Jacobi transform

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

The Jacobi transform of a function $F(x)$ is

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

The inverse Jacobi transform is given by

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

Some Jacobi transform pairs

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