In mathematics, the indefinite product operator is the inverse operator of . It is like a discrete version of the indefinite product integral. Some authors use term discrete multiplicative integration
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.
If is a period of function then
Connection to indefinite sum
Indefinite product can be expressed in terms of indefinite sum:
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.
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)
- 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".
- Algorithms for Nonlinear Higher Order Difference Equations, Manuel Kauers
- 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