Visible point vector summations from hypercube and hyperpyramid lattices (Q1273335)

Summary: New identities are given, based on ideas related to visible (from the origin) point vectors. Each result was found from summing on vpv lattices dividing space into radial regions from the origin. This is related to recent work by the author in which new partition type infinite products were derived. Also recently, \textit{M. Baake}, \textit{U. Grimm}, and \textit{D. H. Warrington} [J. Phys. A, Math. Gen. 27, No. 8, 2668-2674 (1994)] and \textit{R. Mosseri} [J. Phys. A, Math. Gen. 25, No. 1, L25--L29 (1992; Zbl 0771.11026)] considered the 2-D 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 \textit{M. L. Glasser} and \textit{I. J. Zucker} [Theoretical chemistry, Adv. and perspect. 5, 67-139 (1980; Zbl 0638.33007)], and in \textit{B. W. Ninham}, \textit{M. L. Glasser}, \textit{B. D. Hughes}, and \textit{N. E. Frankel} [Physica A 186, 441-481 (1992)] involving Mellin and Möbius inversions are applicable also to the current paper.