Rational difference equation

<p>A rational difference equation is a nonlinear <a href="/facts/Difference_equation/JolXC71M">difference equation</a> of the form
</p>

x
          
            n
            +
            1
          
        
        =
        
          
            
              α
              +
              
                ∑
                
                  i
                  =
                  0
                
                
                  k
                
              
              
                β
                
                  i
                
              
              
                x
                
                  n
                  −
                  i
                
              
            
            
              A
              +
              
                ∑
                
                  i
                  =
                  0
                
                
                  k
                
              
              
                B
                
                  i
                
              
              
                x
                
                  n
                  −
                  i
                
              
            
          
        
         
        ,
      
    
    {\displaystyle x_{n+1}={\frac {\alpha +\sum _{i=0}^{k}\beta _{i}x_{n-i}}{A+\sum _{i=0}^{k}B_{i}x_{n-i}}}~,}

<p>where the initial conditions 
  
    
      
        
          x
          
            0
          
        
        ,
        
          x
          
            −
            1
          
        
        ,
        …
        ,
        
          x
          
            −
            k
          
        
      
    
    {\displaystyle x_{0},x_{-1},\dots ,x_{-k}}
  
 are such that the denominator  never vanishes for any n.
</p>

Rational difference equation open-in-new

Rational difference equation