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In this paper, we first prove that for any strong divisibility sequences$\boldsymbol{a} = \left(a_n\right)_{n\geq 1}$, we have the identity:$\mathrm{lcm} \left\lbrace \binom{n}{0}_{\bf{a}}, \binom{n}{1}_{\bf{a}},\dots,\binom{n}{n}_{\bf{a}} \right\rbrace = \frac{\mathrm{lcm} \left(a_1,\dots , a_n, a_{n+1}\right)}{a_{n+1}}$ $\left(\forall n \geq 1\right)$, generalizing theidentity of Farhi (obtained in 2009 for $a_n=n$). Then, we derive from this onesome other interesting identities. Finally, we apply those identities toestimate the least common multiple of the consecutive terms of some Lucassequences. Denoting by $\left(F_n\right)_n$ the usual Fibonacci sequence, weprove for example that for all $n \geq 1$, we have \[\Phi^{\frac{n^2}{4}-\frac{9}{4}} \leq \mathrm{lcm}\left(F_1,\dots,F_n\right)\leq \Phi^{\frac{n^2}{3}+\frac{4n}{3}} , \] where $\Phi$ denotes the goldenratio.

Identities and estimations involving the least common multiple of strong divisibility sequences