Indefinite product explained

In mathematics, the indefinite product operator is the inverse operator of $Q(f(x))\; =\; \backslash frac$. 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 $\backslash prod\_x\; f(x)\; =\; F(x)$, then

(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.^[1] e.g.

\prod ^{n} _{k =1} 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 $\backslash prod\; \_x\; f(x)$. 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} (x +1)/(x) = C x \prod _{x} (x + a)/(x) = (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 ^{(x)/((x -1 ) 2)} \prod _{x} a ^{(1)/(x)} = C a ^{(\Gamma'(x))/(\Gamma(x))} \prod _{x} x ^{x=} C e ^{\zeta ^{\prime(-1,x)-\zeta} ^{\prime(-1)}} = Ce ^{\psi ^{( -2 )} (z) + (z ^{2-z})/(2) - (z 2)/(ln) ( 2 \pi)} = C \operatorname{K}(x)

(see K-function)

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

(see Barnes G-function)

\prod _{x} \operatorname{sexp} _{a(x)} = (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 + (b)/(a) \right) \prod _{x} ax ^{2+bx} = Ca ^{x} \Gamma (x) \Gamma \left(x + (b)/(a) \right) \prod _{x} x ^{2+1} = C \Gamma (x - i) \Gamma (x + i) \prod _{x} x + (1)/(x) = (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
• Fractal derivative

Further reading

• http://reference.wolfram.com/mathematica/ref/Product.html -Indefinite products with Mathematica
• https://archive.today/20070812032158/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

External links

• Non-Newtonian calculus website

Notes and References

1. http://www.risc.uni-linz.ac.at/people/mkauers/publications/kauers05c.pdf Algorithms for Nonlinear Higher Order Difference Equations