We construct linear operators factorizing the three bases of symmetric polynomials: monomial symmetric functions mλ(x), elementary symmetric polynomials Eλ(x), and Schur functions sλ(x), into products of univariate polynomials.

We construct linear operators factorizing the three bases of symmetric polynomials: monomial symmetric functions mλ(x), elementary symmetric polynomials Eλ(x), and Schur functions sλ(x), into products of univariate polynomials.

The symmetric Macdonald polynomials are able to be constructed out of the non-symmetric Macdonald polynomials. This allows us to develop the theory of the symmetric Macdonald polynomials by first developing the theory of their non-symmetric counterparts. In taking this approach we are able to obtain new results as well as simpler and more accessible derivations of some of the known fundamental ...

This paper presents some applications using several properties of three important symmetric polynomials: elementary symmetric polynomials, complete symmetric polynomials and the power sum symmetric polynomials. The applications includes a simple proof of El-Mikkawy conjecture in [M.E.A. El-Mikkawy, Appl. Math. Comput. 146 (2003) 759–769] and a very easy proof of the Newton–Girard formula. In ad...

The family of general Jacobi polynomials P (α,β) n where α, β ∈ C can be characterised by complex (nonhermitian) orthogonality relations (cf. [15]). The special subclass of Jacobi polynomials P (α,β) n where α, β ∈ R are classical and the real orthogonality, quasi-orthogonality as well as related properties, such as the behaviour of the n real zeros, have been well studied. There is another spe...

Newman polynomials are those with all coefficients in {0, 1}. We consider here the problem of finding Newman polynomials P such that all the coefficients of P 2 are so small as possible for deg P and P (1) given. A set A ⊂ [1, N ] is called a B2[g] sequence if every integer n has at most g distinct representations as n = a1 + a2 with a1, a2 ∈ A and a1 ≤ a2. Gang Yu [4] introduced a new idea to ...

f(T1, . . . , Tn) = f(Tσ(1), . . . , Tσ(n)) for all σ ∈ Sn. Example 1. The sum T1 + · · ·+ Tn and product T1 · · ·Tn are symmetric, as are the power sums T r 1 + · · ·+ T r n for any r ≥ 1. As a measure of how symmetric a polynomial is, we introduce an action of Sn on F [T1, . . . , Tn]: (σf)(T1, . . . , Tn) = f(Tσ−1(1), . . . , Tσ−1(n)). We need σ−1 rather than σ on the right side so this is a...

When τ is a quasi-definite moment functional onP , the vector space of all real polynomials, we consider a symmetric bilinear form φ(·, ·) on P ×P defined by φ(p, q) = λp(a)q(a)+ μp(b)q(b)+ 〈τ, p′q ′〉, where λ,μ, a, and b are real numbers. We first find a necessary and sufficient condition for φ(·, ·) to be quasi-definite. When τ is a semi-classical moment functional, we discuss algebraic prope...

‎let $g_{i} $ be a subgroup of $ s_{m_{i}}‎ ,‎ 1 leq i leq k$‎. ‎suppose $chi_{i}$ is an irreducible complex character of $g_{i}$‎. ‎we consider $ g_{1}times cdots times g_{k} $ as subgroup of $ s_{m} $‎, ‎where $ m=m_{1}+cdots‎ +‎m_{k} $‎. ‎in this paper‎, ‎we give a formula for the dimension of $h_{d}(g_{1}times cdots times g_{k}‎, ‎chi_{1}timescdots times chi_{k})$ and investigate the existe...

Multisymmetric polynomials are the r-fold diagonal invariants of the symmetric group Sn. Elementary multisymmetric polynomials are analogues of the elementary symmetric polynomials, in the multisymmetric setting. In this paper, we give a necessary and sufficient condition on a ring A for the algebra of multisymmetric polynomials with coefficients in A to be generated by the elementary multisymm...