m
x
P
ℓ
m
(
x
)
{\displaystyle (x^{2}-1){P_{\ell }^{m}}'(x)=-(\ell +m)(\ell -m+1){\sqrt {1-x^{2}}}P_{\ell }^{m-1}(x)-mxP_{\ell }^{m}(x)}
P
ℓ
+
1
ℓ
+
1
(
x
)
=
−
(
2
ℓ
+
1
)
1
−
x
2
P
ℓ
ℓ
(
x
)
{\displaystyle P_{\ell +1}^{\ell +1}(x)=-(2\ell +1){\sqrt {1-x^{2}}}P_{\ell }^{\ell }(x)}
P
ℓ
ℓ
(
x
)
=
(
−
1
)
l
(
2
ℓ
−
1
)
!
!
(
1
−
x
2
)
(
l
/
2
)
{\displaystyle P_{\ell }^{\ell }(x)=(-1)^{l}(2\ell -1)!!(1-x^{2})^{(l/2)}}
P
ℓ
+
1
ℓ
(
x
)
=
x
(
2
ℓ
+
1
)
P
ℓ
ℓ
(
x
)
{\displaystyle P_{\ell +1}^{\ell }(x)=x(2\ell +1)P_{\ell }^{\ell }(x)}
уреди
Придружени Лежандрови полиноми могу да се параметризирају помоћу углова, тј.
x
=
cos
θ
{\displaystyle x=\cos \theta }
P
ℓ
m
(
cos
θ
)
=
(
−
1
)
m
(
sin
θ
)
m
d
m
d
(
cos
θ
)
m
(
P
ℓ
(
cos
θ
)
)
{\displaystyle P_{\ell }^{m}(\cos \theta )=(-1)^{m}(\sin \theta )^{m}\ {\frac {d^{m}}{d(\cos \theta )^{m}}}\left(P_{\ell }(\cos \theta )\right)\,}
Онда добијамо да је првих неколико полинома:
P
0
0
(
cos
θ
)
=
1
P
1
0
(
cos
θ
)
=
cos
θ
P
1
1
(
cos
θ
)
=
−
sin
θ
P
2
0
(
cos
θ
)
=
1
2
(
3
cos
2
θ
−
1
)
P
2
1
(
cos
θ
)
=
−
3
cos
θ
sin
θ
P
2
2
(
cos
θ
)
=
3
sin
2
θ
P
3
0
(
cos
θ
)
=
1
2
(
5
cos
3
θ
−
3
cos
θ
)
P
3
1
(
cos
θ
)
=
−
3
2
(
5
cos
2
θ
−
1
)
sin
θ
P
3
2
(
cos
θ
)
=
15
cos
θ
sin
2
θ
P
3
3
(
cos
θ
)
=
−
15
sin
3
θ
{\displaystyle {\begin{aligned}P_{0}^{0}(\cos \theta )&=1\\[8pt]P_{1}^{0}(\cos \theta )&=\cos \theta \\[8pt]P_{1}^{1}(\cos \theta )&=-\sin \theta \\[8pt]P_{2}^{0}(\cos \theta )&={\tfrac {1}{2}}(3\cos ^{2}\theta -1)\\[8pt]P_{2}^{1}(\cos \theta )&=-3\cos \theta \sin \theta \\[8pt]P_{2}^{2}(\cos \theta )&=3\sin ^{2}\theta \\[8pt]P_{3}^{0}(\cos \theta )&={\tfrac {1}{2}}(5\cos ^{3}\theta -3\cos \theta )\\[8pt]P_{3}^{1}(\cos \theta )&=-{\tfrac {3}{2}}(5\cos ^{2}\theta -1)\sin \theta \\[8pt]P_{3}^{2}(\cos \theta )&=15\cos \theta \sin ^{2}\theta \\[8pt]P_{3}^{3}(\cos \theta )&=-15\sin ^{3}\theta \\[8pt]\end{aligned}}}
За фиксниm ,
P
ℓ
m
(
cos
θ
)
{\displaystyle P_{\ell }^{m}(\cos \theta )}
[
0
,
π
]
{\displaystyle [0,\pi ]}
sin
θ
{\displaystyle \sin \theta }
∫
0
π
P
k
m
(
cos
θ
)
P
ℓ
m
(
cos
θ
)
sin
θ
d
θ
=
2
(
ℓ
+
m
)
!
(
2
ℓ
+
1
)
(
ℓ
−
m
)
!
δ
k
,
ℓ
{\displaystyle \int _{0}^{\pi }P_{k}^{m}(\cos \theta )P_{\ell }^{m}(\cos \theta )\,\sin \theta \,d\theta ={\frac {2(\ell +m)!}{(2\ell +1)(\ell -m)!}}\ \delta _{k,\ell }}
Такође за фиксни ℓ:
∫
0
π
P
ℓ
m
(
cos
θ
)
P
ℓ
n
(
cos
θ
)
csc
θ
d
θ
=
{
0
if
m
≠
n
(
ℓ
+
m
)
!
m
(
ℓ
−
m
)
!
if
m
=
n
≠
0
∞
if
m
=
n
=
0
{\displaystyle \int _{0}^{\pi }P_{\ell }^{m}(\cos \theta )P_{\ell }^{n}(\cos \theta )\csc \theta \,d\theta ={\begin{cases}0&{\text{if }}m\neq n\\{\frac {(\ell +m)!}{m(\ell -m)!}}&{\text{if }}m=n\neq 0\\\infty &{\text{if }}m=n=0\end{cases}}}
P
ℓ
m
(
cos
θ
)
{\displaystyle P_{\ell }^{m}(\cos \theta )}
d
2
y
d
θ
2
+
cot
θ
d
y
d
θ
+
[
λ
−
m
2
sin
2
θ
]
y
=
0
{\displaystyle {\frac {d^{2}y}{d\theta ^{2}}}+\cot \theta {\frac {dy}{d\theta }}+\left[\lambda -{\frac {m^{2}}{\sin ^{2}\theta }}\right]\,y=0\,}
За
m
≥
0
{\displaystyle m\geq 0}
λ
=
ℓ
(
ℓ
+
1
)
{\displaystyle \lambda =\ell (\ell +1)\,}
ℓ
≥
m
{\displaystyle \ell \geq m}
P
ℓ
m
(
cos
θ
)
{\displaystyle P_{\ell }^{m}(\cos \theta )}
Придружени Лежандрови полиноми сусрећу се у многим проблемима физике са сферном симетријом.
Једначина
∇
2
ψ
+
λ
ψ
=
0
{\displaystyle \nabla ^{2}\psi +\lambda \psi =0}
лапласијана у сферним координатама:
∇
2
ψ
=
∂
2
ψ
∂
θ
2
+
cot
θ
∂
ψ
∂
θ
+
csc
2
θ
∂
2
ψ
∂
ϕ
2
.
{\displaystyle \nabla ^{2}\psi ={\frac {\partial ^{2}\psi }{\partial \theta ^{2}}}+\cot \theta {\frac {\partial \psi }{\partial \theta }}+\csc ^{2}\theta {\frac {\partial ^{2}\psi }{\partial \phi ^{2}}}.}
Парцијална диференцијална једначина
∇
2
ψ
+
λ
ψ
=
0
{\displaystyle \nabla ^{2}\psi +\lambda \psi =0}
∂
2
ψ
∂
θ
2
+
cot
θ
∂
ψ
∂
θ
+
csc
2
θ
∂
2
ψ
∂
ϕ
2
+
λ
ψ
=
0
{\displaystyle {\frac {\partial ^{2}\psi }{\partial \theta ^{2}}}+\cot \theta {\frac {\partial \psi }{\partial \theta }}+\csc ^{2}\theta {\frac {\partial ^{2}\psi }{\partial \phi ^{2}}}+\lambda \psi =0}
Решава се сепарацијом варијабли по θ и φ, тако да је φ део облика
sin
(
m
ϕ
)
{\displaystyle \sin(m\phi )}
cos
(
m
ϕ
)
{\displaystyle \cos(m\phi )}
d
2
y
d
θ
2
+
cot
θ
d
y
d
θ
+
[
λ
−
m
2
sin
2
θ
]
y
=
0
{\displaystyle {\frac {d^{2}y}{d\theta ^{2}}}+\cot \theta {\frac {dy}{d\theta }}+\left[\lambda -{\frac {m^{2}}{\sin ^{2}\theta }}\right]\,y=0\,}
за коју су решења придружени Лежандрови полиноми
P
ℓ
m
(
cos
θ
)
{\displaystyle P_{\ell }^{m}(\cos \theta )}
ℓ
≥
m
{\displaystyle \ell {\geq }m}
λ
=
ℓ
(
ℓ
+
1
)
{\displaystyle \lambda =\ell (\ell +1)}
На тај начин добили смо да су једначина:
∇
2
ψ
+
λ
ψ
=
0
{\displaystyle \nabla ^{2}\psi +\lambda \psi =0}
има несингуларна решења само за
λ
=
ℓ
(
ℓ
+
1
)
{\displaystyle \lambda =\ell (\ell +1)}
P
ℓ
m
(
cos
θ
)
cos
(
m
ϕ
)
0
≤
m
≤
ℓ
{\displaystyle P_{\ell }^{m}(\cos \theta )\ \cos(m\phi )\ \ \ \ 0\leq m\leq \ell }
и
P
ℓ
m
(
cos
θ
)
sin
(
m
ϕ
)
0
<
m
≤
ℓ
.
{\displaystyle P_{\ell }^{m}(\cos \theta )\ \sin(m\phi )\ \ \ \ 0<m\leq \ell .}
За сваки
ℓ
{\displaystyle \ell }
2
ℓ
+
1
{\displaystyle 2\ell +1}
m и они су ортогонални.
Решења се обично пишу у облику:
Y
ℓ
,
m
(
θ
,
ϕ
)
=
(
2
ℓ
+
1
)
(
ℓ
−
m
)
!
4
π
(
ℓ
+
m
)
!
P
ℓ
m
(
cos
θ
)
e
i
m
ϕ
−
ℓ
≤
m
≤
ℓ
.
{\displaystyle Y_{\ell ,m}(\theta ,\phi )={\sqrt {\frac {(2\ell +1)(\ell -m)!}{4\pi (\ell +m)!}}}\ P_{\ell }^{m}(\cos \theta )\ e^{im\phi }\qquad -\ell \leq m\leq \ell .}
При томе та решења
Y
ℓ
,
m
(
θ
,
ϕ
)
{\displaystyle Y_{\ell ,m}(\theta ,\phi )}
сферни хармоници .
![{\displaystyle (1-x^{2})\,y''-2xy'+\left(\ell [\ell +1]-{\frac {m^{2}}{1-x^{2}}}\right)\,y=0,\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6716f9d1977e6e496dfe7e5ba97b5da42965b9)
![{\displaystyle P_{\ell }(x)={\frac {1}{2^{\ell }\,\ell !}}\ {\frac {d^{\ell }}{dx^{\ell }}}\left[(x^{2}-1)^{\ell }\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43ca96dae138d4dc21c8eb5ee22e0f3c09d8b2ae)
![{\displaystyle P_{\lambda }^{\mu }(z)={\frac {1}{\Gamma (1-\mu )}}\left[{\frac {1+z}{1-z}}\right]^{\mu /2}\,_{2}F_{1}(-\lambda ,\lambda +1;1-\mu ;{\frac {1-z}{2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a33abd8dbaad6ee01cef4e0d064a2ebd78aad53a)
![{\displaystyle 2mxP_{\ell }^{m}(x)=-{\sqrt {1-x^{2}}}\left[P_{\ell }^{m+1}(x)+(\ell +m)(\ell -m+1)P_{\ell }^{m-1}(x)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16b8c51959e61b723fca74ad217ca195809a70d1)
![{\displaystyle {\sqrt {1-x^{2}}}P_{\ell }^{m}(x)={\frac {1}{2\ell +1}}\left[P_{\ell -1}^{m+1}(x)-P_{\ell +1}^{m+1}(x)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e801cdb77fcad6f2295e63b31913df8fb61bc4d8)
![{\displaystyle {\sqrt {1-x^{2}}}P_{\ell }^{m}(x)={\frac {1}{2\ell +1}}\left[(\ell -m+1)(\ell -m+2)P_{\ell +1}^{m-1}(x)-(\ell +m-1)(\ell +m)P_{\ell -1}^{m-1}(x)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa0b88f261e938f80c8b93576b8bf3e290727722)
![{\displaystyle {\sqrt {1-x^{2}}}{P_{\ell }^{m}}'(x)={\frac {1}{2}}\left[(\ell +m)(\ell -m+1)P_{\ell }^{m-1}(x)-P_{\ell }^{m+1}(x)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2de809e31787d4d71c52128c036caa699e9e07ca)
![{\displaystyle (1-x^{2}){P_{\ell }^{m}}'(x)={\frac {1}{2\ell +1}}\left[(\ell +1)(\ell +m)P_{l-1}^{m}(x)-l(l-m+1)P_{l+1}^{m}(x)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e951fe445cfb45a8dff7dec73382b6036e8b858)
![{\displaystyle {\begin{aligned}P_{0}^{0}(\cos \theta )&=1\\[8pt]P_{1}^{0}(\cos \theta )&=\cos \theta \\[8pt]P_{1}^{1}(\cos \theta )&=-\sin \theta \\[8pt]P_{2}^{0}(\cos \theta )&={\tfrac {1}{2}}(3\cos ^{2}\theta -1)\\[8pt]P_{2}^{1}(\cos \theta )&=-3\cos \theta \sin \theta \\[8pt]P_{2}^{2}(\cos \theta )&=3\sin ^{2}\theta \\[8pt]P_{3}^{0}(\cos \theta )&={\tfrac {1}{2}}(5\cos ^{3}\theta -3\cos \theta )\\[8pt]P_{3}^{1}(\cos \theta )&=-{\tfrac {3}{2}}(5\cos ^{2}\theta -1)\sin \theta \\[8pt]P_{3}^{2}(\cos \theta )&=15\cos \theta \sin ^{2}\theta \\[8pt]P_{3}^{3}(\cos \theta )&=-15\sin ^{3}\theta \\[8pt]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20ccc176ee7825b517e022fff6a21a5c4dfd87ec)
![{\displaystyle [0,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2a912eda6ef1afe46a81b518fe9da64a832751)
![{\displaystyle {\frac {d^{2}y}{d\theta ^{2}}}+\cot \theta {\frac {dy}{d\theta }}+\left[\lambda -{\frac {m^{2}}{\sin ^{2}\theta }}\right]\,y=0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf734246861cd768b5d9f61be5f5f3b0f3f502f5)
