Congruences for q-Lucas Numbers

@article{Pan2013CongruencesFQ,
  title={Congruences for q-Lucas Numbers},
  author={Hao Pan},
  journal={Electr. J. Comb.},
  year={2013},
  volume={20},
  pages={P29}
}
  • Hao Pan
  • Published 2013 in Electr. J. Comb.
For α, β, γ, δ ∈ Z and ν = (α, β, γ, δ), the q-Fibonacci numbers are given by F ν 0 (q) = 0, F ν 1 (q) = 1 and F ν n+1(q) = q αn−βF ν n (q) + q γn−δF ν n−1(q) for n > 1. And define the q-Lucas number Ln(q) = F ν n+1(q)+ q γ−δF ν∗ n−1(q), where ν∗ = (α, β− α, γ, δ − γ). Suppose that α = 0 and γ is prime to n, or α = γ is prime to n. We prove that Ln(q) ≡ (−1) (mod Φn(q)) for n > 3, where Φn(q) is the n-th cyclotomic polynomial. A similar congruence for q-Pell-Lucas numbers is also established. 

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