Moments of Dirichlet Splines and Their Applications to Hypergeometric Functions

Dirichlet averages of multivariate functions are employed for a derivation of basic recurrence formulas for the moments of multivariate Dirichlet splines. An algorithm for computing the moments of multivariate simplex splines is presented. Applications to hypergeometric functions of several variables are discussed. Introduction. In [9], H.B. Curry and I.J. Schoenberg have pointed out that univariate B-splines can be constructed from volumes of slices of convex polyhedra. An extension of this idea to the case of multivariate splines is due to C. de Boor [1]. Since the geometric construction is too complicated to be used in numerical computations, some basic recurrence formulas for these functions have been found (see [3,8,10,11,14,17,18,19,22,23]). Multivariate B-splines (also called simplex splines) have been studied extensively over the past thirteen years by many researchers. These functions have been found useful for some applications of data fitting, computer aided geometric design, and mathematical statistics. In [29] the author addressed some new problems where the simplex splines could play a prominent role. Further generalizations of multivariate simplex splines appear in [19] and [13]. Here and thereafter we call these functions Dirichlet splines. Our choice of terminology is motivated by the fact that the distributional definition of the Dirichlet splines (see (2.2)) involves the Dirichlet density function. It has been demonstrated that this class of splines is well designed for some problems of mathematical statistics (see [13], [19], and references therein). Applications to the theory of multivariate convex functions are reported in [26, 27]. Recently we have noticed that there is a simple relationship between univariate Dirichlet splines and some special functions such as R-hypergeometric functions and confluent hypergeometric functions