On the least common multiple of several consecutive values of a polynomial

In this note we prove the periodicity of an arithmetic function that is the quotient of the product of $k+1$ values (where $k \geq 1$) of a polynomial $f \in {\mathbb Z}[x]$ at $k + 1$ consecutive integers ${f(n) f(n + 1) \cdots f(n + k)}$ and the least common multiple of the corresponding integers ${f(n),f(n + 1),\dots,f(n + k)}$. We show that this function is periodic if and only if no difference between two roots of $f$ is a positive integer smaller than or equal to $k$. This implies an asymptotic formula for the least common multiple of $f(n),f(n+1),\dots,f(n+k)$ and extends some earlier results in this area from linear and quadratic polynomials $f$ to polynomials of arbitrary degree $d$. A period in terms of the reduced resultants of $f(x)$ and $f(x+\ell)$, where $1 \leq \ell \leq k$, is given explicitly, as well as few examples of $f$ when the smallest period can be established.