Binomial polynomial
= Binomial polynomial
{title2=$B_n(X)=\binom Xn$}
For $n\geq0$, define $B_0(X)=1$ and $B_n(X)=X(X-1)\cdots(X-n+1)/n!$. The <binomial coefficient> formula shows $B_n(m)\in\mathbb Z$ for every nonnegative integer $m$. Since these integers are dense in the <p-adic integers>, <continuity> of the <polynomial> implies $B_n(\mathbb Z_p)\subseteq\mathbb Z_p$, so it is an <integer-valued polynomial> there and $\|B_n\|_\infty=1$. The <Pascal's identity> gives $\Delta B_{n+1}=B_n$ for the <forward difference operator>.