The Wikipedia page for the digamma function states that the following holds: $$\psi(s+1) = -\gamma - \sum_{k=1}^\infty \frac{(-1)^k}{k}\binom{s}{k}.$$ Is there any textbook/paper where this formula is given?
2025-01-13 02:14:15.1736734455
Newton Series for the Digamma Function
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See the result in the display immediately before the words "third Retkes identity" at the English Wikipedia page for Harmonic number. It is the derivation $H_n = \dots = \sum_{k=1}^{n}(-1)^{k-1} \frac{1}{k} \binom{n}{k}$, which gives you the less obvious piece in the identity you see.