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This paper is devoted to establish nontrivial effective lower bounds for theleast common multiple of consecutive terms of a sequence ${(u_n)}_{n \in\mathbb{N}}$ whose general term has the form $u_n = r {[n]}_q + u_0$, where $q, r$ are positive integers and $u_0$ is a non-negative integer such that$\mathrm{gcd}(u_0 , r) = \mathrm{gcd}(u_1 , q) = 1$. For such a sequence, weshow that for all positive integer $n$, we have $\mathrm{lcm}\{u_1 , u_2 ,\dots , u_n\} \geq c_1 \cdot c_2^n \cdot q^{\frac{n^2}{4}}$, where $c_1$ and$c_2$ are positive constants depending only on $q , r$ and $u_0$. This can beconsidered as a $q$-analog of the lower bounds already obtained by the author(in 2005) and by Hong and Feng (in 2006) for the arithmetic progressions.

Nontrivial effective lower bounds for the least common multiple of a $q$-arithmetic progression