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.

Thus


 * $$Q\left( \prod_x f(x) \right) = f(x) \, .$$

More explicitly, if $\prod_x f(x) = F(x) $, then


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

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


 * $$\prod _x f(Tx)=C f(Tx)^{x-1} $$

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


 * $$\prod _x f(x)= \exp \left(\sum _x \ln f(x)\right) $$

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.


 * $$\prod_{k=1}^n f(k)$$.

Rules

 * $$\prod _x f(x)g(x) = \prod _x f(x)\prod _x g(x) $$


 * $$\prod _x f(x)^a = \left(\prod _x f(x)\right)^a $$


 * $$\prod _x a^{f(x)} = a^{\sum _x f(x)} $$

List of indefinite products
This is a list of indefinite products $\prod _x f(x) $. Not all functions have an indefinite product which can be expressed in elementary functions.


 * $$\prod _x a = C a^x $$


 * $$\prod _x x = C\, \Gamma (x) $$


 * $$\prod _x \frac{x+1}{x} = C x$$


 * $$\prod _x \frac{x+a}{x} = \frac{C\,\Gamma (x+a)}{\Gamma (x)}$$


 * $$\prod _x x^a = C\, \Gamma (x)^a $$


 * $$\prod _x ax = C a^x \Gamma (x) $$


 * $$\prod _x a^x = C a^{\frac{x}{2} (x-1)} $$


 * $$\prod _x a^{\frac{1}{x}} = C a^{\frac{\Gamma'(x)}{\Gamma(x)}} $$


 * $$\prod _x x^x= C\, e^{\zeta^\prime(-1,x)-\zeta^\prime(-1)}= C\,e^{\psi^{(-2)}(z)+\frac{z^2-z}{2}-\frac z2 \ln (2\pi)}= C\, \operatorname{K}(x) $$


 * (see K-function)


 * $$\prod _x \Gamma(x) = \frac{C\,\Gamma(x)^{x-1}}{\operatorname{K}(x)} = C\,\Gamma(x)^{x-1} e^{\frac z2 \ln (2\pi)-\frac{z^2-z}{2}-\psi^{(-2)}(z)}= C\, \operatorname{G}(x) $$


 * (see Barnes G-function)


 * $$\prod _x \operatorname{sexp}_a(x) = \frac{C\, (\operatorname{sexp}_a (x))'}{\operatorname{sexp}_a (x)(\ln a)^x} $$


 * (see super-exponential function)


 * $$\prod _x x+a = C\,\Gamma (x+a) $$


 * $$\prod _x ax+b = C\, a^x \Gamma \left(x+\frac{b}{a}\right) $$


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


 * $$\prod _x x^2+1 = C\, \Gamma (x-i) \Gamma (x+i) $$


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


 * $$\prod _x \csc x \sin (x+1) = C \sin x $$


 * $$\prod _x \sec x \cos (x+1) = C \cos x $$


 * $$\prod _x \cot x \tan (x+1) = C \tan x $$


 * $$\prod _x \tan x \cot (x+1) = C \cot x $$