Three-term recurrence relation

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, and especially in <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a>, a homogeneous linear three-term recurrence relation (TTRR, the qualifiers "homogeneous linear" are usually taken for granted) is a <a href="/facts/Recurrence_relation/JolXC71M">recurrence relation</a> of the form
</p>

y
          
            n
            +
            1
          
        
        =
        
          a
          
            n
          
        
        
          y
          
            n
          
        
        +
        
          b
          
            n
          
        
        
          y
          
            n
            −
            1
          
        
      
    
    {\displaystyle y_{n+1}=a_{n}y_{n}+b_{n}y_{n-1}}
  
 for 
  
    
      
        n
        =
        1
        ,
        2
        ,
        .
        .
        .
        ,
      
    
    {\displaystyle n=1,2,...,}

<p>where the sequences 
  
    
      
        {
        
          a
          
            n
          
        
        }
      
    
    {\displaystyle \{a_{n}\}}
  
 and 
  
    
      
        {
        
          b
          
            n
          
        
        }
      
    
    {\displaystyle \{b_{n}\}}
  
, together with the initial values 
  
    
      
        
          y
          
            0
          
        
        ,
        
          y
          
            1
          
        
      
    
    {\displaystyle y_{0},y_{1}}
  
 govern the evolution of the sequence 
  
    
      
        {
        
          y
          
            n
          
        
        }
      
    
    {\displaystyle \{y_{n}\}}
  
.
</p>

Three-term recurrence relation open-in-new

Three-term recurrence relation