25 research outputs found
On sums of powers of zeros of polynomials
Due to Girard's (sometimes called Waring's) formula the sum of the th
power of the zeros of every one variable polynomial of degree , ,
can be given explicitly in terms of the coefficients of the monic polynomial. This formula is closely related to a known \par
\noindent variable generalization of Chebyshev's polynomials of the first
kind, . 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 .\par Classical orthogonal polynomials are taken as
examples. The results for ordinary Chebyshev and
polynomials are presented in detail. This will correct a statement about power
sums of zeros of Chebyshev's polynomials found in the literature. For the
various cases (Jacobi, Laguerre, Hermite) these moment generating functions
provide solutions to certain Riccati equations
On the Equivalence of Three Complete Cyclic Systems of Integers
The system of coaches by Hilton and Pedersen, the system of cyclic sequences
of Schick, and Braendli-Bayne, related to diagonals in regular (2 n)-gons, and
the system of modified modular doubling sequences elaborated in this paper are
proved to be equivalent. The latter system employs the modified modular
equivalence used by Braendli-Bayne. A sequence of Euler tours related on
Schick's cycles of diagonals is also presented.Comment: 25 pages with 5 figures and 2 table
Cantor's List of Real Algebraic Numbers of Heights 1 to 7
Cantor gave in his fundamental article an elegant proof of the countability
of real algebraic numbers based on a positive integer height, denoted by him as
N, of integer and irreducible polynomials of given degree (denoted by him as n)
with relative prime coefficients. The finite number of real algebraic numbers
with given height he called phi(N), and gave the first three instances.\pn Here
we give a systematic list for the real algebraic numbers of height, which we
denote by n, for n from 1 to 7 and polynomials of degree k.Comment: 13 pages, 9 table
The Measure of the Orthogonal Polynomials Related to Fibonacci Chains: The Periodic Case
The spectral measure for the two families of orthogonal polynomial systems
related to periodic chains with N-particle elementary unit and nearest
neighbour harmonic interaction is computed using two different methods. The
interest is in the orthogonal polynomials related to Fibonacci chains in the
periodic approximation. The relation of the measure to appropriately defined
Green's functions is established.Comment: 19 pages, TeX, 3 scanned figures, uuencoded file, original figures on
request, some misprints corrected, tbp: J. Phys.