\[P_{n}(z)=\sum_{k=0}^{n}\ell_{k}(z)f_{k}=\omega_{n+1}(z)\sum_{k=0}^{n}\frac{%
\widetilde{\omega}_{k}}{z-z_{k}}f_{k}=\ifrac{\sum_{k=0}^{n}\frac{\widetilde{%
\omega}_{k}f_{k}}{z-z_{k}}}{\sum_{k=0}^{n}\frac{\widetilde{\omega}_{k}}{z-z_{k%
}}},\]