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)}$$