The RationalNormalForms package is used to solve the following problems:

Construct the polynomial normal form of a rational function.

Construct the rational canonical forms of a rational function.

Construct a minimal representation of a hypergeometric term.

Each command in the RationalNormalForms package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

The long form, RationalNormalForms:-command, is always available. The short form can be used after loading the package.

The following is a list of available commands.

To display the help page for a particular RationalNormalForms command, see Getting Help with a Command in a Package.

$\mathrm{with}\left(\mathrm{RationalNormalForms}\right)\:$

$F≔\frac{\left(n^2-1\right)\left(3n+1\right)!}{\left(n+3\right)!\left(2n+7\right)!}$

$F≔\frac{\left(n^2-1\right)\left(3n+1\right)!}{\left(n+3\right)!\left(2n+7\right)!}$

$\mathrm{IsHypergeometricTerm}\left(F\,n\,'\mathrm{certificate}'\right)$

$\mathrm{true}$

$\mathrm{certificate}$

$\frac{3\left(3n+2\right)\left(3n+4\right)n\left(n+2\right)}{2\left(2n+9\right){\left(n+4\right)}^2\left(n-1\right)}$

$z,r,s,u,v≔\mathrm{RationalCanonicalForm}\left[1\right]\left(\mathrm{certificate}\,n\right)$

$z,r,s,u,v≔\frac{27}{4},\left(n+\frac{2}{3}\right)\left(n+\frac{4}{3}\right),\left(n+\frac{9}{2}\right)\left(n+4\right),n-1,\left(n+3\right)\left(n+2\right)$

$\mathrm{MinimalRepresentation}\left[1\right]\left(F\,n\,k\right)$

$\frac{{\left(\frac{27}{4}\right)}^n\left(n-1\right)\left(\prod _{k=2}^{n-1}\frac{\left(k+\frac{2}{3}\right)\left(k+\frac{4}{3}\right)}{\left(k+\frac{9}{2}\right)\left(k+4\right)}\right)}{721710\left(n+3\right)\left(n+2\right)}$

Abramov, S., and Petkovsek, M. "Canonical Representations of Hypergeometric Terms." FPSAC. 2000.