q-theta function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the q-theta function (or modified Jacobi theta function) is a type of <a href="/facts/Q-series/keASLfpe">q-series</a> which is used to define <a href="/facts/Elliptic_hypergeometric_series/w3mEH4eT">elliptic hypergeometric series</a>.
 It is given by

θ
        (
        z
        ;
        q
        )
        :=
        
          ∏
          
            n
            =
            0
          
          
            ∞
          
        
        (
        1
        −
        
          q
          
            n
          
        
        z
        )
        
          (
          
            1
            −
            
              q
              
                n
                +
                1
              
            
            
              /
            
            z
          
          )
        
      
    
    {\displaystyle \theta (z;q):=\prod _{n=0}^{\infty }(1-q^{n}z)\left(1-q^{n+1}/z\right)}

where one takes 0 ≤ |q| < 1. It obeys the identities

θ
        (
        z
        ;
        q
        )
        =
        θ
        
          (
          
            
              
                q
                z
              
            
            ;
            q
          
          )
        
        =
        −
        z
        θ
        
          (
          
            
              
                1
                z
              
            
            ;
            q
          
          )
        
        .
      
    
    {\displaystyle \theta (z;q)=\theta \left({\frac {q}{z}};q\right)=-z\theta \left({\frac {1}{z}};q\right).}

It may also be expressed as:

θ
        (
        z
        ;
        q
        )
        =
        (
        z
        ;
        q
        
          )
          
            ∞
          
        
        (
        q
        
          /
        
        z
        ;
        q
        
          )
          
            ∞
          
        
      
    
    {\displaystyle \theta (z;q)=(z;q)_{\infty }(q/z;q)_{\infty }}

where 
 
 
 
 (
 ⋅
 ⋅
 
 )
 
 ∞
 
 
 
 
 {\displaystyle (\cdot \cdot )_{\infty }}
 
 is the <a href="/facts/Q-Pochhammer_symbol/keASLfpe">q-Pochhammer symbol</a>.

q-theta function open-in-new

q-theta function