Visible Point Vector (vpv) Identities

Geoff Campbell


The following identities I found whilst experimenting with properties of series transformations.

The second and third formulas below are bivariate & trivariate examples of the so-called "Visible Point Vector" identities or simply "vpv" identities. They have interpretations in terms of total exertion required to jump on 2-D & 3-D stepping stones, taking all possible paths from one integer lattice point to another. The "effort" involved with each jump from one stepping stone to another is given by a coefficient based on whether or not the "destination" stone is "visible" from the "jump from" stone. In fact "all possible paths" corresponds to all possible vector partitions. We use the notation (a, b, c) to mean the greatest common divisor of a, b, and c. Similarly for 4-tuples, 5-tuples, etc.










These identities and others were published in the International Journal of Mathematics and Mathematical Sciences in December 1994. In those papers I coined the phrase "companion identities" to designate the fact that often these identities appear naturally as identities summed on symmetries of radial lattice point partitions which ray from the origin. (see CAMPBELL, G. B.: "Infinite products over visible lattice points", Internat. J. Math. & Math. Sci., Vol 17, No 4, 1994, 637-654; and " A closer look at some new identities", Internat. J. Math. & Math. Sci., Vol 21, No 3, 1998)

The following results I called hyperpyramid identities because the higher dimensional lattice sums considered are enclosed within infinitely extended hyperpyramids. Some of these will appear in 1998 in the International Journal of Mathematics and Mathematical Sciences. They are already preprinted at the Australian National University.











In particular, this leads to






Each side of the 4-variate infinite product above has the expansion involving determinants in the coefficients,






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