A Closer Look At Some New Identities

Mathematics Research Report MRR95-055

A Closer Look At Some New Identities

Geoffrey B. Campbell

Abstract: We obtain infinite products related to the concept of visible from the origin point vectors. Among these is $$\prod^{\infty}_{k = 3} \ (1 - z^k)^{\varphi_{3} (k) /k} = {\frac{1}{1 - z}} \ {exp} \ \left( {\frac{z^3}{2(1 - z)^2}} - {\frac{1}{2}} z \ - {\frac{1}{2z (1 - z)}} \right), \ \ \ |z| &le 1, $$ \noindent in which $\varphi_3 (k)$ 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 $(a, b, k)$.

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