On sums of powers of zeros of polynomials

Abstract

Due to Girard's (sometimes called Waring's) formula the sum of the rth power of the zeros of every one variable polynomial of degree N, P N ( x), can be given explicitly in terms of the coefficients of the monic P ̃ N(x) polynomial. This formula is closely related to a known N − 1 variable generalisation of Chebyshev's polynomials of the first kind, T r ( N − 1) . The generating function of these power sums (or moments) is known to involve the logarithmic derivative of the considered polynomial. This entails a simple formula for the Stieltjes transform of the distribution of zeros. Perron-Stieltjes inversion can be used to find this distribution, e.g., for N → ∞. Classical orthogonal polynomials are taken as examples. The results for ordinary Chebyshev T N ( x) and U N ( x) polynomials are presented in detail. This will correct a statement about power sums of zeros of Chebyshev's T-polynomials found in the literature. For the various cases (Jacobi, Laguerre, Hermite) these moment generating functions provide solutions to certain Riccati equations.