Indefinite product

In mathematics, the indefinite product operator is the inverse operator of Q(f(x)) = \frac{f(x%2B1)}{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%2B1)}{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.

Contents

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%2B1}{x} = C x
\prod _x \frac{x%2Ba}{x} = \frac{C\,\Gamma (x%2Ba)}{\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)%2B\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%2Ba = C\,\Gamma (x%2Ba) \,
\prod _x ax%2Bb = C\, a^x \Gamma \left(x%2B\frac{b}{a}\right) \,
\prod _x ax^2%2Bbx = C\,a^x \Gamma (x) \Gamma \left(x%2B\frac{b}{a}\right) \,
\prod _x x^2%2B1 = C\, \Gamma (x-i) \Gamma (x%2Bi)
\prod _x x%2B\frac {1}{x} = \frac{C\, \Gamma (x-i) \Gamma (x%2Bi)}{\Gamma (x)}
\prod _x \csc x \sin (x%2B1) = C \sin x \,
\prod _x \sec x \cos (x%2B1) = C \cos x \,
\prod _x \cot x \tan (x%2B1) = C \tan x \,
\prod _x \tan x \cot (x%2B1) = C \cot x \,

See also

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