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

Polynomial Points

E. F. Cornelius, Jr.
College of Engineering and Science
University of Detroit, Mercy
Detroit, MI 48221-3038

Phill Schultz
School of Mathematics and Statistics
The University of Western Australia
Nedlands 6009


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.

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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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