Non-periodic continued fractions for quadratic irrationalities (Q2810678)

A well-known result states that any quadratic irrational has a continued fraction expansion which is periodic after a certain stage and the converse was also proved. This theorem was proved by Lagrange in 1770. The author investigates here a continued fraction of the form NEWLINENEWLINE\[NEWLINE F(x):= x+\frac{1}{1/x}+\frac{1}{x^2}+\frac{1}{1/x^2}+\cdots+\frac{1}{x^k}+\frac{1}{1/x^k}+\cdots, \; x \in \mathbb{N}, NEWLINE\]NEWLINE NEWLINEwhich is briefly written as follows NEWLINENEWLINE\[NEWLINE F(x):=\left[x,\frac{1}{x},x^2,\frac{1}{x^2}, x^3, \frac{1}{x^3}, \cdots, x^k, \frac{1}{x^k}, \cdots \right]. NEWLINE\]NEWLINE NEWLINEThe following theorems are proved in this article.NEWLINENEWLINENEWLINETheorem. For \(k \geq 1\), let NEWLINENEWLINE\[NEWLINE F_k(x):=\left[x,\frac{1}{x},x^2,\frac{1}{x^2}, x^3, \frac{1}{x^3}, \cdots, x^k, \frac{1}{x^k} \right], NEWLINE\]NEWLINE and NEWLINENEWLINE\[NEWLINE F_k^*(x):=\left[x,\frac{1}{x},x^2,\frac{1}{x^2}, x^3, \frac{1}{x^3}, \cdots, x^k \right], NEWLINE\]NEWLINE NEWLINEbe truncations of \(F(z)\) to given odd and even convergents, respectively. Then \(F_k(x)\) and \(F_k^*(x)\) converge to the positive root of \(P(x,z)=z^2-(2x-1)z-x\).NEWLINENEWLINENEWLINETheorem. The continued fraction NEWLINE\[NEWLINE F(x,s):=\left[x,\frac{s}{x},x^2,\frac{s}{x^2}, x^3, \frac{s}{x^3}, \cdots, x^k, \frac{s}{x^k}, \cdots \right], \qquad s \in \mathbb{N}, NEWLINE\]NEWLINE converges to the positive root of the quadratic \(P(x,s,z)=sz^2-((s+1)x-1)z-x\).