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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.

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  • $\begingroup$ The series inside is a Gauss hypergeometric function of $-1/s$. Therefore you need the asymptotics of this function at $-\infty$ which can be obtained using connection formulas relating $_2F_1(-1/s)$ with $_2F_1(s)$. This will presumably produce $s^n\ln s$ and polygamma. $\endgroup$ – Start wearing purple Jan 2 '16 at 12:26

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