Does this reduce down to the PolyGamma function?
$H_n=$ $$\lim_{s\to 0} \, \left(-\frac{\left(\frac{1}{s}+1\right)^n (s+1)^{-n} \left(\sum _{k=0}^{\infty } \frac{\left(-\frac{1}{s}\right)^k \left(\prod _{i=0}^{k-1} (i+n)\right) \left(\prod _{i=0}^{k-1} (i+n)\right)}{k! \left(\prod _{i=0}^{k-1} (i+n+1)\right)}\right)}{n}+\frac{(s+1)^{-n}}{n}+s-\log (s)\right)$$
Mathematica tells me it reduces down to:
$$\frac{1}{n}+\psi ^{(0)}(n)+\gamma$$
but I don't know where to start to make that happen.