On the Legendre constant of \(\alpha \)-continued fractions (Q626836)

The author proves the existence of the Legendre constant of \(\alpha\)-continued fractions, or the \(\alpha\)-Legendre constant for \(0<\alpha<1/2\). Its existence for \(1/2\leq\alpha\leq 1\) is proven by \textit{C. Kraaikamp} in [Acta Arith. 57, 1--39 (1991; Zbl 0721.11029)]. She also gives its upper and lower estimates for each \(\alpha\) with \(0<\alpha<\sqrt{2}-1\) as follows. If \(1/(k+1)<\alpha<1/k\) for an integer \(k\), then the \(\alpha\)-Legendre constant is between \(1/(k+2)\) and \(1/k\).