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 ${\left(U_n\right)}_{n\ge 0}$ and the second Lucas sequence ${\left(V_n\right)}_{n\ge 0}$ are defined respectively by $U_{n+2}=a{U}_{n+1}+b{U}_n,U_0=0,U_1=1$ and $V_{n+2}=a{V}_{n+1}+b{V}_n,V_0=2,V_1=a$, where $n\ge 0$. Let $m$ be an integer with $\gcd (m,b)=1$. Then the smallest positive integer $k$ satisfying $m\mid U_k$ is called the order of appearance of $m$ in the first Lucas sequence $(U_n{)}_{n\ge 0}$, denoted by $\tau (m)$, i.e., $\tau (m):=\min \{k\ge 1:m\mid U_k\}$. When $a>0$ and $\mathrm{\Delta }=a^2+4b>0$, we give explicit formulae for $\tau (U_m{V}_n),\tau (U_m{U}_n)$, $\tau (V_m{V}_n)$ and $\tau (U_n{U}_{n+p}{U}_{n+2p})$, thus generalizing the results of Irmak and Ray.