We discuss Lagrange interpolation on two sets of nodes in two dimensions where the coordinates of the nodes are Chebyshev points having either the same or opposite parity. We use a formula of Xu for Lagrange polynomials to obtain a general interpolation theorem for bivariate polynomials at either set of Chebyshev nodes. An extra term must be added to the interpolation formula to handle all poly...

We show how symmetric orthogonal polynomials can be linked to polynomials associated with certain orthogonal L-polynomials. We provide some examples to illustrate the results obtained. Finally as an application, we derive information regarding the orthogonal polynomials associated with the weight function (1 + kx2)(l x2)~1/2, k>0.

We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with posit...

Recently I.Macdonald defined a family of systems of orthogonal symmetric polynomials depending of two parameters q, k which interpolate between Schur’s symmetric functions and certain spherical functions on SL(n) over the real and p-adic fields [M]. These polynomials are labeled by dominant integral weights of SL(n), and (as was shown by I.Macdonald) are uniquely defined by two conditions: 1) t...

We study a new kind of symmetric polynomials P_n(x_1,...,x_m) degree n in m real variables, which have arisen the theory numerical semigroups. establish their basic properties and find representation through power sums E_k=\sum_{j=1}^m x_j^k. observe visual similarity between normalized P_n(x_1,...,x_m)/\chi_m, where \chi_m=\prod_{j=1}^m x_j, polynomial part partition function W(s,{d_1,...,d_m}...