$t$-Discriminants with parameters

D. V. Tretyakov ; V. Piskunova

Taurida Journal of Computer Science Theory and Mathematics, no. 2 (2019), pp. 77-99

Résumé

Quadratic irrationalities which have continues fractions decomposes of next forms: \begin{gather*} \alpha(h,t)=\frac{\sqrt{D}-b}{a}= [q_{0},\overline{q_{1},q_{2},...,q_{n},h,q_{n},...q_{2},q_{1},tq_{0}}],\\ \alpha_{1}(h,t)=\frac{\sqrt{D_{1}}-b_{1}}{a_{1}}= [q_{0},\overline{q_{1},q_{2},...,q_{n},h,h,q_{n},...q_{2},q_{1},tq_{0}}],\\ \alpha_{2}(h_1,h_2,t)=\frac{\sqrt{D_{2}}-b_{2}}{a_{2}}= [q_{0},\overline{q_{1},q_{2},...,q_{n},h_{1},h_{2},q_{n},...q_{2},q_{1},tq_{0}}] \end{gather*} are considered in this paper. $h, \ h_{1}, \ h_{2}, \ t \geq 2$ are natural parameters and number system $\langle q_{1},q_{2},...,q_{n},q_{n},...q_{2},q_{1}\rangle$ is palindrome. Formulas for calculating $D, \ D_{i}, \ a, \ a_{i}, \ b, \ b_{i}, \ i=1,2$ are obtained. Monotone irrationalities properties with respect to parameters are investigated. Case $t=2$ is previously considered. In first of two cases indicated monotonicity is depend on “semiperiod” length $n$ for everyone $t \geq 2$. In third case for everyone $t \geq 2$ the monotone dependence is a more complicated. For fixed $h_{1}$ $\alpha_{2}$ is monotonically increasing (decreasing) with respect to $h_{2}$ and for fixed $h_{2}$ $\alpha_{2}$ is monotonically decreasing (increasing) with respect to $h_{1}$ depending on “semiperiod” length $n$. The monotonicity with respect to parameter $t \geq 2$ investigated too. Obtained dependence is rather different and is not depending on “semiperiod”. Oblique asymptote is found in all cases. Every considered case is illustrated by examples.

Détail

Mots-clés : t-discriminants
Keywords: continued periodic fractions with parameters, monotonicity.