Rings of power series in the binomial polynomials

Abstract

The study of the ring of all formal series $a_{0}+a_{1}\binom{x}{1}+a_{2} \binom{x}{2}+\cdots$ with integer coefficients, denoted by $\mathbb{Z}[\hspace{-1.6pt}[\binom{x}{1},\binom{x}{2},\dots]\hspace{-1.6pt}]$, or $\mathbb{Z}[\hspace{-1.6pt}[\binom{x}{n}]\hspace{-1.6pt}]_{n\geq0}$ for short, is motivated by the elementary number theoretical properties of the binomial coefficients. The binomial polynomials as well as the binomial coefficients and their generalizations can be found in different branches of mathematics, e.g. in algebra, analysis, combinatorics and in topology. The question of finding the remainder when dividing $\binom{n}{k}$ by a prime (Lucas's 1878 theorem) leads to base-$p$ expansions in the binomial coefficients and the consideration of integer-valued polynomials with rational coefficients. And although the study of these polynomials dates back to the seventeenth century, the study of this set as a ring began in 1936 with Skolem. More generally, the bijective correspondence between the set of functions defined on the set of non-negative integers and the series $a_{0}+a_{1}\binom{x}{1}+a_{2}\binom{x}{2}+\cdots$ is used by Mahler in the field of $p$-adic analysis and naturally leads to the expansion of Skolem's approach and the definition of $\mathbb{Z}[\hspace{-1.6pt}[\binom{x}{1},\binom{x}{2},\dots]\hspace{-1.6pt}]$ or in fact of $R[\hspace{-1.6pt}[\binom{x}{1},\binom{x}{2},\dots]\hspace{-1.6pt}]$ (with $R$ any ring). Iwasawa used such series in connection with the $p$-adic $L$-functions but without considering the ring.