The MinimalRepresentation[1](H,n,k) and MinimalRepresentation[2](H,n,k) functions construct the first and second minimal representations for H, where H be a hypergeometric term in n. respectively.

If $H\left(n\right)$ is a hypergeometric term such that $\frac{H\left(n+1\right)}{H\left(n\right)}=R\left(n\right)$, a rational function in n for all $n_0\le n$, then $H\left(n\right)=H\left(n_0\right)\left(\prod _{k=n_0}^{n-1}R\left(k\right)\right)$, $n_0\le n$. If $z,r,s,u,v$ is a rational normal form of $R\left(n\right)$, then $H\left(n\right)=\frac{H\left(n_0\right)z^{n}F\left(n\right)\left(\prod _{k=n_0}^{n-1}\frac{r\left(k\right)}{s\left(k\right)}\right)}{z^{n_0}F\left(n_0\right)}$, where $F=\frac{u}{v}$.

Note: $r\left(k\right)$ and $s\left(k\right)$ are of minimal possible degrees.

The first and second minimal representations of $H\left(n\right)$ are constructed from the first and second canonical forms of $R\left(n\right)$, respectively.

This function is part of the RationalNormalForms package, and so it can be used in the form MinimalRepresentation(..) only after executing the command with(RationalNormalForms). However, it can always be accessed through the long form of the command by using RationalNormalForms[MinimalRepresentation](..).

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

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

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

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

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

$\mathrm{MinimalRepresentation}\left[2\right]\left(H\,n\,k\right)$

$\frac{4{\left(\frac{27}{4}\right)}^n\left(\prod _{k=2}^{n-1}\frac{\left(k+\frac{2}{3}\right)\left(k+\frac{4}{3}\right)}{\left(k+\frac{9}{2}\right)\left(k-1\right)}\right)}{24057{\left(n+3\right)}^2{\left(n+2\right)}^2\left(n+1\right)n}$

Abramov, S., and Petkovsek, M. "Canonical Representations of Hypergeometric Terms." In Proceedings of FPSAC '01, 1-10. Edited by H. Barcelo and V. Welker. Tucson: University of Arizona Press, 2001.