阿佩尔函数
阿佩爾函數是法國數學家（Paul Apell) 在 1880 年為推廣高斯超幾何函數而創建的一組雙變數函數，定義如下

  
    
      
        
          F
          
            1
          
        
        (
        a
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          b
          
            1
          
        
        ,
        
          b
          
            2
          
        
        ,
        c
        ;
        x
        ,
        y
        )
        =
        
          ∑
          
            m
            ,
            n
            =
            0
          
          
            ∞
          
        
        
          
            
              (a)
                
                  m
                  +
                  n
                
              
              (
              
                b
                
                  1
                
              
              
                )
                
                  m
                
              
              (
              
                b
                
                  2
                
              
              
                )
                
                  n
                
              
            
            
              (c)
                
                  m
                  +
                  n
                
              
              
              m
              !
              
              n
              !
            
          
        
        
        
          x
          
            m
          
        
        
          y
          
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        ,
      
    
    {\displaystyle F_{1}(a,b_{1},b_{2},c;x,y)=\sum _{m,n=0}^{\infty}{\frac {(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m} y^{n}~,}
  

  
    
      
        
          F
          
            2
          
        
        (
        a
        ,
        
          b
          
            1
          
        
        ,
        
          b
          
            2
          
        
        ,
        
          c
          
            1
          
        
        ,
        
          c
          
            2
          
        
        ;
        x
        ,
        y
        )
        =
        
          ∑
          
            m
            ,
            n
            =
            0
          
          
            ∞
          
        
        
          
            
              (a)
                
                  m
                  +
                  n
                
              
              (
              
                b
                
                  1
                
              
              
                )
                
                  m
                
              
              (
              
                b
                
                  2
                
              
              
                )
                
                  n
                
              
            
            
              (
              
                c
                
                  1
                
              
              
                )
                
                  m
                
              
              (
              
                c
                
                  2
                
              
              
                )
                
                  n
                
              
              
              m
              !
              
              n
              !
            
          
        
        
        
          x
          
            m
          
        
        
          y
          
            n
          
        
         
        ,
      
    
    {\displaystyle F_{2}(a,b_{1},b_{2},c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty}{\frac {(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m} y^{n}~,}
  

  
    
      
        
          F
          
            3
          
        
        (
        
          a
          
            1
          
        
        ,
        
          a
          
            2
          
        
        ,
        
          b
          
            1
          
        
        ,
        
          b
          
            2
          
        
        ,
        c
        ;
        x
        ,
        y
        )
        =
        
          ∑
          
            m
            ,
            n
            =
            0
          
          
            ∞
          
        
        
          
            
              (
              
                a
                
                  1
                
              
              
                )
                
                  m
                
              
              (
              
                a
                
                  2
                
              
              
                )
                
                  n
                
              
              (
              
                b
                
                  1
                
              
              
                )
                
                  m
                
              
              (
              
                b
                
                  2
                
              
              
                )
                
                  n
                
              
            
            
              (c)
                
                  m
                  +
                  n
                
              
              
              m
              !
              
              n
              !
            
          
        
        
        
          x
          
            m
          
        
        
          y
          
            n
          
        
         
        ,
      
    
    {\displaystyle F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)=\sum _{m,n=0}^{\infty}{\frac {(a_{1})_{m}(a_{2})_{n}(b_{1})_{m}(b_{2})_{n}}{(c)_{m+n}\,m!\,n!}}\,x^{m} y^{n}~,}
  

  
    
      
        
          F
          
            4
          
        
        (
        a
        ,
        b
        ,
        
          c
          
            1
          
        
        ,
        
          c
          
            2
          
        
        ;
        x
        ,
        y
        )
        =
        
          ∑
          
            m
            ,
            n
            =
            0
          
          
            ∞
          
        
        
          
            
              (a)
                
                  m
                  +
                  n
                
              
              (b)
                
                  m
                  +
                  n
                
              
            
            
              (
              
                c
                
                  1
                
              
              
                )
                
                  m
                
              
              (
              
                c
                
                  2
                
              
              
                )
                
                  n
                
              
              
              m
              !
              
              n
              !
            
          
        
        
        
          x
          
            m
          
        
        
          y
          
            n
          
        
         
        .
      
    
    {\displaystyle F_{4}(a,b,c_{1},c_{2};x,y)=\sum _{m,n=0}^{\infty}{\frac {(a)_{m+n}(b)_{m+n}}{(c_{1})_{m}(c_{2})_{n}\,m!\,n!}}\,x^{m} y^{n}~.}
  
其中的符號
  
    
      
        :
        (a)
          
            m
            +
            n
          
        
      
    
    {\displaystyle :(a)_{m+n}}
  是階乘冪
阿佩爾函數是嫪麗切拉函數和 Kampé_de_Fériet 函數的特例。