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t -discriminants with parameters.

, t -discriminants with parameters. // Taurida Journal of Computer Science Theory and Mathematics, – 2019. – T.18. – №2. – P. 77-
logo DOI https://doi.org/10.37279/1729-3901-2019-2-43-77-99

Quadratic irrationalities which have continues fractions decomposes of next forms:
$$α(h, t) = \frac{\sqrt{D} − b}{a} = [q_0, \overline{q_1, q_2, ..., q_n, h, q_n, ...q_2, q_1, tq_0}],$$
$$α_1(h, t) = \frac{\sqrt{D_1} − b_1}{a_1} = [q_0, \overline{q_1, q_2, ..., q_n, h, h, q_n, ...q_2, q_1, tq_0}],$$
$$α_2(h_1, h_2, t) = \frac{\sqrt{D_2} − b_2}{a_2} = [q_0, \overline{q_1, q_2, ..., q_n, h_1, h_2, q_n, ...q_2, q_1, tq_0}]$$
are considered in this paper. $h, h_1, h_2, t ≥ 2$ 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 $t ≥ 2$.
In third case for everyone $t ≥ 2$ the monotone dependence is a more complicated. For fixed $h_1 α_2$ is monotonically increasing (decreasing) with respect to $h_2$ and for fixed $h_2 α_2$ is monotonically decreasing (increasing) with respect to $h_1$ depending on "semiperiod" length $n$.
The monotonicity with respect to parameter $t ≥ 2$ 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.
Keywords: t-discriminants, continued periodic fractions with parameters, monotonicity

UDC: 
511.41