Cassini and Catalan identities

Cassini's identity (sometimes called Simson's identity) and Catalan's identity are <a href="/facts/Identity_(mathematics)/LR46j4uR">mathematical identities</a> for the <a href="/facts/Fibonacci_number/mqxpAUQD">Fibonacci numbers</a>. Cassini's identity, a special case of Catalan's identity, states that for the nth Fibonacci number,

F
          
            n
            −
            1
          
        
        
          F
          
            n
            +
            1
          
        
        −
        
          F
          
            n
          
          
            2
          
        
        =
        (
        −
        1
        
          )
          
            n
          
        
        .
      
    
    {\displaystyle F_{n-1}F_{n+1}-F_{n}^{2}=(-1)^{n}.}

Note here 
 
 
 
 
 F
 
 0
 
 
 
 
 {\displaystyle F_{0}}
 
 is taken to be 0, and 
 
 
 
 
 F
 
 1
 
 
 
 
 {\displaystyle F_{1}}
 
 is taken to be 1.
Catalan's identity generalizes this:

F
          
            n
          
          
            2
          
        
        −
        
          F
          
            n
            −
            r
          
        
        
          F
          
            n
            +
            r
          
        
        =
        (
        −
        1
        
          )
          
            n
            −
            r
          
        
        
          F
          
            r
          
          
            2
          
        
        .
      
    
    {\displaystyle F_{n}^{2}-F_{n-r}F_{n+r}=(-1)^{n-r}F_{r}^{2}.}

Vajda's identity generalizes this:

F
          
            n
            +
            i
          
        
        
          F
          
            n
            +
            j
          
        
        −
        
          F
          
            n
          
        
        
          F
          
            n
            +
            i
            +
            j
          
        
        =
        (
        −
        1
        
          )
          
            n
          
        
        
          F
          
            i
          
        
        
          F
          
            j
          
        
        .
      
    
    {\displaystyle F_{n+i}F_{n+j}-F_{n}F_{n+i+j}=(-1)^{n}F_{i}F_{j}.}

Cassini and Catalan identities open-in-new

Cassini and Catalan identities