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\begin{document}

\begin{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  $\p$ be the set of such sequences and  $\Pii$ the additive  group
of all infinite sequences of integers. Then $\p$ is a subgroup of
$\Pii$ and   $\Pii/\p \cong \prod_{n=2}^\infty \Z/n!\Z$.  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.
\end{abstract}

\end{document}
