Solution of an integral with strange imprecision of gamma functions
Solution 1:
I don't think closed form is possible, but here is (possibly) a more controlled series formula for the function: $$ I(\alpha, m, n) = \sum_{j=0}^{\infty} {}^{m+j-1}C_{j} \,\,(\alpha(n-1))^{-j} \left[\frac{\Gamma (j+n) - \Gamma (j+n, \alpha(n-1)(1-\epsilon))}{\Gamma(n) - \Gamma(n, (n-1)\alpha/(1+\epsilon))}\right] $$
I got this using the variable substitution $z=(y-1)/y$ and carrying out a Taylor expansion in powers of $z$.