Jacobi transform

In mathematics, Jacobi transform is an integral transform named after the mathematician Carl Gustav Jacob Jacobi, which uses Jacobi polynomials [math]\displaystyle{ P_n^{\alpha,\beta}(x) }[/math] as kernels of the transform .

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

[math]\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 }[/math]

The inverse Jacobi transform is given by

[math]\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)} }[/math]

Some Jacobi transform pairs

[math]\displaystyle{ F(x)\, }[/math]	[math]\displaystyle{ f^{\alpha,\beta}(n)\, }[/math]
[math]\displaystyle{ x^m, \ m\lt n \, }[/math]	[math]\displaystyle{ 0 }[/math]
[math]\displaystyle{ x^n \, }[/math]	[math]\displaystyle{ n!(\alpha+\beta+2n+1)\delta_n }[/math]
[math]\displaystyle{ P_m^{\alpha,\beta}(x) \, }[/math]	[math]\displaystyle{ \delta_n \delta_{m, n} }[/math]
[math]\displaystyle{ (1+x)^{a-\beta} \, }[/math]	[math]\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)} }[/math]
[math]\displaystyle{ (1-x)^{\sigma-\alpha}, \ \Re \sigma\gt -1 \, }[/math]	[math]\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)} }[/math]
[math]\displaystyle{ (1-x)^{\sigma-\beta}P_m^{\alpha,\sigma}(x), \ \Re \sigma\gt -1 \, }[/math]	[math]\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)} }[/math]
[math]\displaystyle{ 2^{\alpha+\beta}Q^{-1}(1-z+Q)^{-\alpha}(1+z+Q)^{-\beta},\ Q=(1-2xz+z^2)^{1/2},\ |z|\lt 1\, }[/math]	[math]\displaystyle{ \sum_{n=0}^\infty \delta_n z^n }[/math]
[math]\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) \, }[/math]	[math]\displaystyle{ -n(n+\alpha+\beta+1)f^{\alpha,\beta}(n) }[/math]
[math]\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\}^kF(x) \, }[/math]	[math]\displaystyle{ (-1)^kn^k(n+\alpha+\beta+1)^kf^{\alpha,\beta}(n) }[/math]