It is well known that the family of Hahn polynomials {hα,β n (x;N)}n≥0 is orthogonal with respect to a certain weight function up to N . In this paper we present a factorization for Hahn polynomials for a degree higher than N and we prove that these polynomials can be characterized by a ∆-Sobolev orthogonality. We also present an analogous result for dual-Hahn, Krawtchouk, and Racah polynomials...

We introduce and study deformations of finite-dimensional modules over rational Cherednik algebras. Our main tool is a generalization of usual harmonic polynomials for each Coxeter groups — the so-called quasiharmonic polynomials. A surprising application of this approach is the construction of canonical elementary symmetric polynomials and their deformations for all Coxeter groups.

We present a method, based on series expansions and symmetric polynomials, by which a mean of two variables can be extended to several variables. We apply it mainly to the logarithmic mean.

This paper studies the elementary symmetric polynomials Sk(x) for x ∈ Rn. We show that if |Sk(x)|, |Sk+1(x)| are small for some k > 0 then |Sl(x)| is also small for all l > k. We use this to prove probability tail bounds for the symmetric polynomials when the inputs are only t-wise independent, that may be useful in the context of derandomization. We also provide examples of t-wise independent ...

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 give several characterizations of discrete Sugeno integrals over bounded distributive lattices, as particular cases of lattice polynomial functions, that is functions which can be represented in the language of bounded lattices using variables and constants. We also consider the subclass of term functions as well as the classes of symmetric polynomial functions and weighted minimum and maxim...

Every symmetric polynomial p = p(x) = p(x1, . . . , xg) (with real coefficients) in g noncommuting variables x1, . . . , xg can be written as a sum and difference of squares of noncommutative polynomials:

We determine the coefficients of the classes of highest weight in the conjugacy class expansion of the monomial symmetric polynomials evaluated at the Jucys-Murphy elements. We apply our result, along with other properties of Jucys-Murphy elements, to give streamlined derivations of the first-order asymptotics and character expansion of the Weingarten function for the unitary group.

The Boolean lattice 2[n] is the power set of [n] ordered by inclusion. A chain c0 ⊂ · · · ⊂ ck in 2[n] is rank-symmetric, if |ci|+ |ck−i| = n for i = 0, . . . , k; and it is symmetric, if |ci| = (n− k)/2 + i. We prove that there exist a bijection p : [n] → [n] and a partial ordering < on [n](>n/2) satisfying the following properties: • ⊂ is an extension of < on [n](>n/2); • if C ⊂ [n](>n/2) is ...