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

Abstract

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.