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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.

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    $\begingroup$ en.wikipedia.org/wiki/Falling_and_rising_factorials $\endgroup$
    – Wojowu
    Commented Sep 5, 2019 at 21:18
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    $\begingroup$ Please don't write "Solved" in your title like you did. Instead, you should answer your question yourself (or let Wojowu do it), or delete the question if you don't think it will be useful to anyone. $\endgroup$
    – Arnaud D.
    Commented Sep 5, 2019 at 22:05
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    $\begingroup$ Hmmm it's interesting that there are two close votes for lack of context, yet I don't see how much more context you could have for a question simply asking for notation. $\endgroup$ Commented Sep 5, 2019 at 23:31
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    $\begingroup$ I refrained from accepting an answer because they both seemed equally good. I've now accepted one for the sake of marking the question as closed. $\endgroup$
    – mjc
    Commented Sep 6, 2019 at 0:42
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    $\begingroup$ @GerryMyerson There were no answers when I posted that comment. $\endgroup$
    – Arnaud D.
    Commented Sep 6, 2019 at 7:37

2 Answers 2

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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:

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

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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 $$ \left(\frac32\right)^{\overline{3}} = \frac{105}{8} $$

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    $\begingroup$ I would disagree that "Pochammer symbol" sounds any more arcane or advanced than "factorial symbol", "summation symbol" or many other things we use regularly in mathematics. Things are given names to make it easy to refer to them and since it's necessary to explain the term "rising factorial" it's no clearer than "Pochammer symbol" in this context. $\endgroup$
    – postmortes
    Commented Sep 6, 2019 at 13:15
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    $\begingroup$ @postmortes - "Rising factorial" is at least somewhat descriptive, even though it may require explanation. "Pochhammer symbol" is not descriptive at all; a person's name tells nothing about the idea. It may suggest that the idea is so complicated or obscure that it can't be given a short descriptive name. $\endgroup$
    – mr_e_man
    Commented Sep 12, 2019 at 23:55
  • $\begingroup$ @hardmath -- Right you are, I had the incorrect spelling. And that extra "h" does clarify everything considerably. $\endgroup$ Commented Sep 13, 2019 at 16:49

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