Matematiikassa Jacobin polynomit ovat luokka ortogonaalisia polynomeja . Ne saadaan hypergeometrisista sarjoista , missä sarjasta otetaan mukaan vain äärellisen monta termiä:
P
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n
n
!
2
F
1
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−
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α
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{\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {(\alpha +1)_{n}}{n!}}\,_{2}F_{1}\left(-n,1+\alpha +\beta +n;\alpha +1;{\frac {1-z}{2}}\right),}
missä
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α
+
1
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n
{\displaystyle (\alpha +1)_{n}}
Pochhammerin symboli) , (Abramowitz & Stegun p561 .) ja siten sillä on olemassa eksplisiittinen lauseke
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n
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∑
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Γ
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{\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {\Gamma (\alpha +n+1)}{n!\Gamma (\alpha +\beta +n+1)}}\sum _{m=0}^{n}{n \choose m}{\frac {\Gamma (\alpha +\beta +n+m+1)}{\Gamma (\alpha +m+1)}}\left({\frac {z-1}{2}}\right)^{m},}
missä
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1
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=
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{\displaystyle P_{n}^{(\alpha ,\beta )}(1)={n+\alpha \choose n}.}
Tässä kokonaisluvulle
n
{\displaystyle n\,}
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Γ
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{\displaystyle {z \choose n}={\frac {\Gamma (z+1)}{\Gamma (n+1)\Gamma (z-n+1)}},}
ja
Γ
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)
{\displaystyle \Gamma (z)\,}
Gammafunktio , jolle
1
/
Γ
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n
+
1
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=
0
{\displaystyle 1/\Gamma (n+1)=0\,}
n
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−
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,
−
2
,
…
{\displaystyle n=-1,-2,\dots \,}
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z
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=
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for
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<
0.
{\displaystyle {z \choose n}=0\quad {\hbox{for}}\quad n<0.}
Polynomit toteuttavat ortogonaalisuusehdon
∫
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1
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d
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+
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+
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2
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+
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+
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+
1
Γ
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+
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Γ
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+
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Γ
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n
!
δ
n
m
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{\displaystyle \int _{-1}^{1}(1-x)^{\alpha }(1+x)^{\beta }P_{m}^{(\alpha ,\beta )}(x)P_{n}^{(\alpha ,\beta )}(x)\;dx={\frac {2^{\alpha +\beta +1}}{2n+\alpha +\beta +1}}{\frac {\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{\Gamma (n+\alpha +\beta +1)n!}}\delta _{nm},}
kun
α
>
−
1
{\displaystyle \alpha >-1}
β
>
−
1
{\displaystyle \beta >-1}
Polynomeilla on symmetrisyysehto
P
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P
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{\displaystyle P_{n}^{(\alpha ,\beta )}(-z)=(-1)^{n}P_{n}^{(\beta ,\alpha )}(z),}
P
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{\displaystyle P_{n}^{(\alpha ,\beta )}(-1)=(-1)^{n}{n+\beta \choose n}.}
Reaaliluvuilla
x
{\displaystyle x}
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{\displaystyle P_{n}^{(\alpha ,\beta )}(x)=\sum _{s}{n+\alpha \choose s}{n+\beta \choose n-s}\left({\frac {x-1}{2}}\right)^{n-s}\left({\frac {x+1}{2}}\right)^{s},}
missä
s
≥
0
{\displaystyle s\geq 0\,}
n
−
s
≥
0
{\displaystyle n-s\geq 0\,}
n
{\displaystyle n}
n
+
α
{\displaystyle n+\alpha }
n
+
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{\displaystyle n+\beta }
n
+
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+
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{\displaystyle n+\alpha +\beta }
P
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=
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−
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]
−
1
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2
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n
−
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+
1
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s
.
{\displaystyle P_{n}^{(\alpha ,\beta )}(x)=(n+\alpha )!(n+\beta )!\sum _{s}\left[s!(n+\alpha -s)!(\beta +s)!(n-s)!\right]^{-1}\left({\frac {x-1}{2}}\right)^{n-s}\left({\frac {x+1}{2}}\right)^{s}.}
Summa voidaan laajentaa kaikille kokonaislukuarvoille, joilla kertomien argumentit ovat epänegatiivisia.
Tämä mahdollistaa Wignerin D-matriisin esittämisen Jacobin polynomiel avulla
d
m
′
m
j
(
ϕ
)
{\displaystyle d_{m'm}^{j}(\phi )\;}
0
≤
ϕ
≤
4
π
{\displaystyle 0\leq \phi \leq 4\pi }
Angular Momentum in Quantum Physics , Addison-Wesley, Reading, (1981)
d
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1
/
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sin
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m
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cos
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m
+
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′
P
j
−
m
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−
m
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+
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)
(
cos
ϕ
)
.
{\displaystyle d_{m'm}^{j}(\phi )=\left[{\frac {(j+m)!(j-m)!}{(j+m')!(j-m')!}}\right]^{1/2}\left(\sin {\frac {\phi }{2}}\right)^{m-m'}\left(\cos {\frac {\phi }{2}}\right)^{m+m'}P_{j-m}^{(m-m',m+m')}(\cos \phi ).}
Jacobin polynomien k :nnes derivaatta johtaa esitykseen
d
k
d
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k
P
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=
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2
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Γ
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{\displaystyle {\frac {\mathrm {d} ^{k}}{\mathrm {d} z^{k}}}P_{n}^{(\alpha ,\beta )}(z)={\frac {\Gamma (\alpha +\beta +n+1+k)}{2^{k}\Gamma (\alpha +\beta +n+1)}}P_{n-k}^{(\alpha +k,\beta +k)}(z).}
Jacobi polynomiat
P
n
(
α
,
β
)
{\displaystyle P_{n}^{(\alpha ,\beta )}}
(
1
−
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2
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y
″
+
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−
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−
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+
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+
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x
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y
′
+
n
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+
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+
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+
1
)
y
=
0.
{\displaystyle (1-x^{2})y''+(\beta -\alpha -(\alpha +\beta +2)x)y'+n(n+\alpha +\beta +1)y=0.\,}
![{\displaystyle P_{n}^{(\alpha ,\beta )}(x)=(n+\alpha )!(n+\beta )!\sum _{s}\left[s!(n+\alpha -s)!(\beta +s)!(n-s)!\right]^{-1}\left({\frac {x-1}{2}}\right)^{n-s}\left({\frac {x+1}{2}}\right)^{s}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d82730cbf4eb92d90abc188c0de9c6aed3dac0c)
![{\displaystyle d_{m'm}^{j}(\phi )=\left[{\frac {(j+m)!(j-m)!}{(j+m')!(j-m')!}}\right]^{1/2}\left(\sin {\frac {\phi }{2}}\right)^{m-m'}\left(\cos {\frac {\phi }{2}}\right)^{m+m'}P_{j-m}^{(m-m',m+m')}(\cos \phi ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2abcaf41d9baf29c60160c839c3d9e062c80ce9e)
