RegularGammaForm - Maple Help

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SumTools[Hypergeometric]

 RegularGammaForm
 construct the regular Gamma-function representation of a hypergeometric term

 Calling Sequence RegularGammaForm(H, n)

Parameters

 H - hypergeometric term of n n - variable

Description

 • Let H be a hypergeometric term of n, R be the certificate of H, and n0 be an integer such that R has neither a pole nor a zero for all $\mathrm{n0}\le n$. The RegularGammaForm(H,n) calling sequence returns the multiplicative decomposition of the form $H\left(\mathrm{n0}\right)\left({\prod }_{k=\mathrm{n0}}^{n-1}R\left(k\right)\right)$ where the product is expressed in terms of a product of the Gamma function of the form $\mathrm{\Gamma }\left(n-c\right)$ where c is a constant and their reciprocals.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right):$
 > $H≔\mathrm{Product}\left(\frac{\frac{1}{2}\left(3{k}^{2}+6k+4\right)\left(2k+3\right)\left(4k+5\right)\left(k+1\right)\left(4k+3\right)}{k\left(4k-1\right)\left(2k-1\right)\left(4k-3\right)\left(2k+5\right)\left(k+2\right)\left(3{k}^{2}+1\right)},k=1..n-1\right)$
 ${H}{≔}{\prod }_{{k}{=}{1}}^{{n}{-}{1}}{}\frac{\left({3}{}{{k}}^{{2}}{+}{6}{}{k}{+}{4}\right){}\left({2}{}{k}{+}{3}\right){}\left({4}{}{k}{+}{5}\right){}\left({k}{+}{1}\right){}\left({4}{}{k}{+}{3}\right)}{{2}{}{k}{}\left({4}{}{k}{-}{1}\right){}\left({2}{}{k}{-}{1}\right){}\left({4}{}{k}{-}{3}\right){}\left({2}{}{k}{+}{5}\right){}\left({k}{+}{2}\right){}\left({3}{}{{k}}^{{2}}{+}{1}\right)}$ (1)
 > $\mathrm{RegularGammaForm}\left(H,n\right)$
 $\frac{{64}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{2}}\right)}^{{n}}{}{\mathrm{\Gamma }}{}\left({n}{+}{1}{-}\frac{{I}{}\sqrt{{3}}}{{3}}\right){}{\mathrm{\Gamma }}{}\left({n}{+}{1}{+}\frac{{I}{}\sqrt{{3}}}{{3}}\right){}{\mathrm{\Gamma }}{}\left({n}{+}\frac{{3}}{{2}}\right){}{\mathrm{\Gamma }}{}\left({n}{+}\frac{{5}}{{4}}\right){}{\mathrm{\Gamma }}{}\left({n}{+}{1}\right){}{\mathrm{\Gamma }}{}\left({n}{+}\frac{{3}}{{4}}\right)}{{{2}}^{{n}}{}{\mathrm{\Gamma }}{}\left({n}\right){}{\mathrm{\Gamma }}{}\left({n}{-}\frac{{1}}{{4}}\right){}{\mathrm{\Gamma }}{}\left({n}{-}\frac{{1}}{{2}}\right){}{\mathrm{\Gamma }}{}\left({n}{-}\frac{{3}}{{4}}\right){}{\mathrm{\Gamma }}{}\left({n}{+}\frac{{5}}{{2}}\right){}{\mathrm{\Gamma }}{}\left({n}{+}{2}\right){}{\mathrm{\Gamma }}{}\left({n}{-}\frac{{I}{}\sqrt{{3}}}{{3}}\right){}{\mathrm{\Gamma }}{}\left({n}{+}\frac{{I}{}\sqrt{{3}}}{{3}}\right)}$ (2)

Compare the number of Gamma-function values returned from RegularGammaForm with that of any one of the four efficient representations of the input hypergeometric term $H\left(n\right)$:

 > $\mathrm{EfficientRepresentation}\left[1\right]\left(H,n\right)$
 $\frac{{64}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{4}}\right)}^{{n}}{}\left({{n}}^{{2}}{+}\frac{{1}}{{3}}\right){}{n}{}\left({n}{-}\frac{{1}}{{4}}\right){}\left({n}{+}\frac{{1}}{{2}}\right){}\left({n}{+}\frac{{1}}{{4}}\right){}\left({n}{-}\frac{{1}}{{2}}\right){}\left({n}{-}\frac{{3}}{{4}}\right)}{{\mathrm{\Gamma }}{}\left({n}{+}\frac{{5}}{{2}}\right){}{\mathrm{\Gamma }}{}\left({n}{+}{2}\right)}$ (3)
 > $\mathrm{EfficientRepresentation}\left[2\right]\left(H,n\right)$
 $\frac{{64}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{4}}\right)}^{{n}}{}\left({{n}}^{{2}}{+}\frac{{1}}{{3}}\right){}\left({n}{-}\frac{{1}}{{4}}\right){}\left({n}{+}\frac{{1}}{{4}}\right){}\left({n}{-}\frac{{3}}{{4}}\right)}{\left({n}{+}\frac{{3}}{{2}}\right){}\left({n}{+}{1}\right){}{\mathrm{\Gamma }}{}\left({n}\right){}{\mathrm{\Gamma }}{}\left({n}{-}\frac{{1}}{{2}}\right)}$ (4)
 > $\mathrm{EfficientRepresentation}\left[3\right]\left(H,n\right)$
 $\frac{{64}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{4}}\right)}^{{n}}{}\left({{n}}^{{2}}{+}\frac{{1}}{{3}}\right){}\left({n}{+}\frac{{1}}{{4}}\right){}\left({n}{-}\frac{{3}}{{4}}\right){}{n}{}\left({n}{-}\frac{{1}}{{4}}\right)}{\left({n}{+}\frac{{3}}{{2}}\right){}{\mathrm{\Gamma }}{}\left({n}{-}\frac{{1}}{{2}}\right){}{\mathrm{\Gamma }}{}\left({n}{+}{2}\right)}$ (5)
 > $\mathrm{EfficientRepresentation}\left[4\right]\left(H,n\right)$
 $\frac{{64}{}\sqrt{{\mathrm{\pi }}}{}{\left(\frac{{1}}{{4}}\right)}^{{n}}{}\left({{n}}^{{2}}{+}\frac{{1}}{{3}}\right){}\left({n}{-}\frac{{1}}{{4}}\right){}\left({n}{+}\frac{{1}}{{4}}\right){}\left({n}{-}\frac{{3}}{{4}}\right)}{\left({n}{+}\frac{{3}}{{2}}\right){}\left({n}{+}{1}\right){}{\mathrm{\Gamma }}{}\left({n}\right){}{\mathrm{\Gamma }}{}\left({n}{-}\frac{{1}}{{2}}\right)}$ (6)