Lucas sequences with cyclotomic root field

Christian Ballot

GDML_Books, (2013), p.

Résumé

A pair of Lucas sequences Uₙ = (αⁿ-βⁿ)/(α-β) and Vₙ = αⁿ + βⁿ is famously associated with each polynomial x² - Px + Q ∈ ℤ[x] with roots α and β. It is the purpose of this paper to show that when the root field of x² - Px + Q is either ℚ(i), or ℚ(ω), where $ω = e^{2πi/6}$, there are respectively two and four other second-order integral recurring sequences of characteristic polynomial x² - Px + Q that are of the same kinship as the U and V Lucas sequences. These are, when ℚ(α,β) = ℚ(i), the G and the H sequences with Gₙ = [(1-i)αⁿ + (1+i)α̅ⁿ]/2, Hₙ = [(1+i)αⁿ + (1-i)α̅ⁿ]/2, and, when ℚ(α,β) = ℚ(ω), the S, T, Y and Z sequences given by Sₙ = (ωαⁿ - ω̅α̅ⁿ)/√(-3), Tₙ = (ω²αⁿ - ω̅²α̅ⁿ)/√(-3), Yₙ = ω̅αⁿ + ωα̅ⁿ, Zₙ = ωαⁿ + ω̅α̅ⁿ, where α̅ = β and $ω̅ = e^{-2πi/6}$. 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.

Détail

Zbl 1306.11017

EUDML-ID : urn:eudml:doc:285988