If $\alpha=e^{2 \pi i / 7}$ and $f(x)=A_{0}+\sum_{k=1}^{20} A_{k} x^{k} \\$, then find the value of, $f(x)+f(\alpha x)+\ldots+f\left(\alpha^{6} x\right)$ independent of $\alpha .$

Solution:

Given that, $\alpha=\mathrm{e}^{2 \pi / 7} \\ $ and, $f(\mathrm{x})=\mathrm{A}_{0}+\sum_{k=1}^{20} A_{k} x^{k} \\ $

Then, $f\left(\alpha^{n} x\right)=\mathrm{A}_{0}+\sum_{k=1}^{6} A_{k} \alpha^{n k} x^{k} \\ $

$\therefore \quad \mathrm{S}=f(x)+f(\alpha \mathrm{x})+\ldots \ldots . .+f\left(\alpha^{6} \mathrm{x}\right)=\sum_{k=1}^{6} f\left(\alpha^{n} x\right) \\ $

$=\sum_{k=1}^{6}\left\{A_{0}+\sum_{k=1}^{20} A_{k} \alpha^{n k} x^{k}\right\} \\ $

$ =7 …