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 the identity of Farhi (obtained in 2009 for $a_n=n$). Then, we derive from this one some other interesting identities. Finally, we apply those identities to estimate the least common multiple of the consecutive terms of some Lucas sequences. Denoting by $\left(F_n\right)_n$ the usual Fibonacci sequence, we prove 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 golden ratio.
Categories: math.NT