János Pintz

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

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:

$\prod _{x}f(x)=\exp (\sum _x\ln f(x))$

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. e.g.

$\prod _{k=1}^{n}f(k)$.

Rules

$\prod _{x}f(x)g(x)=\prod _{x}f(x)\prod _{x}g(x)$

$\prod _{x}f(x{)}^a={(\prod _{x}f(x))}^a$

$\prod _x{a}^{f(x)}=a^{\sum _{x}f(x)}$

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.

$\prod _{x}a=C{a}^x$

$\prod _{x}x=C\mathrm{\Gamma }(x)$

$\prod _x\frac{x+1}{x}=Cx$

$\prod _x\frac{x+a}{x}=\frac{C\mathrm{\Gamma }(x+a)}{\mathrm{\Gamma }(x)}$

$\prod _x{x}^a=C\mathrm{\Gamma }(x{)}^a$

$\prod _{x}ax=C{a}^x\mathrm{\Gamma }(x)$

$\prod _x{a}^x=C{a}^{\frac{x}{2}(x-1)}$

$\prod _x{a}^{\frac{1}{x}}=C{a}^{\frac{{\mathrm{\Gamma }}^{\prime }(x)}{\mathrm{\Gamma }(x)}}$

$\prod _x{x}^x=C{e}^{{\zeta }^{\prime }(-1,x)-{\zeta }^{\prime }(-1)}=C{e}^{{\psi }^{(-2)}(z)+\frac{z^2-z}{2}-\frac{z}{2}\ln (2\pi )}=C\mathrm{K}(x)$

(see K-function)

$\prod _x\mathrm{\Gamma }(x)=\frac{C\mathrm{\Gamma }(x{)}^{x-1}}{\mathrm{K}(x)}=C\mathrm{\Gamma }(x{)}^{x-1}{e}^{\frac{z}{2}\ln (2\pi )-\frac{z^2-z}{2}-{\psi }^{(-2)}(z)}=C\mathrm{G}(x)$

(see Barnes G-function)

$\prod _x{\mathrm{sexp}}_a(x)=\frac{C({\mathrm{sexp}}_a(x){)}^{\prime }}{{\mathrm{sexp}}_a(x)(\ln a{)}^x}$

(see super-exponential function)

$\prod _{x}x+a=C\mathrm{\Gamma }(x+a)$

$\prod _{x}ax+b=C{a}^x\mathrm{\Gamma }(x+\frac{b}{a})$

$\prod _{x}a{x}^2+bx=C{a}^x\mathrm{\Gamma }(x)\mathrm{\Gamma }(x+\frac{b}{a})$

$\prod _x{x}^2+1=C\mathrm{\Gamma }(x-i)\mathrm{\Gamma }(x+i)$

$\prod _{x}x+\frac{1}{x}=\frac{C\mathrm{\Gamma }(x-i)\mathrm{\Gamma }(x+i)}{\mathrm{\Gamma }(x)}$

$\prod _x\csc x\sin (x+1)=C\sin x$

$\prod _x\sec x\cos (x+1)=C\cos x$

$\prod _x\cot x\tan (x+1)=C\tan x$

$\prod _x\tan x\cot (x+1)=C\cot x$

See also

• Indefinite sum
• Product integral
• List of derivatives and integrals in alternative calculi

References

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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