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.

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

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

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

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

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

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

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

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

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

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

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

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

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