Ring generated by binomial polynomials

Definition

Let $R$ be a commutative unital ring of characteristic zero. Let $K$ be the quotient ring of $R$ by the multiplicative subset of nonzero integers. Then, the ring generated by binomial polynomials over $R$ is the subring of $K[x]$ comprising all $R$-linear combinations of the polynomials:

${\binom {x}{r}}={\frac {x(x-1)(x-2)\dots (x-r+1)}{r!}}$.

where $r\geq 0$ (for $r=0$, this is the constant polynomial $1$).

Equivalently, it is the tensor product with $R$ of the ring generated by binomial polynomials over the rational integers, i.e., the ring generated by binomial polynomials over $\mathbb {Z}$.

Equivalently, it is the ring $\operatorname {Int} (\mathbb {Z} ,R)$: the ring of all polynomials $f\in K[x]$ such that $f(\mathbb {Z} )\subseteq R$.

Facts

• Ring of integer-valued polynomials is contained in ring generated by binomial polynomials
• Ring of integer-valued polynomials over rational integers equals ring generated by binomial polynomials