Gauss’ digamma theorem

Theorem (Gauss).

$$\mathrm{\Psi }(x+n)=\frac{1}{x}+\frac{1}{x+1}+\mathrm{\cdots }+\frac{1}{x+n-1}+\mathrm{\Psi }(x),n=1,2,3,\mathrm{\dots }$$

$$\mathrm{\Psi }\left(\frac{p}{q}\right)=-\gamma -\frac{\pi }{2}\cot \frac{\pi p}{q}-\ln q+\sum _{n=1}^{q-1}\cos \frac{2\pi np}{q}\ln \left(2\sin \frac{\pi n}{q}\right)$$

where .