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In mathematics, Jacobi transform is an integral transform named after the mathematician Carl Gustav Jacob Jacobi , which uses Jacobi polynomials
P
n
α
,
β
(
x
)
{\displaystyle P_{n}^{\alpha ,\beta }(x)}
[ 1] [ 2] [ 3] [ 4]
The Jacobi transform of a function
F
(
x
)
{\displaystyle F(x)}
[ 5]
J
{
F
(
x
)
}
=
f
α
,
β
(
n
)
=
∫
−
1
1
(
1
−
x
)
α
(
1
+
x
)
β
P
n
α
,
β
(
x
)
F
(
x
)
d
x
{\displaystyle J\{F(x)\}=f^{\alpha ,\beta }(n)=\int _{-1}^{1}(1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha ,\beta }(x)\ F(x)\ dx}
The inverse Jacobi transform is given by
J
−
1
{
f
α
,
β
(
n
)
}
=
F
(
x
)
=
∑
n
=
0
∞
1
δ
n
f
α
,
β
(
n
)
P
n
α
,
β
(
x
)
,
where
δ
n
=
2
α
+
β
+
1
Γ
(
n
+
α
+
1
)
Γ
(
n
+
β
+
1
)
n
!
(
α
+
β
+
2
n
+
1
)
Γ
(
n
+
α
+
β
+
1
)
{\displaystyle J^{-1}\{f^{\alpha ,\beta }(n)\}=F(x)=\sum _{n=0}^{\infty }{\frac {1}{\delta _{n}}}f^{\alpha ,\beta }(n)P_{n}^{\alpha ,\beta }(x),\quad {\text{where}}\quad \delta _{n}={\frac {2^{\alpha +\beta +1}\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{n!(\alpha +\beta +2n+1)\Gamma (n+\alpha +\beta +1)}}}
Some Jacobi transform pairs
F
(
x
)
{\displaystyle F(x)\,}
f
α
,
β
(
n
)
{\displaystyle f^{\alpha ,\beta }(n)\,}
x
m
,
m
<
n
{\displaystyle x^{m},\ m<n\,}
0
{\displaystyle 0}
x
n
{\displaystyle x^{n}\,}
n
!
(
α
+
β
+
2
n
+
1
)
δ
n
{\displaystyle n!(\alpha +\beta +2n+1)\delta _{n}}
P
m
α
,
β
(
x
)
{\displaystyle P_{m}^{\alpha ,\beta }(x)\,}
δ
n
δ
m
,
n
{\displaystyle \delta _{n}\delta _{m,n}}
(
1
+
x
)
a
−
β
{\displaystyle (1+x)^{a-\beta }\,}
(
n
+
α
n
)
2
α
+
a
+
1
Γ
(
a
+
1
)
Γ
(
α
+
1
)
Γ
(
a
−
β
+
1
)
Γ
(
α
+
a
+
n
+
2
)
Γ
(
a
−
β
+
n
+
1
)
{\displaystyle {\binom {n+\alpha }{n}}2^{\alpha +a+1}{\frac {\Gamma (a+1)\Gamma (\alpha +1)\Gamma (a-\beta +1)}{\Gamma (\alpha +a+n+2)\Gamma (a-\beta +n+1)}}}
(
1
−
x
)
σ
−
α
,
ℜ
σ
>
−
1
{\displaystyle (1-x)^{\sigma -\alpha },\ \Re \sigma >-1\,}
2
σ
+
β
+
1
n
!
Γ
(
α
−
σ
)
Γ
(
σ
+
1
)
Γ
(
n
+
β
+
1
)
Γ
(
α
−
σ
+
n
)
Γ
(
β
+
σ
+
n
+
2
)
{\displaystyle {\frac {2^{\sigma +\beta +1}}{n!\Gamma (\alpha -\sigma )}}{\frac {\Gamma (\sigma +1)\Gamma (n+\beta +1)\Gamma (\alpha -\sigma +n)}{\Gamma (\beta +\sigma +n+2)}}}
(
1
−
x
)
σ
−
β
P
m
α
,
σ
(
x
)
,
ℜ
σ
>
−
1
{\displaystyle (1-x)^{\sigma -\beta }P_{m}^{\alpha ,\sigma }(x),\ \Re \sigma >-1\,}
2
α
+
σ
+
1
m
!
(
n
−
m
)
!
Γ
(
n
+
α
+
1
)
Γ
(
α
+
β
+
m
+
n
+
1
)
Γ
(
σ
+
m
+
1
)
Γ
(
α
−
β
+
1
)
Γ
(
α
+
β
+
n
+
1
)
Γ
(
α
+
σ
+
m
+
n
+
2
)
Γ
(
α
−
β
+
m
+
1
)
{\displaystyle {\frac {2^{\alpha +\sigma +1}}{m!(n-m)!}}{\frac {\Gamma (n+\alpha +1)\Gamma (\alpha +\beta +m+n+1)\Gamma (\sigma +m+1)\Gamma (\alpha -\beta +1)}{\Gamma (\alpha +\beta +n+1)\Gamma (\alpha +\sigma +m+n+2)\Gamma (\alpha -\beta +m+1)}}}
Some more Jacobi transform pairs
F
(
x
)
{\displaystyle F(x)\,}
f
α
,
β
(
n
)
{\displaystyle f^{\alpha ,\beta }(n)\,}
2
α
+
β
Q
−
1
(
1
−
z
+
Q
)
−
α
(
1
+
z
+
Q
)
−
β
,
Q
=
(
1
−
2
x
z
+
z
2
)
1
/
2
,
|
z
|
<
1
{\displaystyle 2^{\alpha +\beta }Q^{-1}(1-z+Q)^{-\alpha }(1+z+Q)^{-\beta },\ Q=(1-2xz+z^{2})^{1/2},\ |z|<1\,}
∑
n
=
0
∞
δ
n
z
n
{\displaystyle \sum _{n=0}^{\infty }\delta _{n}z^{n}}
(
1
−
x
)
−
α
(
1
+
x
)
−
β
d
d
x
[
(
1
−
x
)
α
+
1
(
1
+
x
)
β
+
1
d
d
x
]
F
(
x
)
{\displaystyle (1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]F(x)\,}
−
n
(
n
+
α
+
β
+
1
)
f
α
,
β
(
n
)
{\displaystyle -n(n+\alpha +\beta +1)f^{\alpha ,\beta }(n)}
{
(
1
−
x
)
−
α
(
1
+
x
)
−
β
d
d
x
[
(
1
−
x
)
α
+
1
(
1
+
x
)
β
+
1
d
d
x
]
}
k
F
(
x
)
{\displaystyle \left\{(1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]\right\}^{k}F(x)\,}
(
−
1
)
k
n
k
(
n
+
α
+
β
+
1
)
k
f
α
,
β
(
n
)
{\displaystyle (-1)^{k}n^{k}(n+\alpha +\beta +1)^{k}f^{\alpha ,\beta }(n)}
References
^ Debnath, L. "On Jacobi Transform." Bull. Cal. Math. Soc 55.3 (1963): 113-120.
^ Debnath, L. "SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS BY JACOBI TRANSFORM." BULLETIN OF THE CALCUTTA MATHEMATICAL SOCIETY 59.3-4 (1967): 155.
^ Scott, E. J. "Jacobi transforms." (1953).
^ Shen, Jie; Wang, Yingwei; Xia, Jianlin (2019). "Fast structured Jacobi-Jacobi transforms" . Math. Comp . 88 (318): 1743– 1772. doi :10.1090/mcom/3377 ^ Debnath, Lokenath, and Dambaru Bhatta. Integral transforms and their applications. CRC press, 2014.
![{\displaystyle (1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]F(x)\,}](/media/api/rest_v1/media/math/render/svg/6cf3f023f7d443653e8a4664e7c2e71e76cb4e07)
![{\displaystyle \left\{(1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]\right\}^{k}F(x)\,}](/media/api/rest_v1/media/math/render/svg/f37708dcabf8079699227184c4cbfbcac88a868d)
