Indefinite product

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

Thus

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

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

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

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.