Let H be a hypergeometric term of n. The EfficientRepresentation[i](H,n) calling sequence constructs the ith efficient representation of H of the form $H\left(n\right)={\mathrm{\alpha }}^{n}V\left(n\right)Q\left(n\right)$ where alpha is a constant, $Q\left(n\right)$ is a product of Gamma-function values and their reciprocals. Additionally,

$Q\left(n\right)$ has the minimal number of factors,

$V\left(n\right)$ is a rational function which is minimal in one sense or another, depending on the particular rational canonical form chosen to represent the certificate of $H\left(n\right)$.

If $i=1$ then $\mathrm{degree}\left(\mathrm{denom}\left(V\right)\right)$ is minimal;

if $i=2$ then $\mathrm{degree}\left(\mathrm{numer}\left(V\right)\right)$ is minimal;

if $i=3$ then $\mathrm{degree}\left(\mathrm{numer}\left(V\right)\right)+\mathrm{degree}\left(\mathrm{denom}\left(V\right)\right)$ is minimal, and $\mathrm{degree}\left(\mathrm{denom}\left(V\right)\right)$ is minimal;

if $i=4$ then $\mathrm{degree}\left(\mathrm{numer}\left(V\right)\right)+\mathrm{degree}\left(\mathrm{denom}\left(V\right)\right)$ is minimal, and $\mathrm{degree}\left(\mathrm{numer}\left(V\right)\right)$ is minimal.

If EfficientRepresentation is called without an index, the first efficient representation is constructed.

$\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right)\:$

$H≔\mathrm{Product}\left(\frac{\frac{1}{2}\left(3k^2+6k+4\right)\left(2k+3\right)\left(4k+5\right)\left(k+1\right)\left(4k+3\right)}{k\left(4k-1\right)\left(2k-1\right)\left(4k-3\right)\left(2k+5\right)\left(k+2\right)\left(3k^2+1\right)}\,k=1..n-1\right)$

$H≔\prod _{k=1}^{n-1}\frac{\left(3k^2+6k+4\right)\left(2k+3\right)\left(4k+5\right)\left(k+1\right)\left(4k+3\right)}{2k\left(4k-1\right)\left(2k-1\right)\left(4k-3\right)\left(2k+5\right)\left(k+2\right)\left(3k^2+1\right)}$

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

$\frac{64\sqrt{\mathrm{\pi }}{\left(\frac{1}{4}\right)}^n\left(n^2+\frac{1}{3}\right)n\left(n-\frac{1}{4}\right)\left(n+\frac{1}{2}\right)\left(n+\frac{1}{4}\right)\left(n-\frac{1}{2}\right)\left(n-\frac{3}{4}\right)}{\mathrm{\Gamma }\left(n+\frac{5}{2}\right)\mathrm{\Gamma }\left(n+2\right)}$

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

$\frac{64\sqrt{\mathrm{\pi }}{\left(\frac{1}{4}\right)}^n\left(n^2+\frac{1}{3}\right)\left(n-\frac{1}{4}\right)\left(n+\frac{1}{4}\right)\left(n-\frac{3}{4}\right)}{\left(n+\frac{3}{2}\right)\left(n+1\right)\mathrm{\Gamma }\left(n\right)\mathrm{\Gamma }\left(n-\frac{1}{2}\right)}$

$\mathrm{EfficientRepresentation}\left[3\right]\left(H\,n\right)$

$\frac{64\sqrt{\mathrm{\pi }}{\left(\frac{1}{4}\right)}^n\left(n-\frac{1}{4}\right)n\left(n^2+\frac{1}{3}\right)\left(n+\frac{1}{4}\right)\left(n-\frac{3}{4}\right)}{\left(n+\frac{3}{2}\right)\mathrm{\Gamma }\left(n-\frac{1}{2}\right)\mathrm{\Gamma }\left(n+2\right)}$

$\mathrm{EfficientRepresentation}\left[4\right]\left(H\,n\right)$

$\frac{64\sqrt{\mathrm{\pi }}{\left(\frac{1}{4}\right)}^n\left(n^2+\frac{1}{3}\right)\left(n-\frac{1}{4}\right)\left(n+\frac{1}{4}\right)\left(n-\frac{3}{4}\right)}{\left(n+\frac{3}{2}\right)\left(n+1\right)\mathrm{\Gamma }\left(n\right)\mathrm{\Gamma }\left(n-\frac{1}{2}\right)}$

$\mathrm{RegularGammaForm}\left(H\,n\right)$

$\frac{64\sqrt{\mathrm{\pi }}{\left(\frac{1}{2}\right)}^n\mathrm{\Gamma }\left(n+1-\frac{\mathrm{I}\sqrt{3}}{3}\right)\mathrm{\Gamma }\left(n+1+\frac{\mathrm{I}\sqrt{3}}{3}\right)\mathrm{\Gamma }\left(n+\frac{3}{2}\right)\mathrm{\Gamma }\left(n+\frac{5}{4}\right)\mathrm{\Gamma }\left(n+1\right)\mathrm{\Gamma }\left(n+\frac{3}{4}\right)}{2^n\mathrm{\Gamma }\left(n\right)\mathrm{\Gamma }\left(n-\frac{1}{4}\right)\mathrm{\Gamma }\left(n-\frac{1}{2}\right)\mathrm{\Gamma }\left(n-\frac{3}{4}\right)\mathrm{\Gamma }\left(n+\frac{5}{2}\right)\mathrm{\Gamma }\left(n+2\right)\mathrm{\Gamma }\left(n-\frac{\mathrm{I}\sqrt{3}}{3}\right)\mathrm{\Gamma }\left(n+\frac{\mathrm{I}\sqrt{3}}{3}\right)}$

Abramov, S.A.; Le, H.Q.; and Petkovsek, M. "Rational Canonical Forms and Efficient Representations of Hypergeometric Terms." Proc. ISSAC'2003, pp. 7-14. 2003.