Newton Series for the Digamma Function 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?

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.