Arithmetical properties of the leaping convergents of \(e^{1/s}\)
The regular continued fraction expansion of \(e^{1/s}\) for \(s\geq 2\) is \([1;\,\overline{s(2k-1)-1,\,1,\,1}]_{k=1}^\infty\). It is natural to take the 3-contraction of this expansion. In theorem 1 it is proved that this contraction is \[ e^{1/s}=1+\frac{2}{2s-1}\,{\quad\atop +}\,\frac 1{6s}\,{\quad\atop +}\,\frac 1{10 s}\,{\quad\atop +}\,\frac 1{14 s}\,{\quad\atop +\dots} \] and that its canonical approximants \(P_n/Q_n\) satisfy \[ \sum_{t=0}^r\frac{P_tQ_{r-t}}{t!(r-t)!}=(4s)^r. \] Inspired by this the author also derives the \((2\ell+1)\)-contraction of the regular continued fraction \([a_0;\,\overline{c_0+dk,\,c_1,\,\dots,\,c_{2\ell}}]_{k=0}^\infty\).
scientific article; zbMATH DE number 2124813