Positivity conditions for bihomogeneous polynomials

Positivity Of Trigonometri Polynomials

A positivity conjecture for Jack polynomials

We present a positivity conjecture for the coefficients of the development of Jack polynomials in terms of power sums. This extends Stanley’s ex-conjecture about normalized characters of the symmetric group. We prove this conjecture for partitions having a rectangular shape.

Positivity of Turán determinants for orthogonal polynomials

The orthogonal polynomials pn satisfy Turán’s inequality if p 2 n(x)− pn−1(x)pn+1(x) ≥ 0 for n ≥ 1 and for all x in the interval of orthogonality. We give general criteria for orthogonal polynomials to satisfy Turán’s inequality. This yields the known results for classical orthogonal polynomials as well as new results, for example, for the q–ultraspherical polynomials.

Two positivity conjectures for Kerov polynomials

Kerov polynomials express the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as polynomials in the “free cumulants” of the associated Young diagram. We present two positivity conjectures for their coefficients. The latter are stronger than the positivity conjecture of Kerov-Biane, recently proved by Féray. 1 Kerov polynomials 1.1 Characters A ...

Positivity Theorems for Solid-Angle Polynomials

For a lattice polytope P , define AP(t) as the sum of the solid angles of all the integer points in the dilate tP . Ehrhart and Macdonald proved that AP(t) is a polynomial in the positive integer variable t. We study the numerator polynomial of the solid-angle series ∑ t≥0 AP(t)z . In particular, we examine nonnegativity of its coefficients, monotonicity and unimodality questions, and study ext...