Diophantine analysis around \([1,2,3,\dots]\) (Q6606727)

The authors study the Diophantine properties of the number \(\mathfrak{z}\) defined by the continued fraction \([1;2,3,4, 5,\ldots]\), ie the \(n\)-th partial quotient of \(\mathfrak{z}\) is \(n+1\) for every integer \(n\ge 0\). The transcendence of \(\mathfrak{z}\) was proved in 1929 by \textit{C. L. Siegel} [Abh. Preuß. Akad. Wiss., Phys.-Math. Kl. 1929, No. 1, 70 S. (1929; JFM 56.0180.05)] as a consequence of his celebrated results on the transcendence of values of the modified Bessel's functions\N\[\NI_{\alpha}(x):=\sum_{k=0}^\infty \frac{(x/2)^{2k+\alpha}}{k!\Gamma(\alpha+k+1)}\N\]\Nat non-zero algebraic arguments, using the identity \(\mathfrak{z}=I_0(2)/I_1(2)\).\N\NLet \(p_n/q_n\) denote the \(n\)-th convergent of the continued fraction of \(\mathfrak{z}\). Let also consider the quadratic number \(Q_n:=[\overline{1;2,3, \ldots, n}]\) (where the pattern \(1,2,3, \ldots, n\) is repeated indefinitely). The two main results of the paper are the following: As \(n\to +\infty\), \N\[\N\left\vert \mathfrak{z}-\frac{p_n}{q_n}\right\vert \sim \frac{\ln\ln(q_n)}{q_n^2\ln(q_n)}\N\]\Nand, for any even integer \(n\ge 2\), \N\[\N\left\vert \mathfrak{z}-Q_n\right\vert \ll \frac{1}{n^2H(Q_n)^2},\N\]\Nwhere \(H(Q_n)\ll q_{n-1}/n\) is the height of the normalized minimal polynomial of \(Q_n\). These results follow from a careful study of the sequences \((p_n/q_n)_n\) and \((Q_n)_n\), by means of the identity \N\[\NQ_n=\frac{p_{n-1}-q_{n-2}+\sqrt{(p_{n-1}-q_{n-2})^2+4q_{n-1}p_{n-2}}}{2q_{n-1}}, \; n\ge 1.\N\]\NThe authors conclude the paper with explicit evaluations of various ``error sums'' series, such as \N\[\N\mathcal{E}_{fac}(\mathfrak{z}):=\sum_{n=0}^\infty \frac{(-1)^n}{n!}(q_n\mathfrak{z}-p_n)= \frac{I_2(2\sqrt{2})}{2I_1(2)}\N\]\Nand \N\[\N\mathcal{E}_{fac}^*(\mathfrak{z}):=\sum_{n=0}^\infty \frac{1}{n!}(q_n\mathfrak{z}-p_n)= \frac{1}{2I_1(2)}.\N\]\N\NFor the entire collection see [Zbl 1530.11003].