Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6 |

College of Engineering and Science

University of Detroit, Mercy

Detroit, MI 48221-3038

USA

Phill Schultz

School of Mathematics and Statistics

The University of Western Australia

Nedlands 6009

Australia

**Abstract:**

We determine the infinite sequences of integers that can be
generated by polynomials with integral coefficients, in the sense that
for each finite initial segment of length there is an integral
polynomial of degree such that
for
.

Let be the set of such sequences and the additive group of all infinite sequences of integers. Then is a subgroup of and . 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.

Received October 30 2006;
revised version received March 22 2007.
Published in *Journal of Integer Sequences* April 2 2007.
Minor revisions, April 17 2008.

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