\[\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(z\right)=\ln z\sum_{s=0}^{n-1}(-1)^{%
s}\frac{a_{s}}{z^{s+1}}+\sum_{s=0}^{n-1}(-1)^{s}\frac{\widetilde{d}_{s}}{z^{s+%
1}}+R_{n}(z),\]