Abstract:

We determine the infinite sequences $(a_k)$ of integers that can be generated by polynomials with integral coefficients, in the sense that for each finite initial segment of length $n$ there is an integral polynomial $f_n(x)$ of degree $<n$ such that $a_k=f_n(k)$ for $k=0,1,\dots, n-1$.

Let ${\mathbf P}$ be the set of such sequences and ${\mathbf \Pi}$ the additive group of all infinite sequences of integers. Then ${\mathbf P}$ is a subgroup of ${\mathbf \Pi}$ and ${\mathbf \Pi}/{\mathbf P}\cong \prod_{n=2}^\infty {\mathbb{Z}}/n!{\mathbb{Z}}$. The methods and results are applied to familiar families of polynomials such as Chebyshev polynomials and shifted Legendre polynomials.

The results are achieved by extending Lagrange interpolation polynomials to power series, using a special basis for the group of integral polynomials, called the integral root basis.