An integer-valued polynomial on a subset $S$ of $\mathbb{Z}$ is a polynomial $f(x) \in \mathbb{Q}[x]$ with the property $f(S) \subseteq \mathbb{Z}$. This article describes the ring of such polynomials in the special case that $S$ is the Fibonacci numbers. An algorithm is described for finding a regular basis, i.e., an ordered sequence of polynomials, the $n$th one of degree $n$, with which any such polynomial can be expressed as a unique integer linear combination.

@article{RationalPolynomialsThatTakeIntegerValuesattheFibonacciNumbers,
title = {Rational Polynomials That Take Integer Values at the Fibonacci Numbers},
author = {Keith Johnson and Kira Scheibelhut},
url = {http://www.jstor.org/stable/10.4169/amer.math.monthly.123.4.338},
abstract = {An integer-valued polynomial on a subset {\$}S{\$} of {\$}\mathbb{\{}Z{\}}{\$} is a polynomial {\$}f(x) \in \mathbb{\{}Q{\}}[x]{\$} with the property {\$}f(S) \subseteq \mathbb{\{}Z{\}}{\$}. This article describes the ring of such polynomials in the special case that {\$}S{\$} is the Fibonacci numbers. An algorithm is described for finding a regular basis, i.e., an ordered sequence of polynomials, the {\$}n{\$}th one of degree {\$}n{\$}, with which any such polynomial can be expressed as a unique integer linear combination.},
comment = {},
urldate = {2016-08-02},
year = 2016,
collections = {Fibonaccinalia,Fun maths facts,Integerology}
}