In this paper, we introduce the modified q-Euler polynomials. The main objective of this paper is to consider symmetric identities of the modified q-Euler polynomials under the symmetric group of degree n. AMS subject classification: 11B68, 11S40, 11S80.

A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of r...

Knop and Sahi simultaneously introduced a family of non-homogeneous, non-symmetric polynomials, Gα(x; q, t). The top homogeneous components of these polynomials are the non-symmetric Macdonald polynomials, Eα(x; q, t). An appropriate Hecke algebra symmetrization of Eα yields the Macdonald polynomials, Pλ(x; q, t). A search for explicit formulas for the polynomials Gα(x; q, t) led to the main re...

Jack polynomials are a remarkable family of polynomials in n variables x = (x1, · · · , xn) with coefficients in the field F := Q(α) where α is an indeterminate. They arise naturally in several statistical, physical, combinatorial, and representation theoretic considerations. The symmetric polynomials ([M1], [St], [LV], [KS]) Jλ = J (α) λ are indexed by partitions λ = (λ1, · · · , λn) where λ1 ...

In this paper we derive the bi-orthogonality relations, diagonal term evaluations and evaluation formulas for the non-symmetric Koornwinder polynomials. For the derivation we use certain representations of the (double) affine Hecke algebra which were originally defined by Noumi and Sahi. We furthermore give the explicit connection between the non-symmetric and the symmetric theory. This leads i...

The Jack (symmetric) polynomials P (α) λ (x) form a class of symmetric polynomials which are indexed by a partition λ and depend rationally on a parameter α. They reduced to the Schur polynomials when α = 1, and to other classical families of symmetric polynomials for several specific parameters. Recently they attracts attention from various points of view, for example the integrable systems an...

The Jack polynomials Jλ(x; α) form a remarkable class of symmetric polynomials. They are indexed by a partition λ and depend on a parameter α. One of their properties is that several classical families of symmetric functions can be obtained by specializing α, e.g., the monomial symmetric functions mλ (α = ∞), the elementary functions eλ′ (α = 0), the Schur functions sλ (α = 1) and finally the t...

Two canonical forms for skew-symmetric matrix polynomials over arbitrary fields are characterized — the Smith form, and its skew-symmetric variant obtained via unimodular congruences. Applications include the analysis of the eigenvalue and elementary divisor structure of products of two skew-symmetric matrices, the derivation of a Smith-McMillan-like canonical form for skew-symmetric rational m...

In this paper we study multivariate polynomial functions in complex variables and the corresponding associated symmetric tensor representations. The focus is on finding conditions under which such complex polynomials/tensors always take real values. We introduce the notion of symmetric conjugate forms and general conjugate forms, and present characteristic conditions for such complex polynomial...

In the theory of symmetric Jack polynomials the coefficients in the expansion of the pth elementary symmetric function ep(z) times a Jack polynomial expressed as a series in Jack polynomials are known explicitly. Here analogues of this result for the non-symmetric Jack polynomials Eη(z) are explored. Necessary conditions for non-zero coefficients in the expansion of ep(z)Eη(z) as a series in no...