On the triplicity among infinite products, infinite series, and continued fractions; and its applications to divergent series
Kiyoshi Sogo
TL;DR
The paper introduces the concept of triplicity $P=S=C$ among infinite products, infinite series, and continued fractions, and demonstrates how this correspondence can be used to evaluate divergent series. It develops systematic methods to generate continued fractions from series via Euler-type, Gauss–Heine-type, Muir–Rogers-type, and Ramanujan-type frameworks, with pivot-variable technology central to the construction. It provides numerous examples, including Gauss’s divergent sums and the Rogers–Ramanujan identities, illustrating non-uniqueness of the $S$ component and the utility for analytic continuation. It further connects triplicity to determinant representations and the Stieltjes moment problem, and discusses extensions, limitations, and open questions in continued fractions and divergent-series analysis.
Abstract
Many identities written by $P=S=C$ are obtained, where $P$ infinite products, $S$ infinite series, and $C$ continued fractions. Such equality is called {\it triplicity}, and it can be used to compute the values of infinite series. It is applied even to obtain sums of divergent series. Many examples of such infinite series are shown, including $1-2+2^3-2^6+\cdots$, which is in Entry 7 of Gauss's diary and its value $0.4275251302\cdots$ is here obtained.