Am I correctly computing $\Delta(x^\overline m)$?

Question

Notation definitions according to Concrete Mathematics:

$\Delta f(x) = f(x+1) - f(x)$

$x^\underline m = x(x-1)...(x-m+1)$, integer $m \ge 0$ (Read aloud as "x to the m falling.")

$x^\overline m = x(x+1)...(x+m-1)$, integer $m \ge 0$ (Read aloud as "x to the m rising.")

---

I derived $\Delta f(x^\underline m)$ for myself, and it is also given in the book, as: $mx^\underline{m-1}$ (A fun result to calculate!)

I then tried to compute $\Delta(x^\overline m)$, which is not shown in the book, and got the slightly more unwieldy result:

$m(x+1)^\overline{m-1}$

The Wikipedia article (which uses different notation), shows the result for the falling factorial, but then only says, "A similar result holds for the rising factorial," without specifying what that result is.

Am I correctly computing $\Delta(x^\overline m)$?

Answer 1

That’s the correct result, so you probably are; the computation is

$$\begin{align*} \Delta(x^{\overline m})&=(x+1)^{\overline m}-x^{\overline m}\\ &=(x+1)^{\overline{m-1}}(x+m)-x(x+1)^{\overline{m-1}}\\ &=m(x+1)^{\overline{m-1}}\;. \end{align*}$$