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 .^[1]^[2]^[3]^[4]

The Jacobi transform of a function $F(x)$ is^[5]

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)}}

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