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...

‎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...

The Askey-Wilson polynomials are orthogonal polynomials in x = cos θ, which are given as a terminating 4φ3 basic hypergeometric series. The non-symmetric AskeyWilson polynomials are Laurent polynomials in z = eiθ, which are given as a sum of two terminating 4φ3’s. They satisfy a biorthogonality relation. In this paper new orthogonality relations for single 4φ3’s which are Laurent polynomials in...

We give two examples where symmetric polynomials play an important rôle in physics: First, the partition functions of ideal quantum gases are closely related to certain symmetric polynomials, and a part of the corresponding theory has a thermodynamical interpretation. Further, the same symmetric polynomials also occur in Berezin’s theory of quantization of phase spaces with constant curvature.

We give a combinatorial formula for the non-symmetric Macdonald polynomials E µ (x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J µ (x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop, that characterizes the non-symmetric Macdonald polynomials.

We give a combinatorial formula for the non-symmetric Macdonald polynomials E µ (x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J µ (x; q, t). We prove the new formula by verifying that it satisfies a recurrence, due to Knop and Sahi, that characterizes the non-symmetric Macdonald polynomials.