We know that, for instance, $$r^3=Ar(r-1)(r-2)+Br(r-1)+Cr+D$$ which can also be written as $$r^3=Ar^\underline{3}+Br^\underline{2}+Cr^\underline{1}+Dr^\underline{0}$$ where $A,B,C,D$ are Stirling numbers of the second kind $\big\lbrace {3\atop k}\big\rbrace$ for $k=3,2,1,0$ respectively, and are also coefficients of falling factorials.
Is there a corresponding formula or name for the coefficients in the case of rising factorials, e.g. $P,Q,R,S$ in $$r^3=Pr(r+1)(r+2)+Qr(r+1)+Rr+S$$ which can also be written as $$r^3=Pr^\overline{3}+Qr^\overline{2}+Rr^\overline{1}+Sr^\overline{0}$$ ?
We have that a monomial can be expressed as a linear combination of falling factorials and the coefficients are the Stirling numbers of the second kind \begin{eqnarray*} r^n= \sum_{k=0}^{n} \big\lbrace {n\atop k}\big\rbrace r^\underline{k}. \end{eqnarray*} In order to express a monomial as a linear combination of rising factorials just replace $\color{red}{r \rightarrow -r}$; note that \begin{eqnarray*} r^\underline{k}=r(r-1)\cdots (r-k+1) \rightarrow (-r)(-r-1) \cdots (-r-k+1) =(-1)^k r^\overline{k}. \end{eqnarray*} So the formula you seek is \begin{eqnarray*} r^n= \sum_{k=0}^{n} (-1)^{n-k} \big\lbrace {n\atop k}\big\rbrace r^\overline{k}. \end{eqnarray*}