Inmathematics, thefalling factorial (sometimes called thedescending factorial,^[1]falling sequential product, orlower factorial) is defined as the polynomial

Therising factorial (sometimes called thePochhammer function,Pochhammer polynomial,ascending factorial,^[1]rising sequential product, orupper factorial) is defined as

The value of each is taken to be 1 (anempty product) when${\displaystyle n=0}$. These symbols are collectively calledfactorial powers.^[2]

ThePochhammer symbol, introduced byLeo August Pochhammer, is the notation${\displaystyle (x{)}_n}$, wheren is anon-negative integer. It may representeither the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used${\displaystyle (x{)}_n}$ with yet another meaning, namely to denote thebinomial coefficient${\displaystyle (\genfrac{}{}{0ex}{}{x}{n})}$.^[3]

In this article, the symbol${\displaystyle (x{)}_n}$ is used to represent the falling factorial, and the symbol${\displaystyle x^{(n)}}$ is used for the rising factorial. These conventions are used incombinatorics,^[4]althoughKnuth's underline and overline notations${\displaystyle x^{\underset{\_}{n}}}$ and${\displaystyle x^{\overline{n}}}$ are increasingly popular.^[2][5]In the theory ofspecial functions (in particular thehypergeometric function) and in the standard reference workAbramowitz and Stegun, the Pochhammer symbol${\displaystyle (x{)}_n}$ is used to represent the rising factorial.^[6][7]

When${\displaystyle x}$ is a positive integer,${\displaystyle (x{)}_n}$ gives the number ofn-permutations (sequences of distinct elements) from anx-element set, or equivalently the number ofinjective functions from a set of size${\displaystyle n}$ to a set of size${\displaystyle x}$. The rising factorial${\displaystyle x^{(n)}}$ gives the number ofpartitions of an${\displaystyle n}$-element set into${\displaystyle x}$ ordered sequences (possibly empty).^[a]

The first few falling factorials are as follows:The first few rising factorials are as follows:The coefficients that appear in the expansions areStirling numbers of the first kind (see below).

When the variable${\displaystyle x}$ is a positive integer, the number${\displaystyle (x{)}_n}$ is equal to the number ofn-permutations from a set ofx items, that is, the number of ways of choosing an ordered list of lengthn consisting of distinct elements drawn from a collection of size${\displaystyle x}$. For example,${\displaystyle (8{)}_3=8\times 7\times 6=336}$ is the number of different podiums—assignments of gold, silver, and bronze medals—possible in an eight-person race. On the other hand,${\displaystyle x^{(n)}}$ is "the number of ways to arrange${\displaystyle n}$ flags on${\displaystyle x}$ flagpoles",^[8]where all flags must be used and each flagpole can have any number of flags. Equivalently, this is the number of ways to partition a set of size${\displaystyle n}$ (the flags) into${\displaystyle x}$ distinguishable parts (the poles), with a linear order on the elements assigned to each part (the order of the flags on a given pole).

The rising and falling factorials are simply related to one another:

Falling and rising factorials of integers are directly related to the ordinaryfactorial:

Rising factorials of half integers are directly related to thedouble factorial:$${\displaystyle \begin{array}{r}{\left[\frac{1}{2}\right]}^{(n)}=\frac{(2n-1)!!}{2^n},{\left[\frac{2m+1}{2}\right]}^{(n)}=\frac{(2(n+m)-1)!!}{2^n(2m-1)!!}.\end{array}}$$

The falling and rising factorials can be used to express abinomial coefficient:

Thus many identities on binomial coefficients carry over to the falling and rising factorials.

The rising and falling factorials are well defined in anyunitalring, and therefore${\displaystyle x}$ can be taken to be, for example, acomplex number, including negative integers, or apolynomial with complex coefficients, or anycomplex-valued function.

The falling factorial can be extended toreal values of${\displaystyle x}$ using thegamma function provided${\displaystyle x}$ and${\displaystyle x+n}$ are real numbers that are not negative integers:$${\displaystyle (x{)}_n=\frac{\mathrm{\Gamma }(x+1)}{\mathrm{\Gamma }(x-n+1)},}$$and so can the rising factorial:$${\displaystyle x^{(n)}=\frac{\mathrm{\Gamma }(x+n)}{\mathrm{\Gamma }(x)}.}$$

Falling factorials appear in multipledifferentiation of simple power functions:$${\displaystyle {\left(\frac{\mathrm{d}}{\mathrm{d}x}\right)}^n{x}^a=(a{)}_n\cdot x^{a-n}.}$$

The rising factorial is also integral to the definition of thehypergeometric function: The hypergeometric function is defined for by thepower series$${\displaystyle {}_2{F}_1(a,b;c;z)=\sum _{n=0}^{\mathrm{\infty }}\frac{a^{(n)}{b}^{(n)}}{c^{(n)}}\frac{z^n}{n!}}$$provided that${\displaystyle c\ne 0,-1,-2,\dots }$. Note, however, that the hypergeometric function literature typically uses the notation${\displaystyle (a{)}_n}$ for rising factorials.

Falling and rising factorials are closely related toStirling numbers. Indeed, expanding the product revealsStirling numbers of the first kind

And the inverse relations usesStirling numbers of the second kind

The falling and rising factorials are related to one another through theLah numbers$L(n,k)=(\genfrac{}{}{0ex}{}{n-1}{k-1})\frac{n!}{k!}$:^[9]

Since the falling factorials are a basis for thepolynomial ring, one can express the product of two of them as alinear combination of falling factorials:^[10]$${\displaystyle (x{)}_m(x{)}_n=\sum _{k=0}^m(\genfrac{}{}{0ex}{}{m}{k})(\genfrac{}{}{0ex}{}{n}{k})k!\cdot (x{)}_{m+n-k}.}$$

The coefficients${\displaystyle (\genfrac{}{}{0ex}{}{m}{k})(\genfrac{}{}{0ex}{}{n}{k})k!}$ are calledconnection coefficients, and have a combinatorial interpretation as the number of ways to identify (or "glue together")k elements each from a set of sizem and a set of sizen.

There is also a connection formula for the ratio of two rising factorials given by$${\displaystyle \frac{x^{(n)}}{x^{(i)}}=(x+i{)}^{(n-i)},\text{for }n\ge i.}$$

Additionally, we can expand generalized exponent laws and negative rising and falling powers through the following identities:^[11](p 52)

Finally,duplication andmultiplication formulas for the falling and rising factorials provide the next relations:

The falling factorial occurs in a formula which representspolynomials using the forwarddifference operator${\displaystyle \mathrm{\Delta }f(x)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}f(x+1)-f(x),}$ which in form is an exact analogue toTaylor's theorem: Compare the series expansion fromumbral calculus

$${\displaystyle f(t)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}{\mathrm{\Delta }}_x^{n}f(x){|}_{x=0}(t{)}_n}$$

with the corresponding series fromdifferential calculus

$${\displaystyle f(t)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{n!}{\left[\frac{\mathrm{d}}{\mathrm{d}x}\right]}^{n}f(x){|}_{x=0}{t}^n.}$$

In this formula and in many other places, the falling factorial${\displaystyle (x{)}_n}$ in the calculus offinite differences plays the role of${\displaystyle x^n}$ in differential calculus. For another example, note the similarity of${\displaystyle \mathrm{\Delta }(x{)}_n=n(x{)}_{n-1}}$ to${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}{x}^n=n{x}^{n-1}.}$

A corresponding relation holds for the rising factorial and the backward difference operator.

The study of analogies of this type is known asumbral calculus. A general theory covering such relations, including the falling and rising factorial functions, is given by the theory ofpolynomial sequences of binomial type andSheffer sequences. Falling and rising factorials are Sheffer sequences of binomial type, as shown by the relations:

where the coefficients are the same as those in thebinomial theorem.

Similarly, the generating function of Pochhammer polynomials then amounts to the umbral exponential,

$${\displaystyle \sum _{n=0}^{\mathrm{\infty }}(x{)}_n\frac{t^n}{n!}={\left(1+t\right)}^x,}$$

since

$${\displaystyle {\mathrm{\Delta }}_x{\left(1+t\right)}^x=t\cdot {\left(1+t\right)}^x.}$$

An alternative notation for the rising factorial$${\displaystyle x^{\overline{m}}\equiv (x{)}_{+m}\equiv (x{)}_m=\stackrel{m\text{ factors}}{\overbrace{x(x+1)\dots (x+m-1)}}\text{for integer }m\ge 0}$$

and for the falling factorial$${\displaystyle x^{\underset{\_}{m}}\equiv (x{)}_{-m}=\stackrel{m\text{ factors}}{\overbrace{x(x-1)\dots (x-m+1)}}\text{for integer }m\ge 0}$$

goes back to A. Capelli (1893) and L. Toscano (1939), respectively.^[2] Graham, Knuth, and Patashnik^{[11](pp 47, 48)}propose to pronounce these expressions as "${\displaystyle x}$ to the${\displaystyle m}$ rising" and "${\displaystyle x}$ to the${\displaystyle m}$ falling", respectively.

An alternative notation for the rising factorial${\displaystyle x^{(n)}}$ is the less common${\displaystyle (x{)}_n^{+}}$. When${\displaystyle (x{)}_n^{+}}$ is used to denote the rising factorial, the notation${\displaystyle (x{)}_n^{-}}$ is typically used for the ordinary falling factorial, to avoid confusion.^[3]

The Pochhammer symbol has a generalized version called thegeneralized Pochhammer symbol, used in multivariateanalysis. There is also aq-analogue, theq-Pochhammer symbol.

For any fixed arithmetic function${\displaystyle f:\mathbb{N}\to \mathbb{C}}$ and symbolic parametersx,t, related generalized factorial products of the form

$${\displaystyle (x{)}_{n,f,t}:=\prod _{k=0}^{n-1}\left(x+\frac{f(k)}{t^k}\right)}$$

may be studied from the point of view of the classes of generalizedStirling numbers of the first kind defined by the following coefficients of the powers ofx in the expansions of(x)_n,f,t and then by the next corresponding triangular recurrence relation:

These coefficients satisfy a number of analogous properties to those for theStirling numbers of the first kind as well as recurrence relations and functional equations related to thef-harmonic numbers,^[12]$${\displaystyle F_n^{(r)}(t):=\sum _{k\le n}\frac{t^k}{f(k{)}^r}.}$$

• Pochhammerk-symbol
• Vandermonde identity

• Weisstein, Eric W."Pochhammer Symbol".MathWorld.

• Gamma and related functions
• Factorial and binomial topics
• Finite differences
• Operations on numbers

• Articles with short description
• Short description matches Wikidata
• Use American English from March 2019
• All Wikipedia articles written in American English