Racah polynomials

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, Racah polynomials are <a href="/facts/Orthogonal_polynomials/9cyRbDYU">orthogonal polynomials</a> named after <a href="/facts/Giulio_Racah/6YIN3WhM">Giulio Racah</a>, as their orthogonality relations are equivalent to his orthogonality relations for <a href="/facts/Racah_coefficient/WCENaaH9">Racah coefficients</a>.
</p><p>The Racah polynomials were first defined by Wilson (1978) and are given by
</p>

p
          
            n
          
        
        (
        x
        (
        x
        +
        γ
        +
        δ
        +
        1
        )
        )
        =

4
          
        
        
          F
          
            3
          
        
        
          [
          
            
              
                
                  
                    −
                    n
                  
                  
                    n
                    +
                    α
                    +
                    β
                    +
                    1
                  
                  
                    −
                    x
                  
                  
                    x
                    +
                    γ
                    +
                    δ
                    +
                    1
                  
                
                
                  
                    α
                    +
                    1
                  
                  
                    γ
                    +
                    1
                  
                  
                    β
                    +
                    δ
                    +
                    1
                  
                
              
            
            ;
            1
          
          ]
        
        .
      
    
    {\displaystyle p_{n}(x(x+\gamma +\delta +1))={}_{4}F_{3}\left[{\begin{matrix}-n&n+\alpha +\beta +1&-x&x+\gamma +\delta +1\\\alpha +1&\gamma +1&\beta +\delta +1\\\end{matrix}};1\right].}

Racah polynomials open-in-new

Racah polynomials