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

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