The IsHypergeometricTerm(H,n) command returns true if $H\left(n\right)$ is a hypergeometric term of n. Otherwise, it returns false.

A function H is hypergeometric of n if $\frac{H\left(n+1\right)}{H\left(n\right)}=R\left(n\right)$, a rational function of n. $R\left(n\right)$ is the certificate of $H\left(n\right)$. If the third optional argument is included, it is assigned the certificate $R\left(n\right)$.

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

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

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

$\mathrm{IsHypergeometricTerm}\left(H\,n\,'\mathrm{certificate}'\right)$

$\mathrm{true}$

$\mathrm{certificate}$

$\frac{3\left(3n+2\right)\left(3n+4\right)n\left(n+2\right)}{2\left(2n+9\right){\left(n+4\right)}^2\left(n-1\right)}$