Jacobsthal and Pell curves. (English) Zbl 0644.10014

For \(\theta\) real, let \[ (i)\quad x=(\alpha^{\theta}-\alpha^{-\theta}\cos\theta\pi)/2\sqrt{2}, \quad y=-\alpha^{-\theta}\sin \theta \pi /2\sqrt{2},\quad \alpha =1+\sqrt{2}; \]
\[ (ii)\quad x=\alpha^{\theta}+\alpha^{-\theta}\cos \theta \pi,\quad y=\alpha^{-\theta}\sin\theta\pi, \quad \alpha =1+\sqrt{2}; \]
\[ (iii)\quad x=(2^{\theta}-\cos \theta \pi)/3,\quad y=-2^{-\theta}\sin \theta \pi /3;\quad (iv)\quad x=2^{\theta}+\cos \theta \pi,\quad y=2^{-\theta}\sin\theta\pi. \] If (x,y) are coordinates of a point in a plane, then (i), (ii), (iii), (iv) are the parametric equations of the Pell curve, the Pell-Lucas curve, the Jacobsthal curve, and the Jacobsthal-Lucas curve, respectively. Elementary properties of these curves are investigated. The (lacunary) locus of the stationary points of each of the curves is obtained.
The theory is generalized to include the Fibonacci curve and the Lucas curve, which are the subject of a separate review. When \(\theta =n\), integer, the x-values in (i), (ii), (iii), and (iv) reduce to the ordinary Pell, Pell-Lucas, Jacobsthal, and Jacobsthal-Lucas numbers, resp.
Reviewer: A.F.Horadam


11B39 Fibonacci and Lucas numbers and polynomials and generalizations
14H45 Special algebraic curves and curves of low genus
11B37 Recurrences