I dont see the transition from $(-z)^k$ in the fist sum to the transition to $(z+2)^k$ in the second sum?
How is that derived?
2026-04-05 17:58:23.1775411903
How to Derive this Digamma identity?
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From $$\psi(1-z)=-\gamma-\sum_{k\geq 1}\zeta(k+1) z^{k} $$ and $\psi(2-z)=\psi(1-z)+\frac{1}{1-z}$ it follows that: $$\psi(2-z) = \frac{1}{1-z}-\gamma-\sum_{k\geq 1}\zeta(k+1) z^{k}=(1-\gamma)+\sum_{k\geq 1}(1-\zeta(k+1))\,z^k\tag{1}$$ now replace $z$ with $z+2$ to get the stated identity.