Example 4 Given that $\alpha^{2005}+\beta^{2005}$ can be expressed as a bivariate polynomial in terms of $\alpha+\beta, \alpha \beta$. Find the sum of the coefficients of this polynomial.
(2005 China Western Mathematical Olympiad)
Since we are asked to find the sum of the coefficients of this polynomial, we should set the variables to 1, i.e., let $\alpha+\beta=1$, $\alpha \beta=1$, so $\alpha, \beta$ are the two roots of the polynomial $f(x)=x^{2}-x+1$. Let $T_{n}=\alpha^{n}+\beta^{n}$, then by Newton's formula, we have
$T_{n}=T_{n-1}-T_{n-2}$ (where $b_{n-1}=1, b_{n-2}=-1$).
Thus, $T_{n}=\left(T_{n-2}-T_{n-3}\right)-T_{n-2}=-T_{n-3}$.

Similarly, $T_{n-3}=-T_{n-6}$.
Therefore, $T_{n}=T_{n-6}$, which means the sequence $\left\{T_{n}\right\}$ is a periodic sequence with a period of 6.
Thus, $T_{2005}=T_{1}=\alpha+\beta=1$.