A summation formula for divergent continued fractions (Q2404288)

The ``\(r/\varphi\)-algorithm'' for summing divergent continued fractions is based on the following ``paradoxical'' formula ([\textit{V. I. Shmoĭlov}, Continued fractions and the \(r/\varphi\)-algorithm (Russian). Taganrog: Izd. Tekhnologicheskiĭ Institut, Yuzhnyĭ Federal'nyĭ Universitet (2012; Zbl 1269.40001)] \[ \lim_{n \to \infty} \frac{\sin (n+1)\varphi}{\sin n\varphi}=e^{i\varphi}. \] Paradoxical expansions of imaginary quantities in divergent continued fractions with real elements have long been known. In this article, these formulas are refined and substantiated on the basis of the theory of uniform distributions, supplemented by certain statements of complex analysis.