Partial fraction expansion for a one-parameter family of meromorphic functions (Q1922329)

The following formula is proved : let \(\gamma \in \mathbb{R}\), then \[ \exp (\gamma z)/(\exp (z) - 1) = f(z, \delta ) + \sum_{k=1}^N \exp (\gamma -k)z, \] for \(\gamma = N+\delta , \;N\in \mathbb{N} \;, 0< \delta \leq 1\), and \[ \exp (\gamma z)/(\exp (z) - 1) = f(z, \delta ) + \sum_{k=0}^{N-1} \exp (\gamma +k)z, \] for \(\gamma = -N+\delta , \;N\in \mathbb{N} \;, 0< \delta \leq 1\), where \(f(z,\delta )=\exp (\delta z)/(\exp (z) - 1)=\frac{1}{z}+c(\delta ) +\sum_{n=1}^{\infty }\frac{2z\cos (2\pi n\delta )-4\pi n\sin (2\pi n\delta )} {z^2+(2\pi n)^2}\), and \(c(\delta )=0\), for \(0<\delta <1\) and \(c(\delta )=1/2\), for \(\delta =1\).