In mathematics, the indefinite product operator is the inverse operator of . It is a discrete version of the geometric integral of geometric calculus.
Thus
More explicitly, if , then
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.
Connection to indefinite sum
Indefinite product can be expressed in terms of indefinite sum:
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.Algorithms for Nonlinear Higher Order Difference Equations, Manuel Kauers e.g.
.
Rules
List of indefinite products
This is a list of indefinite products . Not all functions have an indefinite product which can be expressed in elementary functions.
(see K-function)
(see Barnes G-function)
(see super-exponential function)
See also
Indefinite sum Product integral List of derivatives and integrals in alternative calculi Fractal derivative Viète's formula
References
Further reading 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