Wilson polynomials

<p>In mathematics, Wilson polynomials are a family of <a href="/facts/Orthogonal_polynomials/9cyRbDYU">orthogonal polynomials</a> introduced by <a href="/facts/James_A._Wilson/xxjC8beA">James A. Wilson</a> (1980)
that generalize <a href="/facts/Jacobi_polynomials/5oKJKB1W">Jacobi polynomials</a>, <a href="/facts/Hahn_polynomials/M0cpTwkm">Hahn polynomials</a>, and <a href="/facts/Charlier_polynomials/kMsE2VBG">Charlier polynomials</a>. 
</p><p>They are defined in terms of the <a href="/facts/Generalized_hypergeometric_function/fqXXDHdR">generalized hypergeometric function</a> and the <a href="/facts/Pochhammer_symbol/sAxnmGoC">Pochhammer symbols</a> by
</p>

p
          
            n
          
        
        (
        
          t
          
            2
          
        
        )
        =
        (
        a
        +
        b
        
          )
          
            n
          
        
        (
        a
        +
        c
        
          )
          
            n
          
        
        (
        a
        +
        d
        
          )
          
            n

4
          
        
        
          F
          
            3
          
        
        
          (
          
            
              
                
                  
                    −
                    n
                  
                  
                    a
                    +
                    b
                    +
                    c
                    +
                    d
                    +
                    n
                    −
                    1
                  
                  
                    a
                    −
                    t
                  
                  
                    a
                    +
                    t
                  
                
                
                  
                    a
                    +
                    b
                  
                  
                    a
                    +
                    c
                  
                  
                    a
                    +
                    d
                  
                
              
            
            ;
            1
          
          )
        
        .
      
    
    {\displaystyle p_{n}(t^{2})=(a+b)_{n}(a+c)_{n}(a+d)_{n}{}_{4}F_{3}\left({\begin{matrix}-n&a+b+c+d+n-1&a-t&a+t\\a+b&a+c&a+d\end{matrix}};1\right).}

Wilson polynomials open-in-new

Wilson polynomials