In my derivation (because of derivatives), I have (or I think that it is better) to write the polynomial (1) in the form (2): $$\tag{1} p(n)=p_0+p_1 n+\cdots+p_r n^r $$
$$\tag{2} p(n)=p'_0P(n,0)+p'_1 P(n,1) +\cdots+p'_r P(n,r) $$ where $P(n,k)=n(n-1)\cdots (n-k+1)$. E.g. $P(n,0)=$, $P(n,1)=n$, $P(n,2)=n(n-1)$.
(A) Is there a name for the representation (2)? (B) Is there a method to go from $(p_i)$ to $(p'_i)$ and vice versa? (with method I mean a closed formula like the Newton binomial formula)
At the end I have the advantage that $$ x^k\frac{d}{d^kx}x^n=P(n,k)x^n. $$
I added the tag generating-functions because this is the topic I'm looking at now; and people familiar with it have done ton of times this computations.
Yes it is a well known alternative basis for polynomials $$ P(n,k) = n\left( {n - 1} \right) \cdots \left( {n - k + 1} \right) = \prod\limits_{j = 0}^{k - 1} {\left( {n - j} \right)} = n^{\,\underline {\,j\,} } $$ which is called Falling Factorial
The conversion is given by the Stirling Numbers of 1st kind (unsigned) in square brackets and of 2nd kind in curly brackets. $$ \eqalign{ & x^{\,j} = \sum\limits_{0\, \le \,k\, \le \,j} {\left\{ \matrix{ j \cr k \cr} \right\}\,x^{\,\underline {\,k\,} } } \; = \sum\limits_{0\, \le \,k\, \le \,j} {\left( { - 1} \right)^{\,j - k} \left\{ \matrix{ j \cr k \cr} \right\}\,x^{\,\overline {\,k\,} } } \;\quad {\rm integer }j \ge 0 \cr & x^{\,\underline {\,j\,} } = \sum\limits_{0\, \le \,k\, \le \,j} {\left( { - 1} \right)^{\,j - k} \left[ \matrix{ j \cr k \cr} \right]x^{\,k} } = \;\sum\limits_{0\, \le \,k\, \le \,j} {\sum\limits_{0\, \le \,i\, \le \,k} {\left( { - 1} \right)^{\,j - i} \,\left[ \matrix{ j \cr k \cr} \right]\left\{ \matrix{ k \cr i \cr} \right\}\,x^{\,\overline {\,i\,} } } } \quad {\rm integer }j \ge 0 \cr & x^{\,\overline {\,j\,} } = \sum\limits_{0\, \le \,k\, \le \,j} {\left[ \matrix{ j \cr k \cr} \right]x^{\,k} } \; = \sum\limits_{0\, \le \,k\, \le \,j} {\sum\limits_{0\, \le \,i\, \le \,k} {\left[ \matrix{ j \hfill \cr k \hfill \cr} \right]\left\{ \matrix{ k \cr i \cr} \right\}\,x^{\,\underline {\,i\,} } } } \;\quad {\rm integer j} \ge 0 \cr} $$