Weighted means of B-splines, positivity of divided differences, and complete homogeneous symmetric polynomials

Abstract We employ the fact that 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 μ with | μ − 2 p | 1 / 2 . We also prove results on linear combinations of the classical complete homogeneous symmetric polynomials and on linear combinations of products of such polynomials.