[For the entire collection see Zbl 0695.00019.]
Our aim in this paper is to illustrate how B-splines could be investigated using their relation to the linear transformation S of Dirichlet distributed random variables. In Section 2 we give a divided difference representation of the generalized B-spline in the univariate case, i.e., when \(s=1\) and discuss shortly some of its properties. The probabilistic interpretation of B-splines combined with a result of W. R. van Zwet (1979), concerning linear combinations of order statistics is used in Section 3 to derive an expansion for the integral of a univariate B-spline. In Section 4 an interesting formula which relates multivariate and univariate B-splines with especially chosen knots is derived. Further, a determinant formula for the multivariate B-spline is also given, combining its probabilistic interpretation with a result of M. M. Ali and E. Mead [Bull. Inst. Stat. Res. Tr. 3, 22-41 (1969)] and G. S. Watson [Biometrika 43, 161-168 (1956; Zbl 0074.142)].