Quasi-polynomial

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a quasi-polynomial (pseudo-polynomial) is a generalization of <a href="/facts/Polynomial/Lzak8VVx">polynomials</a>. While the coefficients of a polynomial come from a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a>, the coefficients of quasi-polynomials are instead <a href="/facts/Periodic_function/Sm8lJQsF">periodic functions</a> with integral period. Quasi-polynomials appear throughout much of <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a> as the enumerators for various objects.
</p><p>A quasi-polynomial can be written as 
  
    
      
        q
        (
        k
        )
        =
        
          c
          
            d
          
        
        (
        k
        )
        
          k
          
            d
          
        
        +
        
          c
          
            d
            −
            1
          
        
        (
        k
        )
        
          k
          
            d
            −
            1
          
        
        +
        ⋯
        +
        
          c
          
            0
          
        
        (
        k
        )
      
    
    {\displaystyle q(k)=c_{d}(k)k^{d}+c_{d-1}(k)k^{d-1}+\cdots +c_{0}(k)}
  
, where 
  
    
      
        
          c
          
            i
          
        
        (
        k
        )
      
    
    {\displaystyle c_{i}(k)}
  
 is a periodic function with integral period. If 
  
    
      
        
          c
          
            d
          
        
        (
        k
        )
      
    
    {\displaystyle c_{d}(k)}
  
 is not identically zero, then the degree of 
  
    
      
        q
      
    
    {\displaystyle q}
  
 is 
  
    
      
        d
      
    
    {\displaystyle d}
  
. Equivalently, a function 
  
    
      
        f
        :
        
          N
        
        →
        
          N
        
      
    
    {\displaystyle f\colon \mathbb {N} \to \mathbb {N} }
  
 is a quasi-polynomial if there exist polynomials 
  
    
      
        
          p
          
            0
          
        
        ,
        …
        ,
        
          p
          
            s
            −
            1
          
        
      
    
    {\displaystyle p_{0},\dots ,p_{s-1}}
  
 such that 
  
    
      
        f
        (
        n
        )
        =
        
          p
          
            i
          
        
        (
        n
        )
      
    
    {\displaystyle f(n)=p_{i}(n)}
  
 when 
  
    
      
        i
        ≡
        n
        
          mod
          
            s
          
        
      
    
    {\displaystyle i\equiv n{\bmod {s}}}
  
. The polynomials 
  
    
      
        
          p
          
            i
          
        
      
    
    {\displaystyle p_{i}}
  
 are called the constituents of 
  
    
      
        f
      
    
    {\displaystyle f}
  
.
</p>

Quasi-polynomial open-in-new

Quasi-polynomial