Weighted Means of B-Splines, Positivity of Divided Differences, and Complete Homogeneous Symmetric Polynomials

Abstract: We employ the fact certain divided differences can be written as weighted means of B-splines and hence are positive. These divided differences include the complete homogeneous symmetric polynomials of even degree $2p$, the positivity of which is a classical result by D. B. Hunter. We extend Hunter's result to complete homogeneous symmetric polynomials of fractional degree, which are defined via Jacobi's bialternant formula. We show in particular that these polynomials have positive real part for real degrees $\mu$ with $|\mu-2p|< 1/2$. We also prove a positivity criterion for linear combinations of the classical complete homogeneous symmetric polynomials and a sufficient criterion for the positivity of linear combinations of products of such polynomials.