How can I show the following equation is true?

Question

I was working on a problem and reduced it to showing ‎‎ $\psi(b-j)=-\frac{1}{b}+\psi(j+1)+O(b),$ where $j \in \mathbb{N}$ and $\psi(t)=\frac{d}{dt}\ln \Gamma(t)$. Your suggestion?

Answer 1

For $b\rightarrow 0$ you get with the digamma reflection formula: $$\psi(b-j) = \psi(j+1-b) + \pi \cot\Big((\pi(j+1-b)\Big) =\psi(j+1-b) - \pi \cot(\pi b)$$ $$ \sim \psi(j+1) -\psi'(j+1)b + O(b^2) - \frac{1}{b}+ \frac{1}{3}b + O(b^2)$$ $$\sim - \frac{1}{b} + \psi(j+1) + O(b)$$