The IsHypergeometricTerm(H,n) function returns true if $H\left(n\right)$ is a hypergeometric term in n. Otherwise, false is returned.

An expression H is hypergeometric in n if $\frac{H\left(n+1\right)}{H\left(n\right)}=R\left(n\right)$, a rational function in n. $R\left(n\right)$ is the certificate of $H\left(n\right)$.

If the third optional argument is included, it is assigned to the certificate of $H\left(n\right)$.

This function is part of the RationalNormalForms package, and so it can be used in the form IsHypergeometricTerm(..) only after executing the command with(RationalNormalForms). However, it can always be accessed through the long form of the command by using RationalNormalForms[IsHypergeometricTerm](..).

$\mathrm{with}\left(\mathrm{RationalNormalForms}\right)\:$

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

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

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

$\mathrm{true}$

$\mathrm{certificate}$

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