# Jacobi transform

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In mathematics, '''Jacobi transform''' is an [integral transform](/source/integral_transform) named after the mathematician [Carl Gustav Jacob Jacobi](/source/Carl_Gustav_Jacob_Jacobi), which uses [Jacobi polynomials](/source/Jacobi_polynomials) <math>P_n^{\alpha,\beta}(x)</math> as kernels of the transform 
.<ref>Debnath, L. "On Jacobi Transform." Bull. Cal. Math. Soc 55.3 (1963): 113-120.</ref><ref>Debnath, L. "SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS BY JACOBI TRANSFORM." BULLETIN OF THE CALCUTTA MATHEMATICAL SOCIETY 59.3-4 (1967): 155.</ref><ref>Scott, E. J. "Jacobi transforms." (1953).</ref><ref name ="SWX19">{{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&ndash;1772|doi=10.1090/mcom/3377|doi-access=free}}</ref>

The Jacobi transform of a function <math>F(x)</math> is<ref>Debnath, Lokenath, and Dambaru Bhatta. Integral transforms and their applications. CRC press, 2014.</ref>

:<math>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>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 ==

{| class="wikitable" align="center"
|+ Some Jacobi transform pairs
! scope="col" | <math>F(x)\,</math>
! scope="col" | <math>f^{\alpha,\beta}(n)\,</math>
|-
|<math>x^m, \ m<n \,</math>
|<math>0</math>
|-
|<math>x^n \,</math>
|<math>n!(\alpha+\beta+2n+1)\delta_n</math>
|-
|<math>P_m^{\alpha,\beta}(x) \,</math>
|<math>\delta_n \delta_{m, n}</math>
|-
|<math>(1+x)^{a-\beta} \,</math>
|<math>\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>(1-x)^{\sigma-\alpha}, \ \Re \sigma>-1 \,</math>
|<math>\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>(1-x)^{\sigma-\beta}P_m^{\alpha,\sigma}(x), \ \Re \sigma>-1 \,</math>
|<math>\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>2^{\alpha+\beta}Q^{-1}(1-z+Q)^{-\alpha}(1+z+Q)^{-\beta},\ Q=(1-2xz+z^2)^{1/2},\ |z|<1\,</math>
|<math>\sum_{n=0}^\infty \delta_n z^n</math>
|-
|<math>(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>-n(n+\alpha+\beta+1)f^{\alpha,\beta}(n)</math>
|-
|<math>\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>(-1)^kn^k(n+\alpha+\beta+1)^kf^{\alpha,\beta}(n)</math>
|}

==References==
{{Reflist}}

Category:Integral transforms
Category:Mathematical physics

{{math-physics-stub}}

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Adapted from the Wikipedia article [Jacobi transform](https://en.wikipedia.org/wiki/Jacobi_transform) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Jacobi_transform?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.