On sums of powers of zeros of polynomials

Details On sums of powers of zeros of polynomials On sums of powers of zeros of polynomials

1997-11-30

arxiv.org/abs/math/9711217v1

Mathematics

Classical Analysis and ODEs

Scientific paper

Due to Girard's (sometimes called Waring's) formula the sum of the $r-$th 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 ${\tilde P}_{N}(x)$ polynomial. This formula is closely related to a known \par \noindent $N-1$ variable generalization 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, {\it e.g.} for $N\to \infty$.\par 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.

Affiliated with

Also associated with

No associations

Rating

On sums of powers of zeros of polynomials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On sums of powers of zeros of polynomials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On sums of powers of zeros of polynomials will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-258256