VISIBLE POINT VECTOR SUMMATIONS FROM HYPERCUBE AND HYPERPYRAMID LATTICES

New identities are given, based on ideas related to visible (from the origin) point vectors. Each related to recent work by the author in which new partition type infinite products were derived. Also recently, Baake et al [3] and Mosseri [14] considered the 2-/9 visible lattice points as part of’ an optical experiment in which the so-called Optical Fourier Transform was applied. Many of. the techniques espoused in Glasser and Zucker [11 ], and in Ninham et al 15] involving Mellin and M6bius inversions are applicable also to the current paper.

In recent papers by the author [5][6][7][8][9][10], new infinite product identities were proven.These were termed visible (from the origin) point vector identities, or simply vpv identities.Each result was found from summing on vpv lattices dividing space into radial regions from the origin.In particular, in Campbell [8] the regions considered for the lattice sums were triangular, cubic, hypercubic, pyramidic, or hyperpyramidic depending on the dimension ofthe space.
Recently also in Baake et al [3] and Mosseri [14] the visible lattice points were considered as part of an optical experiment in which the so-called Optical Fourier Transform was applied to the 2-D vpv's.Apostol  [2] gave an excellent introduction to the ideas behind vpv's and calculated their density in space Many of the techniques discussed in Glasser and Zucker 11 ], and in Ninham et al 15] involving Mellin and MObius inversions are applicable to vpv identities from the author's papers [5][6][7][8][9][10] The current paper seeks in part to explore this idea further.It then also adds the perspective that the neatest presentation of these results is in the use of the polylogarithm function Lim(z as described in Lewin [13], and given below in ((1.2) In Andrews' book on partitions [1] an historical perspective is given of generating functions for partitions, and an introduction to works on plane partitions and vector partitions.The identities of the author's papers [5][6][7][8][9][10], especially [7] and within comexts such as those presented in Ninham et al [15], and Glasser and Zucker [11 highlight the possibility of modem application of ideas from this book.Each vpv infinite product turned out to be combinatorially equivalent to a theorem on weighted vector partitions.(see Campbell [6]) 2.

HYPERCUBE IDENTITIES.
The underlying concept for proof ofthese vpv identities comes from the LEMMA 2.1.(see Campbell [5][6][7][8][9][10]) Consider an infinite region raying out of the origin in any Euclidean vector space.The set of all lattice point vectors from the origin in that region is precisely the set of posltive integer multtples of the visible point vectors (vpv's) m that region.
We note that when m in (2.3) is a positive integer, Li_=(z) d--(1-Following easily from theorem 1.2 are the two particular cases, logz (2.4) l" The combinatorial interpretation of (2.5) may be likened to assigning stepping stones to each lattice point in the first quadrant, and likewise assigning a value for the effort involved in each jump to a successive stone.The "effort" in the case of(2.5) has a value of (_l)(O.b)1-1/"b| .(a,b)Jalong the vector <a, b> from any existing stepping stone.The person jumping (we assume) will jump to a stone which is always further away from the origin.In combinatorial terms, (2.5) states that THEOREM 2.3.Let v <a, b> be a vector with integer coordinates such that a >_ 0 and (o.b)(-1/b]Then the b > 1, and assume that in walking from the origin to v we assign an effort of (-1) (a,b) total effort from performing all possible such jumps from the origin to lattice poim v in the first quadrant is given by , where S is the set of unrestricted vector partitions of v with integer coordinates such that a > 0 and b >_ 1, and (2.7) is also equal to the coefficiem of y"z in (1-z) -1m-y).
Another way of interpreting formulae such as (2.6) is to take logarithms of both sides and equate coefficients of like powers of z yielding essentially COROLLARY 2. In this section we give two theorems as appear in Campbell [7][8].The first ofthese is THEOREM 2.5.(see Campbell [8]) If 1, 2 n; (n 1) then for each }x,,x,,_,l,...,lx,,x,,_,...x,I < 1, and b, C, (2.9) The logarithm of this result is quite easy to prove using the technique in Campbell [5][6][7][8][9][10], by summing on the vpv's in the n-space hyperpyramid defined by the inequalities: x <x., x <x x,,_, <x.(2.10) in the first n-space hyperquadrant, and applying lemma 2.1.Alternatively, the logarithmic derivative of both sides of (2.9) gives rise to an easily proven identity, again using lemma 2.1 in the process.(see Campbell  [9]) The corresponding theorem from Campbell [7] was summed very simply over all lattice point vectors in the first hyperquadrant.The 2-D and 3-D cases of that are as follows: COROLLARY 2.6.If lyz] and [z] < 1, and s +  (2.13)This result is again quite easy to prove by summing on the vpv's in the n-space hyperpyramid defined by the inequalities: X <X,, X <In, X,_l <x.(2.14) in the first n-space hyperquadrant, and applying lemma 2.1.
The simplest examples are from setting (hence p, q, r, and s 0) in (2.11) and (2.12), ( These results are given in Campbell [6][7] without the full generality of theorem 2.5, and proven in Campbell [9] from another perspective.Particular cases of (2.15) to (2.18) are examined in Campbell [8], where they are shown to have non-trivial simple cases.For example, if each of v, w, x, and y are set equal to z, then the binomial coefficients appear as exponents in the fight sides of (2.15) to (2.18), and (2.18) for example, becomes (2.19)This particular case was first given in the authoPs paper [8].The right side is easily expanded into a power series in z and the left side when expanded enumerates weighted vpv partitions.

3.
A GENERAL HYPERPYRAMID THEOREM.In Campbell [7] the main theorem of the paper came from summing on all lattice points in the first hyperquadrant.In section 2 this is restricted to hyperpyrarnid lattices with symmetry in n out of the n variables This is evident in viewing (2.9) to (2.13).If we choose to vary the shape of our hyperpyramid the process is identical to the type of multidimensional sum used in the author's [8] to prove theorem 2.2.In [8], which the current paper is sequel to, it becomes clear from this that the general hyperpyramid sums leading to infinite products are included in the THEOREM 3.1.(see Campbell [8]) For conditions of theorem .8, and in addition for a, R we have (3.1)where -1 if a,, <1. (3.2) This theorem counts terms on any possible hyperpyramid lattice in the first hyperquadrant, where the hyperpyramid apex is at the origin.This theorem led to the full range of so called hyperpyramid infinite product identities in [8].Theorems 2.8 and 3.1 are each cases of the more general We see that this theorem counts terms on the same hyperpyramid lattice as does theorem 2.2 which is the case where m 1.

4.
OUTLINE OF THE PROOF OF THEOREM 3.2, AND SOME EXAMPLES.This involves basically the same method given in Campbell [7] and is related also to the method given in [8].We consider only the case with n 2 as it shows the method for the general case, and there is no problem in generalizing the process.We start with the sum valid for the criteria of the We note by illustration that this is easily seen for a 1/2 to be z' z' I y._) z< y.7_ 1 z'< y _ i T l z I y' y . . 7 1i--;-+--I+ +7 I+ +7 I+ +7" I+]7+ +'",   so equating (4.1) and (4.4) yields the n 2 equivalent of theorem 3.2.The method used here can equally apply to the successive cases n 3, 4, 5  Progressing in this manner we see that the method is essentially the same for each dimension.This is due to the symmetry in the non-vertical axes of the lattice.
The cases presented as corollaries of the theorems so far were given in Campbell [8] although other types of corollaries than infinite products could have been given.That is, the first few examples taken with m 2 and successively n 2, 3,   ,j=l j' J k' provided r + s + 2.
a.b< a.b>O,>l (4.4)Both of these appear to be new results A,CKNOWLEDGEMENT.The author thanks Professor George E Andrews for his continued moral support and the Australian National University for their continued association during 1995.