A complete classification of \(\mathbb Z_p\)-sequences corresponding to a polynomial (Q848779)

Let \(p\) be a prime number. In this interesting paper the author characterizes the polynomial sequences over \(\mathbb{Z}/p\mathbb{Z}\) in terms of their minimal period. More precisely, we say that a sequence \(S\) from the integers \(\mathbb{Z}\) into \(\mathbb{Z}/p\mathbb{Z}\) is a polynomial sequence if \(S(n) \equiv T(n) \pmod{p}\), where \(T\) is a real polynomial in one variable such that \(T(z) \in \mathbb{Z}\) for all integer \(z \in \mathbb{Z}.\) The main result of the paper is that a \textit{periodic} sequence from the integers \(\mathbb{Z}\) into \(\mathbb{Z}/p\mathbb{Z}\) is a polynomial sequence if and only if their minimal period \(d\) \(\pmod{p}\) is of the form \(d=p^h\) for some non-negative integer \(h.\) The proofs uses results about binomial coefficients modulo \(p\). The main use of the result is to quickly decide when a given sequence is a polynomial sequence. Besides two examples in the paper, this can be used, e.g., to deduce the well known fact that bell numbers modulo \(p\) are not a polynomial sequence, since in this case \(\gcd(d,p)=1\) since \(d \mid 1 + p + \cdots + p^{p-1}.\)