# expressing canonical base of univariate polynomials in binomial base

Two bases are fairly standard for ${\mathbb Q}[X]$ : the canonical base $(X^j)_{j\geq 0}$ and the binomial base $(b_j(X))_{j\geq 0}$ where $b_j(X)=\binom{X}{j}=\frac{X(X-1)\ldots (X-(j-1))}{j!}$ (thus $b_0(X)=1$, $b_1(X)=X$, $b_2(X)=\frac{X(X-1)}{2}\ldots$ ).

It is not hard to express the canonical base in terms of the binomial base : one has $X^0=b_0(X)$ and for $i>1$,

$$X^i=\sum_{j=1}^p \left(\sum_{t=1}^j c^{ij}_{t}\right) b_j(X), \text{ with } c^{ij}_{t}=\binom{j}{t} (-1)^{j-t}t^i \tag{1}$$

Does this formula (1) have a name ? Or is there a theory with a name where it plays an important role ?

• I would call those bases the monomial basis and a scaled falling factorial power basis, respectively. If anything, binomial basis would seem to suggest $((X-a)^j)_{j\in\Bbb N}$ to me. – Marc van Leeuwen Dec 25 '14 at 6:20

If we write $(x)_k$ for $x(x-1)(x-2)\cdots(x-k+1)$, then $$x^n=\sum_0^nS(n,k)(x)_k$$ where the $S(n,k)$ are the Stirling numbers of the second kind. This looks very much like your formula, more so if we write it as $$x^n=\sum_0^nk!S(n,k){x\choose k}$$ Wikipedia will get you started on these numbers.