Reciprocal polynomial

In <a href="/facts/Algebra/nMdqczjo">algebra</a>, given a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a>

p
        (
        x
        )
        =
        
          a
          
            0
          
        
        +
        
          a
          
            1
          
        
        x
        +
        
          a
          
            2
          
        
        
          x
          
            2
          
        
        +
        ⋯
        +
        
          a
          
            n
          
        
        
          x
          
            n
          
        
        ,
      
    
    {\displaystyle p(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},}

with <a href="/facts/Coefficient/5Hwq2Wuq">coefficients</a> from an arbitrary <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>, its reciprocal polynomial or reflected polynomial, denoted by p∗ or pR, is the polynomial

p
          
            ∗
          
        
        (
        x
        )
        =
        
          a
          
            n
          
        
        +
        
          a
          
            n
            −
            1
          
        
        x
        +
        ⋯
        +
        
          a
          
            0
          
        
        
          x
          
            n
          
        
        =
        
          x
          
            n
          
        
        p
        (
        
          x
          
            −
            1
          
        
        )
        .
      
    
    {\displaystyle p^{*}(x)=a_{n}+a_{n-1}x+\cdots +a_{0}x^{n}=x^{n}p(x^{-1}).}

That is, the coefficients of p∗ are the coefficients of p in reverse order. Reciprocal polynomials arise naturally in <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> as the <a href="/facts/Characteristic_polynomial/7m8roDqo">characteristic polynomial</a> of the <a href="/facts/Inverse_of_a_matrix/fPqXk3V8">inverse of a matrix</a>.
In the special case where the field is the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>, when

p
        (
        z
        )
        =
        
          a
          
            0
          
        
        +
        
          a
          
            1
          
        
        z
        +
        
          a
          
            2
          
        
        
          z
          
            2
          
        
        +
        ⋯
        +
        
          a
          
            n
          
        
        
          z
          
            n
          
        
        ,
      
    
    {\displaystyle p(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots +a_{n}z^{n},}

the conjugate reciprocal polynomial, denoted p†, is defined by,

p
          
            †
          
        
        (
        z
        )
        =
        
          
            
              a
              
                n
              
            
            ¯
          
        
        +
        
          
            
              a
              
                n
                −
                1
              
            
            ¯
          
        
        z
        +
        ⋯
        +
        
          
            
              a
              
                0
              
            
            ¯
          
        
        
          z
          
            n
          
        
        =
        
          z
          
            n
          
        
        
          
            
              p
              (
              
                
                  
                    
                      z
                      ¯
                    
                  
                
                
                  −
                  1
                
              
              )
            
            ¯
          
        
        ,
      
    
    {\displaystyle p^{\dagger }(z)={\overline {a_{n}}}+{\overline {a_{n-1}}}z+\cdots +{\overline {a_{0}}}z^{n}=z^{n}{\overline {p({\bar {z}}^{-1})}},}

where 
 
 
 
 
 
 
 a
 
 i
 
 
 ¯
 
 
 
 
 {\displaystyle {\overline {a_{i}}}}
 
 denotes the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> of 
 
 
 
 
 a
 
 i
 
 
 
 
 {\displaystyle a_{i}}
 
, and is also called the reciprocal polynomial when no confusion can arise.
A polynomial p is called self-reciprocal or palindromic if p(x) = p∗(x).
The coefficients of a self-reciprocal polynomial satisfy ai = an−i for all i.

Reciprocal polynomial open-in-new

Reciprocal polynomial