Visible Point Vector Identity

A set of identities involving $n$-D visible lattice points was discovered by Campbell (1994). Examples include

$$\prod_{\scriptstyle (a,b)=1\atop\scriptstyle a\geq 0, b\leq 1} (1-y^az^b)^{-1/b}=(1-z)^{-1/(1-y)}$$

for $\vert yz\vert, \vert z\vert<1$ and

$$\prod_{\scriptstyle (a,b,c)=1\atop\scriptstyle a,b\geq 0, c\leq 1} (1-x^ay^bz^c)^{-1/c}=(1-z)^{-1/[(1-x)(1-y)]}$$

for $\vert xyz\vert, \vert xz\vert, \vert yz\vert, \vert z\vert<1$.

References

Campbell, G. B. ``Infinite Products Over Visible Lattice Points.'' Internat. J. Math. Math. Sci. 17, 637-654, 1994.

Campbell, G. B. ``Visible Point Vector Identities.'' http://www.geocities.com/CapeCanaveral/Launchpad/9416/vpv.html.