Indefinite product

In mathematics, the indefinite product operator is the inverse operator of $Q(f(x)) = \frac{f(x+1)}{f(x)}$. It is like a discrete version of the indefinite product integral. Some authors use term discrete multiplicative integration^[1]

Thus

$Q( \prod_x f(x) )= f(x) \, .$

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.^[2] 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)

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

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

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

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

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

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

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

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

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

See also

• Indefinite sum
• Product integral
• List of derivatives and integrals in alternative calculi

References

1. N. Aliev, N. Azizi and M. Jahanshahi (2007) "Invariant functions for discrete derivatives and their applications to solve non-homogenous linear and non-linear difference equations".
2. Algorithms for Nonlinear Higher Order Difference Equations, Manuel Kauers

Further reading

• http://reference.wolfram.com/mathematica/ref/Product.html -Indefinite products with Mathematica
• http://www.math.rwth-aachen.de/MapleAnswers/660.html - 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