International Journal of Mathematics and Mathematical Sciences
Volume 21 (1998), Issue 3, Pages 581-586
doi:10.1155/S0161171298000805
A closer look at some new identities
Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra ACT 0200, Australia
Received 23 February 1996
Copyright © 1998 Geoffrey B. Campbell. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We obtain infinite products related to the concept of visible from the origin point vectors.
Among these is
∏k=3∞(1−Z)φ,(k)/k=11−Zexp(Z32(1−Z)2−12Z−12Z(1−Z)), |Z|<1,
in which φ3(k) denotes for fixed k, the number of positive integer solutions of (a,b,k)=1 where
a<b<k, assuming (a,b,k) is the gcd(a,b,k).