If a function exists in closed form, e.g. $\sum\limits_{k \geq 0}z^k = {}_2 F_1 \bigg[{{1\; 1}\atop{1}} \vert z \bigg] = \frac{1}{1-z}$, but in case it doesn't, why bother rewriting it, e.g. $\sum\limits_{k \leq m}\binom{n}{k} = \sum\limits_{k \geq 0} \binom{n}{m-k} = \binom{n}{m} {}_2 F_1 \bigg[{{-m\; 1}\atop{n-m+1}} \vert 1 \bigg] $
since it doesn't yield a closed form or approximation of it?
Hmm, I don't know... because the Gaussian hypergeometric function satisfies a very convenient set of identities?
Also, what André said in the comments. Gauss and others spent a fair bit of time unraveling identities satisfied by this function, and it'd be a damn shame not to make use of our predecessors' effort.