A 3-dimensional Eulerian array

We give a generalization of the identity proved by J.Worpitzky in [4], by expressing each power x n as a linear combination of the images of β m under the powers of the shift operator E (here β m (x) := x m m! ). We encode the coefficients of these linear combinations in a 3-dimensional array -the Eulerian octant -and we find recurrences formulae, explicit expressions and generating functions for its entries. 1 Introduction......Page 1 2 Worpitzky’s identity......Page 2 3 The strength of a vector with respect to an endomorphism......Page 3 4 Proof of Worpitzky’s identity......Page 6 5 The Eulerian octant......Page 7 6 Pages......Page 8 7 Planes......Page 13 8 Panels......Page 16 9 Explicit formulæ......Page 17