On sums of powers of zeros of polynomials

✍ Scribed by Wolfdieter Lang

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 988 KB
• Volume: 89
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(97)00240-9

✦ Synopsis

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~,(x), can be given explicitly in terms of the coefficients of the monic /Sx(x) polynomial. This formula is closely related to a known N -1 variable generalisation of Chebyshev's polynomials of the first kind, Tr/'~-~). 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 /~.(x) and U~,(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.