Infinite products over visible lattice points

Abstract

ABSTRACT. About fifty new multivariate combinatorial identities are given, connected with partition theory, prime products, and Dirichlet series. Connections to Lattice Sums in Chemistry and Physics are alluded to also.

Visible point vector summations from hypercube and hyperpyramid lattices

“…Another way of interpreting formulae such as (2.6) is to take logarithms of both sides and equate In this section we give two theorems as appear in Campbell [7][8]. The first ofthese is THEOREM 2.…”

“…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.…”

“…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 [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.…”

“…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 The differentiated logarithm of this result is quite easy to prove using lemma 2.1 as shown in Campbell [5][6][7][8][9][10] In this and henceforth we use the terminology for the polylogarithm, (see Lewin [13] …”

“…The first ofthese is THEOREM 2. 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:…”