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

$J\{F(x)\}=f^{\alpha ,\beta }(n)=\underset{-1}{\overset{1}{\int }}(1-x{)}^{\alpha }(1+x{)}^{\beta }{P}_n^{\alpha ,\beta }(x)F(x)dx$

The inverse Jacobi transform is given by

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

Some Jacobi transform pairs

Some Jacobi transform pairs

$F(x)$	$f^{\alpha ,\beta }(n)$
$x^m,m<n$	$0$
$x^n$	$n!(\alpha +\beta +2n+1){\delta }_n$
$P_m^{\alpha ,\beta }(x)$	${\delta }_n{\delta }_{m,n}$
$(1+x{)}^{a-\beta }$	$(\genfrac{}{}{0ex}{}{n+\alpha }{n})2^{\alpha +a+1}\frac{\mathrm{\Gamma }(a+1)\mathrm{\Gamma }(\alpha +1)\mathrm{\Gamma }(a-\beta +1)}{\mathrm{\Gamma }(\alpha +a+n+2)\mathrm{\Gamma }(a-\beta +n+1)}$
$(1-x{)}^{\sigma -\alpha },\mathrm{\Re }\sigma >-1$	$\frac{2^{\sigma +\beta +1}}{n!\mathrm{\Gamma }(\alpha -\sigma )}\frac{\mathrm{\Gamma }(\sigma +1)\mathrm{\Gamma }(n+\beta +1)\mathrm{\Gamma }(\alpha -\sigma +n)}{\mathrm{\Gamma }(\beta +\sigma +n+2)}$
$(1-x{)}^{\sigma -\beta }{P}_m^{\alpha ,\sigma }(x),\mathrm{\Re }\sigma >-1$	$\frac{2^{\alpha +\sigma +1}}{m!(n-m)!}\frac{\mathrm{\Gamma }(n+\alpha +1)\mathrm{\Gamma }(\alpha +\beta +m+n+1)\mathrm{\Gamma }(\sigma +m+1)\mathrm{\Gamma }(\alpha -\beta +1)}{\mathrm{\Gamma }(\alpha +\beta +n+1)\mathrm{\Gamma }(\alpha +\sigma +m+n+2)\mathrm{\Gamma }(\alpha -\beta +m+1)}$

Some more Jacobi transform pairs

$F(x)$	$f^{\alpha ,\beta }(n)$
$2^{\alpha +\beta }{Q}^{-1}(1-z+Q{)}^{-\alpha }(1+z+Q{)}^{-\beta },Q=(1-2xz+z^2{)}^{1/2},|z|<1$	$\sum _{n=0}^{\mathrm{\infty }}{\delta }_n{z}^n$
$(1-x{)}^{-\alpha }(1+x{)}^{-\beta }\frac{d}{dx}[(1-x{)}^{\alpha +1}(1+x{)}^{\beta +1}\frac{d}{dx}]F(x)$	$-n(n+\alpha +\beta +1)f^{\alpha ,\beta }(n)$
${\{(1-x{)}^{-\alpha }(1+x{)}^{-\beta }\frac{d}{dx}[(1-x{)}^{\alpha +1}(1+x{)}^{\beta +1}\frac{d}{dx}]\}}^{k}F(x)$	$(-1{)}^k{n}^k(n+\alpha +\beta +1{)}^k{f}^{\alpha ,\beta }(n)$

References

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