Jacobi transform

In mathematics, Jacobi transform is an <a href="/facts/Integral_transform/AB2yjHJJ">integral transform</a> named after the mathematician <a href="/facts/Carl_Gustav_Jacob_Jacobi/4Syeauo5">Carl Gustav Jacob Jacobi</a>, which uses <a href="/facts/Jacobi_polynomials/5oKJKB1W">Jacobi polynomials</a> 
 
 
 
 
 P
 
 n
 
 
 α
 ,
 β
 
 
 (
 x
 )
 
 
 {\displaystyle P_{n}^{\alpha ,\beta }(x)}
 
 as kernels of the transform 
.
The Jacobi transform of a function 
 
 
 
 F
 (
 x
 )
 
 
 {\displaystyle F(x)}
 
 is

J
        {
        F
        (
        x
        )
        }
        =
        
          f
          
            α
            ,
            β
          
        
        (
        n
        )
        =
        
          ∫
          
            −
            1
          
          
            1
          
        
        (
        1
        −
        x
        
          )
          
            α
          
        
         
        (
        1
        +
        x
        
          )
          
            β
          
        
         
        
          P
          
            n
          
          
            α
            ,
            β
          
        
        (
        x
        )
         
        F
        (
        x
        )
         
        d
        x
      
    
    {\displaystyle 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
          
            α
            ,
            β
          
        
        (
        n
        )
        }
        =
        F
        (
        x
        )
        =
        
          ∑
          
            n
            =
            0
          
          
            ∞
          
        
        
          
            1
            
              δ
              
                n
              
            
          
        
        
          f
          
            α
            ,
            β
          
        
        (
        n
        )
        
          P
          
            n
          
          
            α
            ,
            β
          
        
        (
        x
        )
        ,
        
        
          where
        
        
        
          δ
          
            n
          
        
        =
        
          
            
              
                2
                
                  α
                  +
                  β
                  +
                  1
                
              
              Γ
              (
              n
              +
              α
              +
              1
              )
              Γ
              (
              n
              +
              β
              +
              1
              )
            
            
              n
              !
              (
              α
              +
              β
              +
              2
              n
              +
              1
              )
              Γ
              (
              n
              +
              α
              +
              β
              +
              1
              )
            
          
        
      
    
    {\displaystyle 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)}}}

Jacobi transform open-in-new

Jacobi transform