Indefinite product

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

\( Q(\prod _{x}f(x))=f(x)\,. \)

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

F ( x + 1 ) F ( x ) = f ( x ) . {\displaystyle {\frac {F(x+1)}{F(x)}}=f(x)\,.} {\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 is a period of function f(x) then

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

Connection to indefinite sum

Indefinite product can be expressed in terms of indefinite sum:

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

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.[1] e.g.

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

Rules

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

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

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

List of indefinite products

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

\( {\displaystyle \prod _{x}a=Ca^{x}} \)

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

\( \prod _{x}{\frac {x+1}{x}}=Cx \)

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

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

\( {\displaystyle \prod _{x}ax=Ca^{x}\Gamma (x)} \)

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

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

\( {\displaystyle \prod _{x}x^{x}=C\,e^{\zeta ^{\prime }(-1,x)-\zeta ^{\prime }(-1)}=C\,e^{\psi ^{(-2)}(z)+{\frac {z^{2}-z}{2}}-{\frac {z}{2}}\ln(2\pi )}=C\,\operatorname {K} (x)} \)

(see K-function)

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

(see Barnes G-function)

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

(see super-exponential function)

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

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

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

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

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

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

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

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

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