Compact notation for the product $k(k+1)(k+2)\cdots(k+n)$?

Question

Is there a convenient and compact notation for writing the product $A$, below, in a compact way?

$$A=k(k+1)(k+2)\cdots(k+n)$$ i.e. a product of $n$ consecutive integers starting from $k$.

This arises from some diagonals of Pascal's triangle.

Answer 1

In Concrete Mathematics, by Graham, Knuth, and Patashnik, the notation used for "rising powers" is $$ m^{\overline{n}} $$ meaning $$ m(m+1)\cdots (m+n-1) $$ with $n$ terms being multiplied.

Many other authors do use the $m^{(n)}$ notation for it, but the advatage of the one listed first is that it nicely unifies with falling powers $$ m^{\underline{n}} = m(m-1)(m-2)\cdots(m-n+1)$$

Mathematica uses the name "Pochammer" function, which is valid in that Paochammer did early work involving such products, but IMHO that makes it sound like something arcane and advanced.

Note that the words "factorial powers" are connected to both of these: $$ 1^{\overline{n}} = n^{\underline{n}} = n! $$ and that as long as the exponent used is an integer, these can be defined for any real $x$, as in $$ \left(\frac32\right)^{\overline{3}} = \frac{105}{8} $$

Answer 2

As Wojowu mentions, this is the rising factorial, defined by:

$$k^{(n)}=k(k+1)(k+2)\cdots(k+n-1)$$

Using product notation you could also write:

$$\prod_{i=0}^{n-1}(k+i)=k(k+1)(k+2)\cdots(k+n-1)$$

And in terms of the factorial:

$$\frac{(k+n-1)!}{(k-1)!}=k(k+1)(k+2)\cdots(k+n-1)$$