# On calculation of special type numerical series generated by 4-th order recurrent sequences.

Authors:
Муртазаева Д. С., Tretyakov D. V. On calculation of special type numerical series generated by 4-th order recurrent sequences. // Taurida Journal of Computer Science Theory and Mathematics, – 2016. – T.15. – №3. – P. 59- https://doi.org/10.37279/1729-3901-2916-15-3-59-67

Calculation of special type numerical series generated by 4-th order recurrent sequences is considered in this article. All sequences are satisfying of equation $v_{n+2}=av_{n}+bv_{n-2}$, where $a>0$, $b\in \mathbb{R}$, $a^{2}-4b>0$. Initial conditions are connecting in some cases.
Let $\left \{ v_{n} \right \}_{n\geq 1}$ is indicated sequence. One is satisfying equalities:
$v_{2k+1}^{2}-v_{2k+3}v_{2k-1}=b^{k-1}\left ( v_{3}^{2}-v_{5}v_{1} \right )$, $v_{2k+2}^{2}-v_{2k+4}v_{2k}=b^{k-1}\left ( v_{4}^{2}-v_{6}v_{2} \right )$, $k\geq 1$.
If $\left \{ w_{n} \right \}_{n\geq 0}$, $w_{0}=0$, $\left \{ g_{n} \right \}_{n\geq -1}$, $g_{-1}=0$ are sequences which satisfying to equation $h_{n+2}=ah_{n}+bh_{n-2}$, $b>0$, than
$w_{2n}w_{2n+2}-w_{2n-2}w_{2n+4}=-ab^{n-1}w_{2}^{2}$, $g_{2n+1}g_{2n+3}-g_{2n-1}g_{2n+5}=-ab^{n-1}w_{1}^{2}$, $n\geq 1$.
In article proofed next theorems.
Theorem 1.Let $\left \{ v_{n} \right \}_{n\geq 1}$ is sequence which defines the equalities $v_{n+2}=av_{n}-bv_{n-2}$, $n\geq 3$.
Than
$\sum_{n=2}^{\infty }arcctg\left ( \frac{av_{2n+1}^{2}}{\sqrt{b}b^{n-1}\Delta _{1}} \right )=arcctg\left ( \frac{a+\sqrt{a^{2}-4b}}{2\sqrt{b}} \right )-arcctg\left ( \frac{v_{3}}{\sqrt{b}v_{1}} \right )$,
where $\Delta _{1}=v_{3}^{2}-v_{1}v_{5}$,
$\sum_{n=2}^{\infty }arcctg\left ( \frac{av_{2n+2}^{2}}{\sqrt{b}b^{n-1}\Delta _{2}} \right )=arcctg\left ( \frac{a+\sqrt{a^{2}-4b}}{2\sqrt{b}} \right )-arcctg\left ( \frac{v_{4}}{\sqrt{b}v_{2}} \right )$,
where $\Delta _{2}=v_{4}^{2}-v_{2}v_{6}$.
Theorem 2.revious theorem results are correcting with the help from choice of initial conditions.
Theorem 3.Let sequence $\left \{ w_{n} \right \}_{n\geq -1}$ with initial conditions $w_{-1}=0$, $w_{0}=0$ are satisfying to equation $w_{n+2}=aw_{n}+bw_{n-2}$, $b>0$. Than
$\sum_{n=2}^{\infty }arcctg\left ( \frac{w_{2n+2}}{w_{2}a\sqrt{b^{n-1}}} \right )=arcctg\left ( a \right )+arcctg\left ( \frac{a^{2}+b}{b} \right )$,
$\sum_{n=1}^{\infty }arcctg\left ( \frac{w_{2n+1}}{w_{1}a\sqrt{b^{n-1}}} \right )=arcctg\left ( a \right )+arcctg\left ( \frac{a^{2}+b}{\sqrt{b}} \right )$.
These theorems are illustrating by some examples.

Keywords:4-th order recurrent sequences, numerical series of special type, biquadratic sequences, characteristic equation, 4-th order Fibonacci generalized sequences

UDC:
517.52