Generalized Appell polynomials

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Polynomial_sequence/JaH5aGcH">polynomial sequence</a> 
 
 
 
 {
 
 p
 
 n
 
 
 (
 z
 )
 }
 
 
 {\displaystyle \{p_{n}(z)\}}
 
 has a generalized Appell representation if the <a href="/facts/Generating_function/K7UVjnjL">generating function</a> for the <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> takes on a certain form:

K
        (
        z
        ,
        w
        )
        =
        A
        (
        w
        )
        Ψ
        (
        z
        g
        (
        w
        )
        )
        =
        
          ∑
          
            n
            =
            0
          
          
            ∞
          
        
        
          p
          
            n
          
        
        (
        z
        )
        
          w
          
            n
          
        
      
    
    {\displaystyle K(z,w)=A(w)\Psi (zg(w))=\sum _{n=0}^{\infty }p_{n}(z)w^{n}}

where the generating function or <a href="/facts/Kernel_(category_theory)/Tsn0uLfc">kernel</a> 
 
 
 
 K
 (
 z
 ,
 w
 )
 
 
 {\displaystyle K(z,w)}
 
 is composed of the series

A
        (
        w
        )
        =
        
          ∑
          
            n
            =
            0
          
          
            ∞
          
        
        
          a
          
            n
          
        
        
          w
          
            n
          
        
        
      
    
    {\displaystyle A(w)=\sum _{n=0}^{\infty }a_{n}w^{n}\quad }
  
 with 
  
    
      
        
          a
          
            0
          
        
        ≠
        0
      
    
    {\displaystyle a_{0}\neq 0}

and

Ψ
        (
        t
        )
        =
        
          ∑
          
            n
            =
            0
          
          
            ∞
          
        
        
          Ψ
          
            n
          
        
        
          t
          
            n
          
        
        
      
    
    {\displaystyle \Psi (t)=\sum _{n=0}^{\infty }\Psi _{n}t^{n}\quad }
  
 and all 
  
    
      
        
          Ψ
          
            n
          
        
        ≠
        0
      
    
    {\displaystyle \Psi _{n}\neq 0}

and

g
        (
        w
        )
        =
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          g
          
            n
          
        
        
          w
          
            n
          
        
        
      
    
    {\displaystyle g(w)=\sum _{n=1}^{\infty }g_{n}w^{n}\quad }
  
 with 
  
    
      
        
          g
          
            1
          
        
        ≠
        0.
      
    
    {\displaystyle g_{1}\neq 0.}

Given the above, it is not hard to show that 
 
 
 
 
 p
 
 n
 
 
 (
 z
 )
 
 
 {\displaystyle p_{n}(z)}
 
 is a <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">polynomial of degree</a> 
 
 
 
 n
 
 
 {\displaystyle n}
 
.
<a href="/facts/Boas%E2%80%93Buck_polynomials/IN038hGY">Boas–Buck polynomials</a> are a slightly more general class of polynomials.

Generalized Appell polynomials open-in-new

Generalized Appell polynomials