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)=Cf(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=Ca^{x}

\prod _{x}x=C\,\Gamma (x)

\prod _{x}{\frac {x+1}{x}}=Cx

\prod _{x}{\frac {x+a}{x}}={\frac {C\,\Gamma (x+a)}{\Gamma (x)}}

\prod _{x}x^{a}=C\,\Gamma (x)^{a}

\prod _{x}ax=Ca^{x}\Gamma (x)

\prod _{x}a^{x}=Ca^{{\frac {x}{2}}(x-1)}

\prod _{x}a^{\frac {1}{x}}=Ca^{\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 {z}{2}}\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 {z}{2}}\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

References

↑ 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

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

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