---
title: "Indefinite product"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Indefinite_product"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T08:15:08.101960+00:00"
instance: "kb-cron"
---

In mathematics, the indefinite product operator is the inverse operator of 
  
    
      
        Q
        (
        f
        (
        x
        )
        )
        =
        
          
            
              f
              (
              x
              +
              1
              )
            
            
              f
              (
              x
              )
            
          
        
      
    
    {\textstyle Q(f(x))={\frac {f(x+1)}{f(x)}}}
  
. It is a discrete version of the geometric integral of geometric calculus, one of the non-Newtonian calculi.
Thus

  
    
      
        Q
        
          (
          
            
              ∏
              
                x
              
            
            f
            (
            x
            )
          
          )
        
        =
        f
        (
        x
        )
        
        .
      
    
    {\displaystyle Q\left(\prod _{x}f(x)\right)=f(x)\,.}
  

More explicitly, if 
  
    
      
        
          ∏
          
            x
          
        
        f
        (
        x
        )
        =
        F
        (
        x
        )
      
    
    {\textstyle \prod _{x}f(x)=F(x)}
  
, then

  
    
      
        
          
            
              F
              (
              x
              +
              1
              )
            
            
              F
              (
              x
              )
            
          
        
        =
        f
        (
        x
        )
        
        .
      
    
    {\displaystyle {\frac {F(x+1)}{F(x)}}=f(x)\,.}
  

If F(x) is a solution of this functional equation for a given f(x), then so is CF(x) for any constant C. Therefore, each indefinite product actually represents a family of functions, differing by a multiplicative constant.


== Period rule ==
If 
  
    
      
        T
      
    
    {\displaystyle T}
  
 is a period of function 
  
    
      
        f
        (
        x
        )
      
    
    {\displaystyle f(x)}
  
 then

  
    
      
        
          ∏
          
            x
          
        
        f
        (
        T
        x
        )
        =
        C
        f
        (
        T
        x
        
          )
          
            x
            −
            1
          
        
      
    
    {\displaystyle \prod _{x}f(Tx)=Cf(Tx)^{x-1}}
  


== Connection to indefinite sum ==
Indefinite product can be expressed in terms of indefinite sum:

  
    
      
        
          ∏
          
            x
          
        
        f
        (
        x
        )
        =
        exp
        ⁡
        
          (
          
            
              ∑
              
                x
              
            
            ln
            ⁡
            f
            (
            x
            )
          
          )
        
      
    
    {\displaystyle \prod _{x}f(x)=\exp \left(\sum _{x}\ln f(x)\right)}
  


== Alternative usage ==
Some authors use the phrase "indefinite product" in a slightly different but related way to describe a product in which the numerical value of the upper limit is not given.  e.g.

  
    
      
        
          ∏
          
            k
            =
            1
          
          
            n
          
        
        f
        (
        k
        )
      
    
    {\displaystyle \prod _{k=1}^{n}f(k)}
  
.


== Rules ==

  
    
      
        
          ∏
          
            x
          
        
        f
        (
        x
        )
        g
        (
        x
        )
        =
        
          ∏
          
            x
          
        
        f
        (
        x
        )
        
          ∏
          
            x
          
        
        g
        (
        x
        )
      
    
    {\displaystyle \prod _{x}f(x)g(x)=\prod _{x}f(x)\prod _{x}g(x)}
  

  
    
      
        
          ∏
          
            x
          
        
        f
        (
        x
        
          )
          
            a
          
        
        =
        
          
            (
            
              
                ∏
                
                  x
                
              
              f
              (
              x
              )
            
            )
          
          
            a
          
        
      
    
    {\displaystyle \prod _{x}f(x)^{a}=\left(\prod _{x}f(x)\right)^{a}}
  

  
    
      
        
          ∏
          
            x
          
        
        
          a
          
            f
            (
            x
            )
          
        
        =
        
          a
          
            
              ∑
              
                x
              
            
            f
            (
            x
            )
          
        
      
    
    {\displaystyle \prod _{x}a^{f(x)}=a^{\sum _{x}f(x)}}
  


== List of indefinite products ==
This is a list of indefinite products 
  
    
      
        
          ∏
          
            x
          
        
        f
        (
        x
        )
      
    
    {\textstyle \prod _{x}f(x)}
  
. Not all functions have an indefinite product which can be expressed in elementary functions.

  
    
      
        
          ∏
          
            x
          
        
        a
        =
        C
        
          a
          
            x
          
        
      
    
    {\displaystyle \prod _{x}a=Ca^{x}}
  

  
    
      
        
          ∏
          
            x
          
        
        x
        =
        C
        
        Γ
        (
        x
        )
      
    
    {\displaystyle \prod _{x}x=C\,\Gamma (x)}
  

  
    
      
        
          ∏
          
            x
          
        
        
          
            
              x
              +
              1
            
            x
          
        
        =
        C
        x
      
    
    {\displaystyle \prod _{x}{\frac {x+1}{x}}=Cx}
  

  
    
      
        
          ∏
          
            x
          
        
        
          
            
              x
              +
              a
            
            x
          
        
        =
        
          
            
              C
              
              Γ
              (
              x
              +
              a
              )
            
            
              Γ
              (
              x
              )
            
          
        
      
    
    {\displaystyle \prod _{x}{\frac {x+a}{x}}={\frac {C\,\Gamma (x+a)}{\Gamma (x)}}}
  

  
    
      
        
          ∏
          
            x
          
        
        
          x
          
            a
          
        
        =
        C
        
        Γ
        (
        x
        
          )
          
            a
          
        
      
    
    {\displaystyle \prod _{x}x^{a}=C\,\Gamma (x)^{a}}
  

  
    
      
        
          ∏
          
            x
          
        
        a
        x
        =
        C
        
          a
          
            x
          
        
        Γ
        (
        x
        )
      
    
    {\displaystyle \prod _{x}ax=Ca^{x}\Gamma (x)}
  

  
    
      
        
          ∏
          
            x
          
        
        
          a
          
            x
          
        
        =
        C
        
          a
          
            
              
                x
                2
              
            
            (
            x
            −
            1
            )
          
        
      
    
    {\displaystyle \prod _{x}a^{x}=Ca^{{\frac {x}{2}}(x-1)}}
  

  
    
      
        
          ∏
          
            x
          
        
        
          a
          
            
              1
              x
            
          
        
        =
        C
        
          a
          
            
              
                
                  Γ
                  ′
                
                (
                x
                )
              
              
                Γ
                (
                x
                )
              
            
          
        
      
    
    {\displaystyle \prod _{x}a^{\frac {1}{x}}=Ca^{\frac {\Gamma '(x)}{\Gamma (x)}}}
  

  
    
      
        
          ∏
          
            x
          
        
        
          x
          
            x
          
        
        =
        C
        
        
          e
          
            
              ζ
              
                ′
              
            
            (
            −
            1
            ,
            x
            )
            −
            
              ζ
              
                ′
              
            
            (
            −
            1
            )
          
        
        =
        C
        
        
          e
          
            
              ψ
              
                (
                −
                2
                )
              
            
            (
            z
            )
            +
            
              
                
                  
                    z
                    
                      2
                    
                  
                  −
                  z
                
                2
              
            
            −
            
              
                z
                2
              
            
            ln
            ⁡
            (
            2
            π
            )
          
        
        =
        C
        
        K
        ⁡
        (
        x
        )
      
    
    {\displaystyle \prod _{x}x^{x}=C\,e^{\zeta ^{\prime }(-1,x)-\zeta ^{\prime }(-1)}=C\,e^{\psi ^{(-2)}(z)+{\frac {z^{2}-z}{2}}-{\frac {z}{2}}\ln(2\pi )}=C\,\operatorname {K} (x)}
  

(see K-function)

  
    
      
        
          ∏
          
            x
          
        
        Γ
        (
        x
        )
        =
        
          
            
              C
              
              Γ
              (
              x
              
                )
                
                  x
                  −
                  1
                
              
            
            
              K
              ⁡
              (
              x
              )
            
          
        
        =
        C
        
        Γ
        (
        x
        
          )
          
            x
            −
            1
          
        
        
          e
          
            
              
                z
                2
              
            
            ln
            ⁡
            (
            2
            π
            )
            −
            
              
                
                  
                    z
                    
                      2
                    
                  
                  −
                  z
                
                2
              
            
            −
            
              ψ
              
                (
                −
                2
                )
              
            
            (
            z
            )
          
        
        =
        C
        
        G
        ⁡
        (
        x
        )
      
    
    {\displaystyle \prod _{x}\Gamma (x)={\frac {C\,\Gamma (x)^{x-1}}{\operatorname {K} (x)}}=C\,\Gamma (x)^{x-1}e^{{\frac {z}{2}}\ln(2\pi )-{\frac {z^{2}-z}{2}}-\psi ^{(-2)}(z)}=C\,\operatorname {G} (x)}
  

(see Barnes G-function)

  
    
      
        
          ∏
          
            x
          
        
        
          sexp
          
            a
          
        
        ⁡
        (
        x
        )
        =
        
          
            
              C
              
              (
              
                sexp
                
                  a
                
              
              ⁡
              (
              x
              )
              
                )
                ′
              
            
            
              
                sexp
                
                  a
                
              
              ⁡
              (
              x
              )
              (
              ln
              ⁡
              a
              
                )
                
                  x
                
              
            
          
        
      
    
    {\displaystyle \prod _{x}\operatorname {sexp} _{a}(x)={\frac {C\,(\operatorname {sexp} _{a}(x))'}{\operatorname {sexp} _{a}(x)(\ln a)^{x}}}}
  

(see super-exponential function)

  
    
      
        
          ∏
          
            x
          
        
        x
        +
        a
        =
        C
        
        Γ
        (
        x
        +
        a
        )
      
    
    {\displaystyle \prod _{x}x+a=C\,\Gamma (x+a)}
  

  
    
      
        
          ∏
          
            x
          
        
        a
        x
        +
        b
        =
        C
        
        
          a
          
            x
          
        
        Γ
        
          (
          
            x
            +
            
              
                b
                a
              
            
          
          )
        
      
    
    {\displaystyle \prod _{x}ax+b=C\,a^{x}\Gamma \left(x+{\frac {b}{a}}\right)}
  

  
    
      
        
          ∏
          
            x
          
        
        a
        
          x
          
            2
          
        
        +
        b
        x
        =
        C
        
        
          a
          
            x
          
        
        Γ
        (
        x
        )
        Γ
        
          (
          
            x
            +
            
              
                b
                a
              
            
          
          )
        
      
    
    {\displaystyle \prod _{x}ax^{2}+bx=C\,a^{x}\Gamma (x)\Gamma \left(x+{\frac {b}{a}}\right)}
  

  
    
      
        
          ∏
          
            x
          
        
        
          x
          
            2
          
        
        +
        1
        =
        C
        
        Γ
        (
        x
        −
        i
        )
        Γ
        (
        x
        +
        i
        )
      
    
    {\displaystyle \prod _{x}x^{2}+1=C\,\Gamma (x-i)\Gamma (x+i)}
  

  
    
      
        
          ∏
          
            x
          
        
        x
        +
        
          
            1
            x
          
        
        =
        
          
            
              C
              
              Γ
              (
              x
              −
              i
              )
              Γ
              (
              x
              +
              i
              )
            
            
              Γ
              (
              x
              )
            
          
        
      
    
    {\displaystyle \prod _{x}x+{\frac {1}{x}}={\frac {C\,\Gamma (x-i)\Gamma (x+i)}{\Gamma (x)}}}
  

  
    
      
        
          ∏
          
            x
          
        
        csc
        ⁡
        x
        sin
        ⁡
        (
        x
        +
        1
        )
        =
        C
        sin
        ⁡
        x
      
    
    {\displaystyle \prod _{x}\csc x\sin(x+1)=C\sin x}
  

  
    
      
        
          ∏
          
            x
          
        
        sec
        ⁡
        x
        cos
        ⁡
        (
        x
        +
        1
        )
        =
        C
        cos
        ⁡
        x
      
    
    {\displaystyle \prod _{x}\sec x\cos(x+1)=C\cos x}
  

  
    
      
        
          ∏
          
            x
          
        
        cot
        ⁡
        x
        tan
        ⁡
        (
        x
        +
        1
        )
        =
        C
        tan
        ⁡
        x
      
    
    {\displaystyle \prod _{x}\cot x\tan(x+1)=C\tan x}
  

  
    
      
        
          ∏
          
            x
          
        
        tan
        ⁡
        x
        cot
        ⁡
        (
        x
        +
        1
        )
        =
        C
        cot
        ⁡
        x
      
    
    {\displaystyle \prod _{x}\tan x\cot(x+1)=C\cot x}
  


== See also ==
Indefinite sum
Product integral
List of derivatives and integrals in alternative calculi
Fractal derivative
Viète's formula


== References ==


== Further reading ==
http://reference.wolfram.com/mathematica/ref/Product.html -Indefinite products with Mathematica
[1] - bug in Maple V to Maple 8 handling of indefinite product
Markus Müller. How to Add a Non-Integer Number of Terms, and How to Produce Unusual Infinite Summations
Markus Mueller, Dierk Schleicher. Fractional Sums and Euler-like Identities