Lucas sequences with cyclotomic root field

Christian Ballot. Lucas sequences with cyclotomic root field. 2013. <http://eudml.org/doc/285988>.

@book{ChristianBallot2013,
abstract = {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.},
author = {Christian Ballot},
keywords = {Lucas sequences; identities; laws of appearance and repetition; congruences; Wolstenholme congruence; divisibility; prime density},
language = {eng},
title = {Lucas sequences with cyclotomic root field},
url = {http://eudml.org/doc/285988},
year = {2013},
}

TY - BOOK
AU - Christian Ballot
TI - Lucas sequences with cyclotomic root field
PY - 2013
AB - 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.
LA - eng
KW - Lucas sequences; identities; laws of appearance and repetition; congruences; Wolstenholme congruence; divisibility; prime density
UR - http://eudml.org/doc/285988
ER -