I have an input set that needs to call the hypergeom function 137544 times

How many different $m, $n, $N, $m+$n-$N, $i, $n-$i, $m+$i-$N, $N-$i, $m+$n do you have? What happens if you cache the result of logfact? And how big is your $m+$n? Can't you just precalculate k! for 1 <= k <= $m+$n and then use a lookup in the webapplication? Then your calculation just becomes 9 lookups, 7 bignum additions and 1 bignum division.