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

Geoffrey B. Campbell

Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra ACT 0200, Australia

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).