Editing Falling and rising factorials (section)

{{Short description|Mathematical functions}}
{{Use American English|date = March 2019}}
{{Redirect|Rising power|the description of a sovereign state or union of states with significant rising influence in global affairs|emerging power}}
In [[mathematics]], the '''falling factorial''' (sometimes called the '''descending factorial''',<ref name=Steffensen/> '''falling sequential product''', or '''lower factorial''') is defined as the polynomial
<math display="block">
\begin{align}
(x)_n = x^\underline{n} &= \overbrace{x(x-1)(x-2)\cdots(x-n+1)}^{n\text{ factors}} \\
&= \prod_{k=1}^n(x-k+1) = \prod_{k=0}^{n-1}(x-k) .
\end{align}</math>

The '''rising factorial''' (sometimes called the '''Pochhammer function''', '''Pochhammer polynomial''', '''ascending factorial''',<ref name=Steffensen>
{{cite book
 | last = Steffensen | first = J.F. | author-link = Johan Frederik Steffensen
 | date = 17 March 2006
 | title = Interpolation
 | publisher = Dover Publications | edition = 2nd
 | isbn = 0-486-45009-0
 | page = 8
}} — A reprint of the 1950 edition by Chelsea Publishing.
</ref> '''rising sequential product''', or '''upper factorial''') is defined as
<math display="block">
\begin{align}
x^{(n)} = x^\overline{n} &= \overbrace{x(x+1)(x+2)\cdots(x+n-1)}^{n\text{ factors}} \\
&= \prod_{k=1}^n(x+k-1) = \prod_{k=0}^{n-1}(x+k) .
\end{align}</math>

The value of each is taken to be 1 (an [[empty product]]) when <math>n=0</math>. These symbols are collectively called '''factorial powers'''.<ref name="The Art of Computer Programming">
{{cite book
 |last=Knuth |first=D.E. |author-link=Donald Knuth
 |title=[[The Art of Computer Programming]] |edition=3rd
 |volume=1 |page=50
}}
</ref>

The '''Pochhammer symbol''', introduced by [[Leo August Pochhammer]], is the notation <math>(x)_n</math>, where {{mvar|n}} is a [[non-negative integer]]. It may represent ''either'' the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used <math>(x)_n</math> with yet another meaning, namely to denote the [[binomial coefficient]] <math>\tbinom{x}{n}</math>.<ref name=Knuth>
{{cite journal
 |last=Knuth |first=D.E.  |author-link=Donald Knuth
 |year=1992
 |title=Two notes on notation
 |journal=[[American Mathematical Monthly]]
 |volume=99 |issue=5 |pages=403–422
 |arxiv=math/9205211 |doi=10.2307/2325085
 |jstor=2325085 |s2cid=119584305
}} The remark about the Pochhammer symbol is on page 414.
</ref>

In this article, the symbol <math>(x)_n</math> is used to represent the falling factorial, and the symbol <math>x^{(n)}</math> is used for the rising factorial. These conventions are used in [[combinatorics]],<ref>
{{cite book
 |last=Olver |first=P.J. |author-link=Peter J. Olver
 |year=1999
 |title=Classical Invariant Theory
 |publisher=Cambridge University Press
 |isbn=0-521-55821-2 |mr=1694364
 |page=101
}}
</ref>
although [[Donald Knuth|Knuth]]'s underline and overline notations <math>x^\underline{n}</math> and <math>x^\overline{n}</math> are increasingly popular.<ref name="The Art of Computer Programming"/><ref>
{{cite book
 |last1=Harris
 |last2=Hirst
 |last3=Mossinghoff
 |year=2008
 |title=Combinatorics and Graph Theory 
 |publisher=Springer
 |isbn=978-0-387-79710-6
 |at=ch.&nbsp;2
}}
</ref>
In the theory of [[special functions]] (in particular the [[hypergeometric function]]) and in the standard reference work ''[[Abramowitz and Stegun]]'', the Pochhammer symbol <math>(x)_n</math> is used to represent the rising factorial.<ref>
{{cite book
 |editor1=Abramowitz, Milton
 |editor2=Stegun, Irene A.
 |date=December 1972  |orig-date=June 1964
 |title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
 |title-link=Abramowitz and Stegun
 |place=Washington, DC
 |publisher=[[United States Department of Commerce]]
 |series=[[National Bureau of Standards]] [[Applied Mathematics]] Series
 |volume=55
 |lccn=64-60036
 |at=p.&nbsp;256 eqn.&nbsp;6.1.22
}}
</ref><ref>
{{cite book
 |last=Slater |first=Lucy J.
 |year=1966
 |title=Generalized Hypergeometric Functions
 |publisher=Cambridge University Press
 |mr=0201688 |at=Appendix&nbsp;I
}} — Gives a useful list of formulas for manipulating the rising factorial in {{math|(''x''){{sub|''n''}}}} notation.
</ref>
 
When <math>x</math> is a positive integer, <math>(x)_n</math> gives the number of [[k-permutation|{{mvar|n}}-permutations]] (sequences of distinct elements) from an {{mvar|x}}-element set, or equivalently the number of [[injective function]]s from a set of size <math>n</math> to a set of size <math>x</math>.  The rising factorial <math>x^{(n)}</math> gives the number of [[Partition of a set|partitions]] of an <math>n</math>-element set into <math>x</math> ordered sequences (possibly empty).{{efn|Here the parts are distinct; for example, when {{math|1=''x'' = ''n'' = 2}}, the {{math|1=(2){{sup|(2)}} = 6}} partitions are <math>(12, -)</math>, <math>(21, -)</math>, <math>(1, 2)</math>, <math>(2, 1)</math>, <math>(-, 12)</math>, and <math>(-, 21)</math>, where − denotes an empty part.}}