Lucas sequences with cyclotomic root field
Tom 490 / 2013
Streszczenie
A pair of Lucas sequences $U_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$ and $V_n=\alpha^n+\beta^n$ is famously associated with each polynomial $x^2-Px+Q\in\mathbb Z[x]$ with roots $\alpha$ and $\beta$. It is the purpose of this paper to show that when the root field of $x^2-Px+Q$ is either $\mathbb Q(i)$, or $\mathbb Q(\omega)$, where $\omega=e^{2\pi i/6}$, there are respectively two and four other second-order integral recurring sequences of characteristic polynomial $x^2-Px+Q$ that are of the same kinship as the $U$ and $V$ Lucas sequences. These are, when $\mathbb Q(\alpha,\beta)=\mathbb Q(i)$, the $G$ and the $H$ sequences with $$ G_n=[(1-i)\alpha^n+(1+i)\bar\alpha^n]/2,\quad H_n=[(1+i)\alpha^n+(1-i)\bar\alpha^n]/2, $$ and, when $\mathbb Q(\alpha,\beta)=\mathbb Q(\omega)$, the $S$, $T$, $Y$ and $Z$ sequences given by \begin{eqnarray*} S_n &=&(\omega\alpha^n-\bar\omega\bar\alpha^n)/\sqrt{-3},\ T_n & = &(\omega^2\alpha^n-\bar\omega^2\bar\alpha^n)/\sqrt{-3},\ Y_n &=&\bar\omega\alpha^n+\omega\bar\alpha^n,\\ Z_n & = &\omega\alpha^n+\bar\omega\bar\alpha^n, \end{eqnarray*} where $\bar\alpha=\beta$ and $\bar\omega=e^{-2\pi i/6}$. Several themes of the theory of Lucas sequences have been selected and studied to support the claim that the six sequences $G$, $H$, $S$, $T$, $Y$ and $Z$ ought to be viewed as Lucas sequences.
