Jacobi transform explained

as kernels of the transform .^{[1] [2] [3] [4]}

The Jacobi transform of a function

is^[5]

The inverse Jacobi transform is given by

Some Jacobi transform pairs

F(x)	f\alpha,\beta(n)
xm, m<n	0
xn	n!(\alpha+\beta+2n+1)\deltan
\alpha,\beta P m (x)	\deltan\deltam,
(1+x)a-\beta	\binom{n+\alpha}{n}2\alpha+a+1 \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	2\sigma+\beta+1 n!\Gamma(\alpha-\sigma) \Gamma(\sigma+1)\Gamma(n+\beta+1)\Gamma(\alpha-\sigma+n) \Gamma(\beta+\sigma+n+2)
(1-x)\sigma-\beta \alpha,\sigma P m (x), \Re\sigma>-1	2\alpha+\sigma+1 m!(n-m)! \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+\betaQ-1(1-z+Q)-\alpha(1+z+Q)-\beta, Q=(1-2xz+z2)1/2,	z	<1\,	infty \sum n=0 \deltanzn
(1-x)-\alpha(1+x)-\beta d dx \left[(1-x)\alpha+1(1+x)\beta+1 d dx \right]F(x)	-n(n+\alpha+\beta+1)f\alpha,\beta(n)
\left\{(1-x)-\alpha(1+x)-\beta d dx \left[(1-x)\alpha+1(1+x)\beta+1 d dx \right]\right\}kF(x)	(-1)knk(n+\alpha+\beta+1)kf\alpha,\beta(n)

Notes and References

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