Let $p(x)=x^d+\sum_{i=0}^{d-1} a_ix^i$. Then Vieta's formula tells us that the $a_i$ can be expressed as signed elementary symmetric polynomials of the roots $\{\alpha_1,\ldots,\alpha_d\}$ of $p(x)$: $$a_i=e_{d-i}(-\alpha_1,\ldots,-\alpha_d)$$ Now suppose that we write $p(x)=\sum_{i=0}^d b_i T_i(x)$ where $T_i(x)$ is the $i$-th Chebyshev polynomial. Still the $b_i$'s are symmetric polynomials in the $\alpha_i$'s.
I can compute these by hand for small degrees, but I don't see a pattern.
Have they been studied?
What are they?