A note on Hurwitzian numbers (Q1812491)

The number \(x\) is called Hurwitzian if the regular continued fraction expansion of \(x\) can be written as \(x=[a_0;a_1,\dots,a_n,\overline{a_{n+1}(k), \dots,a_{n+p}(k)}]_{k=0}^\infty\), where \(a_{n+1}(k),\dots, a_{n+p}(k)\) are polynomials with rational coefficients which take positive integral values for \(k=0,1,2,\dots\), and at least one of them is not constant. In this note Hurwitzian numbers are defined for three other continued fraction expansions, namely, the nearest integer continued fraction expansion, the backward continued fraction expansion, and \textit{H. Nakada}'s \(\alpha\)-continued fraction expansions [Tokyo J. Math. 4, 399-426 (1981; Zbl 0479.10029)]. It is shown that their set of Hurwitzian numbers coincides with the classical set of Hurwitzian numbers.