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

.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).{{cite journal|last=Shen|first=Jie|last2=Wang|first2=Yingwei|last3=Xia|first3=Jianlin|date=2019|title=Fast structured Jacobi-Jacobi transforms|journal=Math. Comp.|volume=88|issue=318|pages=1743–1772|doi=10.1090/mcom/3377|doi-access=free}}

The Jacobi transform of a function $F(x)$ isDebnath, Lokenath, and Dambaru Bhatta. Integral transforms and their applications. CRC press, 2014.

: $J\{F(x)\} = f^{\alpha,\beta}(n) = \int_{-1}^1 (1-x)^\alpha\ (1+x)^\beta \ P_n^{\alpha,\beta}(x)\ F(x) \ dx$

The inverse Jacobi transform is given by

: $J^{-1}\{f^{\alpha,\beta}(n)\} = F(x) = \sum_{n=0}^\infty \frac{1}{\delta_n} f^{\alpha,\beta}(n) P_n^{\alpha,\beta}(x), \quad \text{where}
\quad \delta_n =\frac{2^{\alpha+\beta+1} \Gamma(n+ \alpha+1) \Gamma(n+\beta+1)}{n! (\alpha+\beta+2n+1) \Gamma(n+ \alpha+\beta+1)}$

Some Jacobi transform pairs

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|+ Some Jacobi transform pairs

! scope="col" | $F(x)\,$

! scope="col" | $f^{\alpha,\beta}(n)\,$

x^m, \ m

| $0$

x^n \,

| $n!(\alpha+\beta+2n+1)\delta_n$

P_m^{\alpha,\beta}(x) \,

| $\delta_n \delta_{m, n}$

(1+x)^{a-\beta} \,

| $\binom{n+\alpha}{n} 2^{\alpha+a+1} \frac{\Gamma(a+1)\Gamma(\alpha+1)\Gamma(a-\beta+1)}{\Gamma(\alpha+a+n+2)\Gamma(a-\beta+n+1)}$

(1-x)^{\sigma-\alpha}, \ \Re \sigma>-1 \,

| $\frac{2^{\sigma+\beta+1}}{n!\Gamma(\alpha-\sigma)}\frac{\Gamma(\sigma+1)\Gamma(n+\beta+1)\Gamma(\alpha-\sigma+n)}{\Gamma(\beta+\sigma+n+2)}$

(1-x)^{\sigma-\beta}P_m^{\alpha,\sigma}(x), \ \Re \sigma>-1 \,

| $\frac{2^{\alpha+\sigma+1}}{m!(n-m)!}\frac{\Gamma(n+\alpha+1)\Gamma(\alpha+\beta+m+n+1)\Gamma(\sigma+m+1)\Gamma(\alpha-\beta+1)}{\Gamma(\alpha+\beta+n+1)\Gamma(\alpha+\sigma+m+n+2)\Gamma(\alpha-\beta+m+1)}$

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|+ Some more Jacobi transform pairs

! scope="col" | $F(x)\,$

! scope="col" | $f^{\alpha,\beta}(n)\,$

2^{\alpha+\beta}Q^{-1}(1-z+Q)^{-\alpha}(1+z+Q)^{-\beta},\ Q=(1-2xz+z^2)^{1/2},\ |z|<1\,

| $\sum_{n=0}^\infty \delta_n z^n$

(1-x)^{-\alpha}(1+x)^{-\beta} \frac{d}{dx}\left[(1-x)^{\alpha+1}(1+x)^{\beta+1} \frac{d}{dx}\right]F(x) \,

| $-n(n+\alpha+\beta+1)f^{\alpha,\beta}(n)$

\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) \,

| $(-1)^kn^k(n+\alpha+\beta+1)^kf^{\alpha,\beta}(n)$

References

Category:Integral transforms

Category:Mathematical physics