Jacobi transform

In mathematics, Jacobi transform is an integral transform named after the mathematician Carl Gustav Jacob Jacobi, which uses Jacobi polynomials ${\displaystyle P_{n}^{\alpha ,\beta }(x)}$ as kernels of the transform [1] [2] [3] [4].

The Jacobi transform of a function ${\displaystyle F(x)}$ is[5]

${\displaystyle J\{F(x)\}=f^{\alpha ,\beta }(n)=\int _{-1}^{1}(1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha ,\beta }(x)\ F(x)\ dx}$

The inverse Jacobi transform is given by

${\displaystyle J^{-1}\{f^{\alpha ,\beta }(n)\}=F(x)=\sum _{n=0}^{\infty }{\frac {1}{\delta _{n}}}f^{\alpha ,\beta }(n)P_{n}^{\alpha ,\beta }(x),\quad {\text{where}}\quad \delta _{n}={\frac {2^{\alpha +\beta +1}\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{n!(\alpha +\beta +2n+1)\Gamma (n+\alpha +\beta +1)}}}$

Some Jacobi transform pairs

${\displaystyle F(x)\,}$	${\displaystyle f^{\alpha ,\beta }(n)\,}$
${\displaystyle x^{m},\ m<n\,}$	${\displaystyle 0}$
${\displaystyle x^{n}\,}$	${\displaystyle n!(\alpha +\beta +2n+1)\delta _{n}}$
${\displaystyle P_{m}^{\alpha ,\beta }(x)\,}$	${\displaystyle \delta _{mn}}$
${\displaystyle (1+x)^{a-\beta }\,}$	${\displaystyle {\binom {n+\alpha }{n}}2^{\alpha +a+1}{\frac {\Gamma (a+1)\Gamma (\alpha +1)\Gamma (a-\beta +1)}{\Gamma (\alpha +a+n+2)\Gamma (a-\beta +n+1)}}}$
${\displaystyle (1-x)^{\sigma -\alpha },\ \Re \sigma >-1\,}$	${\displaystyle {\frac {2^{\sigma +\beta +1}}{n!\Gamma (\alpha -\sigma )}}{\frac {\Gamma (\sigma +1)\Gamma (n+\beta +1)\Gamma (\alpha -\sigma +n)}{\Gamma (\beta +\sigma +n+2)}}}$
${\displaystyle (1-x)^{\sigma -\beta }P_{m}^{\alpha ,\sigma }(x),\ \Re \sigma >-1\,}$	${\displaystyle {\frac {2^{\alpha +\sigma +1}}{m!(n-m)!}}{\frac {\Gamma (n+\alpha +1)\Gamma (\alpha +\beta +m+n+1)\Gamma (\sigma +m+1)\Gamma (\alpha -\beta +1)}{\Gamma (\alpha +\beta +n+1)\Gamma (\alpha +\sigma +m+n+2)\Gamma (\alpha -\beta +m+1)}}}$
${\displaystyle 2^{\alpha +\beta }Q^{-1}(1-z+Q)^{-\alpha }(1+z+Q)^{-\beta },\ Q=(1-2xz+z^{2})^{1/2},\ |z|<1\,}$	${\displaystyle \sum _{n=0}^{\infty }\delta _{n}z^{n}}$
${\displaystyle (1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]F(x)\,}$	${\displaystyle -n(n+\alpha +\beta +1)f^{\alpha ,\beta }(n)}$
${\displaystyle \left\{(1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d}{dx}}\left[(1-x)^{\alpha +1}(1+x)^{\beta +1}{\frac {d}{dx}}\right]\right\}^{k}F(x)\,}$	${\displaystyle (-1)^{k}n^{k}(n+\alpha +\beta +1)^{k}f^{\alpha ,\beta }(n)}$

References

1. Debnath, L. "On Jacobi Transform." Bull. Cal. Math. Soc 55.3 (1963): 113-120.
2. Debnath, L. "SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS BY JACOBI TRANSFORM." BULLETIN OF THE CALCUTTA MATHEMATICAL SOCIETY 59.3-4 (1967): 155.
3. Scott, E. J. "Jacobi transforms." (1953).
4. Shen, Jie; Wang, Yingwei; Xia, Jianlin (2019). "Fast structured Jacobi-Jacobi transforms". Math. Comp. 88 (318): 1743–1772. doi:10.1090/mcom/3377.
5. Debnath, Lokenath, and Dambaru Bhatta. Integral transforms and their applications. CRC press, 2014.