Waring's Formula Research Articles

The Power Sums Involving Fibonacci Polynomials and Their Applications

The Girard and Waring formula and mathematical induction are used to study a problem involving the sums of powers of Fibonacci polynomials in this paper, and we give it interesting divisible properties. As an application of our result, we also prove a generalized conclusion proposed by R. S. Melham.

On the Sums of Powers of Chebyshev Polynomials and Their Divisible Properties

The main purpose of this paper is using mathematical induction and the Girard and Waring formula to study a problem involving the sums of powers of the Chebyshev polynomials and prove some divisible properties. We obtained two interesting congruence results involving Fibonacci numbers and Lucas numbers as some applications of our theorem.

The Waring formula and fusion rings

The potential that generates the cohomology ring of the Grassmannian is given in terms of the elementary symmetric functions using the Waring formula that computes the power sum of roots of an algebraic equation in terms of its coefficients. As a consequence, the fusion potential for su( N) K is obtained. This potential is the explicit Chebyshev polynomial in several variables of the first kind. We also derive the fusion potential for sp( N) K from a reciprocal algebraic equation. This potential is identified with another Chebyshev polynomial in several variables. We display a connection between these fusion potentials and generalized Fibonacci and Lucas numbers. In the case of su( N) K the generating function for the generalized Fibonacci numbers obtained are in agreement with Lascoux using the theory of symmetric functions. For sp( N) K , however, the generalized Fibonacci numbers are obtained from new sequences.

A Note on a Generalization of Waring's Formula

We show Zeng's “corrected” form of the generalized Waring formula is, in fact, equivalent to Konvalina's original form.

On sums of powers of zeros of polynomials

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.

On a Generalization of Waring's Formula

1gives a generalization of Waring's formula. Here we show that Konvalina's extension is a straightforward consequence of the Waring formula itself.

A Generalization of Waring's Formula

Waring's formula for expressing power sum symmetric functions in terms of elementary symmetric functions is generalized to monomial symmetric functions with equal exponents by applying the orthogonality property of Ramanujan sums together with the Möbius function over the set partition lattice.

The combinatorial power of the companion matrix

Using combinatorial methods, we obtain the explicit polynomials for all elements in an arbitrary power of the companion matrix depending on n variables and provide some interesting applications and relationships to Waring's formula on symmetric functions, the general solution to homogeneous linear recurrence relations, the multiplicative inverse of formal power series, the generating function of compositions (of numbers), a unified approach to Chebyshev polynomials including two recently discovered classes that satisfy analogous smallest-norm and orthogonality properties subject to different weight functions, Dickson polynomials of various kinds arising from the theory of finite fields, combinatorial expansions of Toeplitz matrices, and the recent notion of cycle dissections involving a bijective study of Waring's formula.

Cyclic Tableaux and Symmetric Functions

We introduce the notion of cyclic tableaux and develop involutions for Waring's formulas expressing the power sum symmetric function pn in terms of the elementary symmetric function en and the homogeneous symmetric function hn. The coefficients appearing in Waring's formulas are shown to be a cyclic analog of the multinomial coefficients, a fact that seems to have been neglected before. Our involutions also spell out the duality between these two forms of Waring's formulas, which turns out to be exactly the “duality between sets and multisets.” We also present an involution for permutations in cycle notation, leading to probably the simplest combinatorial interpretation of the Möbius function of the partition lattice and a purely combinatorial treatment of the fundamental theorem on symmetric functions. This paper is motivated by Chebyshev polynomials in connection with Waring's formula in two variables.

Remark on the Note "A Generalization of Waring's Formula"

Remark on the Note "A Generalization of Waring's Formula"

A Generalization of Waring's Formula

A Generalization of Waring's Formula