In a series of earlier papers [2, 3, 4, 5], the authors, and their collaborators, studied various patterns in the Pascal Triangle of binomial coefficients; and extended the Pascal Triangle, and the patterns, to encompass negative integral values of n and r in the binomial coefficient

The resulting extension of the Pascal Triangle was described as the Pascal Hexagon, the central portion of which appears in Figure 1(a).

Figure 1a: The Paxal Hexagon, showing values of æ ç è n r s ö ÷ ø with r + s = n, for 4³ n, r, s³ -5. Note that the vertices of all unit equiangular triangles pointing down satisfy the Pascal Identity, æ ç è n r     s ö ÷ ø =  æ ç è n-1 r-1     s ö ÷ ø +  æ ç è n-1 r     s-1 ö ÷ ø except for the triangle in the center, formed by the frontier lines between the sextants; that is æ ç è 0 0     0 ö ÷ ø ¹ æ ç è -1 -1     0 ö ÷ ø +  æ ç è -1 0     -1 ö ÷ ø This doesn't cause trouble in practice, since we would always confine ourselves to a situation in which either r³0 or s³0 for all binomial coefficients entering into the discussion (e.g., in relation to a binomial expansion). Figure 1b: Directions in which n, r, s are constant.

In those earlier papers the following pattern figured prominently: given a 'fundamental parallelogram' P = ABCD

Figure 2 should make it plain why we have described (0.1) as a Star of David Theorem. It is very substantially generalized from the original special case [1], in which no distortion of the star takes place.

We describe the vertices A, C, E, G, as the even vertices, and B, D, F, H as the odd vertices, because the number of different edges traversed between the anchor A and the vertex in question (regardless of the length of the edge) is an even number for A, C, E, G, and an odd number for B, D, F, H. This fact may be seen by counting the number of edges between the anchor and the vertex, on the illustration in Figure 3. Equivalently, the parity may be determined by counting the number of k_i's that are changed in the bottom of the symbol

Figure 3.

The weight of P is given by

W(P) =  product     of     even     vertices product    of     odd     vertices =  A×C×E×G B×D×F×H ,

so that, by a straightforward calculation,

W(P) =  n!(n+a1+a2)!(n+a2+a3)!(n+a1+a3)! (n+a1)!(n+a2)!(n+a3)!(n+a1+a2+a3)!       (0.3)

Thus, for fixed a₁, a₂, a₃, W(P) is independent of k₁, k₂, k₃, and invariant under the action of sliding P in the plane n=constant. Further, the symmetric group S₃ acts on the triple (a₁, a₂, a₃). For any p ÎS₃ let P^* be the p-image of P, with its vertices labelled in the obvious way. Then A is invariant under p and we get a Hyperstar of David Theorem of the form

B*×C×D*×E×F*×G×H* = B×C*×D×E*×F×G*×H      (0.4)

If p is a cyclic permutation of length 3 (as illustrated in Figure 3), then (0.4) is a genuinely new identity. However, if p leaves some a_j fixed, then (0.4) degenerates to a disguised form of (0.1). Thus if p simply exchanges a₁ and a₂, then p leaves both the anchor A and the vertex F fixed (F^*=F), and (0.4) is a simple consequence of the two Star of David identities

Figure 4.

Our other concern is to extend the domain of the multinomial coefficients

æ ç è     n k1 k2 ...  km ö ÷ ø

to permit any integer values of n, k₁, k₂, ..., k_m, subject, of course, to the identity n=k₁+k₂+... +k_m.

The case m=3 is then of special geometrical interest. In this case it is natural to ask "What is the 3-dimensional analogue of the Pascal Hexagon?" We certainly expect the analogue of the Pascal Triangle to be the Pascal Tetrahedron (see Figure 5). In fact, extending the Pascal Tetrahedron in 3-dimensions, in a manner analogous to extending the Pascal Triangle in 2-dimensions, produces a certain semi-regular (Archimedean) polyhedron. Here we are content briefly to identify it as the cuboctahedron, but in an Appendix to this paper we give the complete answer to our question along with many features of the analogy.

A useful piece of information concerning this polyhedron is found in Section 2; namely, for general m, if n is negative, then

These (generalized) multinomial coefficients may then be used to provide power series expansions of (a₁+a₂+... +a_m)ⁿ, for n<0 in suitable domains of R^m.

In Section 1 we describe the properties of weights of fundamental hyperparallelepipeds (hppd's) in dimension m. We formulate and establish a general Hyperstar of David Theorem. In Section 2 we extend the Pascal m-simplex of multinomial coefficients to allow n to be a negative integer, and justify formula (0.6). In Section 3 we discuss generalizations of our results. In particular we point out that the multinomial coefficients may be replaced by suitable separable functions of the variables n, k₁, k₂,...,k_m. We are also concerned in this section with generalizing an interesting observation of Hoggatt-Alexanderson [6] relating to the 'nearest neighbours' of a given multinomial coefficient