Macdonald defined in [M1] a remarkable class of symmetric polynomials Pλ(x; q, t) which depend on two parameters and interpolate between many families of classical symmetric polynomials. For example Pλ(x; t) = Pλ(x; 0, t) are the Hall-Littlewood polynomials which themselves specialize for t = 0 to Schur functions sλ. Also Jack polynomials arise by taking q = t and letting t tend to 1. The Hall-...

Entropy and other fundamental quantities of information theory are customarily expressed and manipulated as functions of probabilities. Here we study the entropy H and subentropy Q as functions of the elementary symmetric polynomials in the probabilities, and reveal a series of remarkable properties. Derivatives of all orders are shown to satisfy a complete monotonicity property. H and Q themse...

We introduce a general class of symmetric polynomials that have saturated Newton polytope and their has integer decomposition property. The covers numerous previously studied polynomials.

Vector-valued Jack polynomials associated to the symmetric group SN are polynomials with multiplicities in an irreducible module of SN and which are simultaneous eigenfunctions of the Cherednik–Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups G(r, p,N) and studi...

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 consider polynomials in two variables which satisfy an admissible second order partial differential equation of the form (*) Auxx + 2Buxy + Cuyy +Dux + Euy = u; and are orthogonal relative to a symmetric bilinear form de…ned by '(p; q) = h ; pqi+ h ; pxqxi ; where A; ; E are polynomials in x and y; is an eigenvalue parameter, and are linear functionals on polynomials. We …nd a condition for ...

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...