Complete homogeneous symmetric polynomials and generalized Vieta's formulas

We generalize the classical Vieta formulas that express the coefficients of a polynomial in terms of all the roots. In particular, we focus on the case when just some of the roots are known. Our formulas are established by exploiting some properties of the complete homogeneous symmetric polynomials. Further, we provide new identities for these polynomials when the variables are in a geometric progression and new identities involving the roots of unity.