The order of appearance of the product of the first and second Lucas numbers

Let a and b be relatively prime integers. Then the first Lucas sequence (Uₙ)n≥0 and the second Lucas sequence (Vₙ)n≥0 are defined respectively by Uₙ₊₂=aUₙ₊₁+bUₙ,\, U₀=0,\,U₁=1 and Vₙ₊₂=aVₙ₊₁+bVₙ,\, V₀=2,\,V₁=a, where n≥0. Let m be an integer with (m,\,b)=1. Then the smallest positive integer k satisfying m Uₖ is called the order of appearance of m in the first Lucas sequence (Uₙ)n≥0, denoted by $τ(m)$, i.e., τ(m):=min≥1:m Uₖ\. When $a>0$ and Δ=a²+4b>0, we give explicit formulae for τ(Uₘ Vₙ), τ(Uₘ Uₙ), τ(Vₘ Vₙ) and τ(UₙUₙ₊ₚUₙ₊₂ₚ), thus generalizing the results of Irmak and Ray.