The AreSimilar(S,T,n) command returns true if $S\left(n\right)$ and $T\left(n\right)$ are similar. Otherwise, it returns false.

Two hypergeometric terms $S\left(n\right)$ and $T\left(n\right)$ are similar if their ratio is a rational function of n.

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

$\mathrm{S1}≔n^2\;$$\mathrm{S2}≔2^n\;$$T≔1$

$\mathrm{S1}≔n^2$

$\mathrm{S2}≔2^n$

$T≔1$

$\mathrm{AreSimilar}\left(\mathrm{S1}\,T\,n\right)$

$\mathrm{true}$

$\mathrm{AreSimilar}\left(\mathrm{S2}\,T\,n\right)$

$\mathrm{false}$

$\mathrm{T1}≔\frac{1}{\left(n+3\right)!\left(2n+7\right)!}\left(3n+1\right)!$

$\mathrm{T1}≔\frac{\left(3n+1\right)!}{\left(n+3\right)!\left(2n+7\right)!}$

$\mathrm{T2}≔\frac{\left(n^2-1\right)\left(3n+1\right)!}{\left(n+3\right)!\left(2n+7\right)!}$

$\mathrm{T2}≔\frac{\left(n^2-1\right)\left(3n+1\right)!}{\left(n+3\right)!\left(2n+7\right)!}$

$\mathrm{AreSimilar}\left(\mathrm{T1}\,\mathrm{T2}\,n\right)$

$\mathrm{true}$