Example 1 Given that $\alpha^{2008}+\beta^{2005}$ can be expressed as a bivariate polynomial in terms of $\alpha+\beta$ and $\alpha \beta$. Find the sum of the coefficients of this polynomial.
(2005, China Western Mathematical Olympiad)
Explanation: Let $S_{n}=\alpha^{n}+\beta^{n}\left(n \in \mathbf{N}_{+}\right)$,
$$
\left(\sigma_{1}, \sigma_{2}\right)=(\alpha+\beta, \alpha \beta) \text {. }
$$

Then $S_{1}=\sigma_{1}$,
$$
S_{2}=\sigma_{1} S_{1}+2(-1)^{3} \sigma_{2}=\sigma_{1}^{2}-2 \sigma_{2}, \cdots \cdots
$$

By Newton's formula, we have
$$
\alpha^{2005}+\beta^{2005}=\sum_{i, j} c_{i, j} \sigma_{1}^{i} \sigma_{2}^{j},
$$

where $c_{i, j}$ are constants.
$$
\begin{array}{l}
\text { Hence } \sum_{i, j} c_{i, j}=\left.\left(\alpha^{2005}+\beta^{2005}\right)\right|_{\sigma_{1}=\sigma_{2}=1} \\
=2 \operatorname{Re}\left(\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}\right)^{2005}=2 \operatorname{Re}\left(\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}\right) \\
=1 .
\end{array}
$$

Therefore, the sum of the coefficients of the required polynomial is 1.