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SumTools[Hypergeometric]

 EfficientRepresentation
 construct the four efficient representations of a hypergeometric term

 Calling Sequence EfficientRepresentation[1](H, n) EfficientRepresentation[2](H, n) EfficientRepresentation[3](H, n) EfficientRepresentation[4](H, n)

Parameters

 H - hypergeometric term of n n - variable

Description

 • 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,
 1 $Q\left(n\right)$ has the minimal number of factors,
 2 $V\left(n\right)$ 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.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}[\mathrm{Hypergeometric}]\right):$
 > $H≔{\prod }_{k=1}^{n-1}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{1\left(3{k}^{2}+6k+4\right)\left(2k+3\right)\left(4k+5\right)\left(k+1\right)\left(4k+3\right)}{2k\left(4k-1\right)\left(2k-1\right)\left(4k-3\right)\left(2k+5\right)\left(k+2\right)\left(3{k}^{2}+1\right)}\right)$
 ${H}{:=}{\prod }_{{k}{=}{1}}^{{n}{-}{1}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left(\frac{{1}}{{2}}{}\frac{\left({3}{}{{k}}^{{2}}{+}{6}{}{k}{+}{4}\right){}\left({2}{}{k}{+}{3}\right){}\left({4}{}{k}{+}{5}\right){}\left({k}{+}{1}\right){}\left({4}{}{k}{+}{3}\right)}{{k}{}\left({4}{}{k}{-}{1}\right){}\left({2}{}{k}{-}{1}\right){}\left({4}{}{k}{-}{3}\right){}\left({2}{}{k}{+}{5}\right){}\left({k}{+}{2}\right){}\left({3}{}{{k}}^{{2}}{+}{1}\right)}\right)$ (1)
 > $\mathrm{EfficientRepresentation}[1]\left(H,n\right)$
 $\frac{{64}{}\sqrt{{\mathrm{π}}}{}{\left(\frac{{1}}{{4}}\right)}^{{n}}{}\left({{n}}^{{2}}{+}\frac{{1}}{{3}}\right){}{n}{}\left({n}{-}\frac{{1}}{{4}}\right){}\left({n}{+}\frac{{1}}{{2}}\right){}\left({n}{+}\frac{{1}}{{4}}\right){}\left({n}{-}\frac{{1}}{{2}}\right){}\left({n}{-}\frac{{3}}{{4}}\right)}{{\mathrm{Γ}}{}\left({n}{+}\frac{{5}}{{2}}\right){}{\mathrm{Γ}}{}\left({n}{+}{2}\right)}$ (2)
 > $\mathrm{EfficientRepresentation}[2]\left(H,n\right)$
 $\frac{{64}{}\sqrt{{\mathrm{π}}}{}{\left(\frac{{1}}{{4}}\right)}^{{n}}{}\left({{n}}^{{2}}{+}\frac{{1}}{{3}}\right){}\left({n}{-}\frac{{1}}{{4}}\right){}\left({n}{+}\frac{{1}}{{4}}\right){}\left({n}{-}\frac{{3}}{{4}}\right)}{\left({n}{+}\frac{{3}}{{2}}\right){}\left({n}{+}{1}\right){}{\mathrm{Γ}}{}\left({n}\right){}{\mathrm{Γ}}{}\left({n}{-}\frac{{1}}{{2}}\right)}$ (3)
 > $\mathrm{EfficientRepresentation}[3]\left(H,n\right)$
 $\frac{{64}{}\sqrt{{\mathrm{π}}}{}{\left(\frac{{1}}{{4}}\right)}^{{n}}{}\left({{n}}^{{2}}{+}\frac{{1}}{{3}}\right){}{n}{}\left({n}{-}\frac{{1}}{{4}}\right){}\left({n}{+}\frac{{1}}{{4}}\right){}\left({n}{-}\frac{{3}}{{4}}\right)}{\left({n}{+}\frac{{3}}{{2}}\right){}{\mathrm{Γ}}{}\left({n}{+}{2}\right){}{\mathrm{Γ}}{}\left({n}{-}\frac{{1}}{{2}}\right)}$ (4)
 > $\mathrm{EfficientRepresentation}[4]\left(H,n\right)$
 $\frac{{64}{}\sqrt{{\mathrm{π}}}{}{\left(\frac{{1}}{{4}}\right)}^{{n}}{}\left({{n}}^{{2}}{+}\frac{{1}}{{3}}\right){}\left({n}{-}\frac{{1}}{{4}}\right){}\left({n}{+}\frac{{1}}{{4}}\right){}\left({n}{-}\frac{{3}}{{4}}\right)}{\left({n}{+}\frac{{3}}{{2}}\right){}\left({n}{+}{1}\right){}{\mathrm{Γ}}{}\left({n}\right){}{\mathrm{Γ}}{}\left({n}{-}\frac{{1}}{{2}}\right)}$ (5)
 > $\mathrm{RegularGammaForm}\left(H,n\right)$
 $\frac{{64}{}\sqrt{{\mathrm{π}}}{}{\left(\frac{{1}}{{2}}\right)}^{{n}}{}{\mathrm{Γ}}{}\left({n}{+}{1}{-}\frac{{1}}{{3}}{}{I}{}\sqrt{{3}}\right){}{\mathrm{Γ}}{}\left({n}{+}{1}{+}\frac{{1}}{{3}}{}{I}{}\sqrt{{3}}\right){}{\mathrm{Γ}}{}\left({n}{+}\frac{{3}}{{2}}\right){}{\mathrm{Γ}}{}\left({n}{+}\frac{{5}}{{4}}\right){}{\mathrm{Γ}}{}\left({n}{+}{1}\right){}{\mathrm{Γ}}{}\left({n}{+}\frac{{3}}{{4}}\right)}{{{2}}^{{n}}{}{\mathrm{Γ}}{}\left({n}\right){}{\mathrm{Γ}}{}\left({n}{-}\frac{{1}}{{4}}\right){}{\mathrm{Γ}}{}\left({n}{-}\frac{{1}}{{2}}\right){}{\mathrm{Γ}}{}\left({n}{-}\frac{{3}}{{4}}\right){}{\mathrm{Γ}}{}\left({n}{+}\frac{{5}}{{2}}\right){}{\mathrm{Γ}}{}\left({n}{+}{2}\right){}{\mathrm{Γ}}{}\left({n}{-}\frac{{1}}{{3}}{}{I}{}\sqrt{{3}}\right){}{\mathrm{Γ}}{}\left({n}{+}\frac{{1}}{{3}}{}{I}{}\sqrt{{3}}\right)}$ (6)

References

 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.