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International Journal of Mathematics and Mathematical Sciences
Volume 21, Issue 3, Pages 581-586
http://dx.doi.org/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

Received 23 February 1996

Copyright © 1998 Hindawi Publishing Corporation. 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).