In mathematics, the falling factorial (sometimes called the descending factorial,[1] falling sequential product, or lower factorial) is defined as the polynomial
![{\displaystyle {\begin{aligned}(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{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/44b41ce701c869eeff2673cab869f7624c5e4d2b)
The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial,[1] rising sequential product, or upper factorial) is defined as
![{\displaystyle {\begin{aligned}(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{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/8d8c99a01e109960f4791a8ff4d214024fc9cc3e)
The value of each is taken to be 1 (an empty product) when
. These symbols are collectively called factorial powers.[2]
The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation
, where 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
with yet another meaning, namely to denote the binomial coefficient
.[3]
In this article, the symbol
is used to represent the falling factorial, and the symbol
is used for the rising factorial. These conventions are used in combinatorics,[4]
although Knuth's underline and overline notations
and
are increasingly popular.[2][5]
In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and Stegun, the Pochhammer symbol
is used to represent the rising factorial.[6][7]
When
is a positive integer,
gives the number of n-permutations (sequences of distinct elements) from an x-element set, or equivalently the number of injective functions from a set of size
to a set of size
. The rising factorial
gives the number of partitions of an
-element set into
ordered sequences (possibly empty).[lower-alpha 1]
The first few falling factorials are as follows:
The first few rising factorials are as follows:
The coefficients that appear in the expansions are Stirling numbers of the first kind (see below).
When the variable
is a positive integer, the number
is equal to the number of n-permutations from a set of x items, that is, the number of ways of choosing an ordered list of length n consisting of distinct elements drawn from a collection of size
. For example,
is the number of different podiums—assignments of gold, silver, and bronze medals—possible in an eight-person race. On the other hand,
is "the number of ways to arrange
flags on
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
(the flags) into
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:
![{\displaystyle {\begin{alignedat}{2}{(x)}_{n}&={(x-n+1)}^{(n)}&&=(-1)^{n}(-x)^{(n)},\\x^{(n)}&={(x+n-1)}_{n}&&=(-1)^{n}(-x)_{n}.\end{alignedat}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/957a7aca4eec029a4b227801f5d4aea50d00c90f)
Falling and rising factorials of integers are directly related to the ordinary factorial:
![{\displaystyle {\begin{aligned}n!&=1^{(n)}=(n)_{n},\\[6pt](m)_{n}&={\frac {m!}{(m-n)!}},\\[6pt]m^{(n)}&={\frac {(m+n-1)!}{(m-1)!}}.\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1abe42997ccb9ab9503b8ab2289e912124ea6708)
Rising factorials of half integers are directly related to the double factorial:
![{\displaystyle {\begin{aligned}\left[{\frac {1}{2}}\right]^{(n)}={\frac {(2n-1)!!}{2^{n}}},\quad \left[{\frac {2m+1}{2}}\right]^{(n)}={\frac {(2(n+m)-1)!!}{2^{n}(2m-1)!!}}.\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/7c8c2067922006bca5c132e731eb2734a57b130b)
The falling and rising factorials can be used to express a binomial coefficient:
![{\displaystyle {\begin{aligned}{\frac {(x)_{n}}{n!}}&={\binom {x}{n}},\\[6pt]{\frac {x^{(n)}}{n!}}&={\binom {x+n-1}{n}}.\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/f7abd360da2887a175116b5e10804c2e4bcd44c7)
Thus many identities on binomial coefficients carry over to the falling and rising factorials.
The rising and falling factorials are well defined in any unital ring, and therefore
can be taken to be, for example, a complex number, including negative integers, or a polynomial with complex coefficients, or any complex-valued function.
Calculus
Falling factorials appear in multiple differentiation of simple power functions:
![{\displaystyle \left({\frac {\mathrm {d} }{\mathrm {d} x}}\right)^{n}x^{a}=(a)_{n}\cdot x^{a-n}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/d29efb8987c98fc749b074c6eb9887acc5545a0d)
The rising factorial is also integral to the definition of the hypergeometric function: The hypergeometric function is defined for
by the power series
provided that
. Note, however, that the hypergeometric function literature typically uses the notation
for rising factorials.
Falling and rising factorials are closely related to Stirling numbers. Indeed, expanding the product reveals Stirling numbers of the first kind![{\displaystyle {\begin{aligned}(x)_{n}&=\sum _{k=0}^{n}s(n,k)x^{k}\\&=\sum _{k=0}^{n}{\begin{bmatrix}n\\k\end{bmatrix}}(-1)^{n-k}x^{k}\\x^{(n)}&=\sum _{k=0}^{n}{\begin{bmatrix}n\\k\end{bmatrix}}x^{k}\\\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/62cc9c9eb31ac6ab4c6ad8f07c09995b7d22f4a5)
And the inverse relations uses Stirling numbers of the second kind![{\displaystyle {\begin{aligned}x^{n}&=\sum _{k=0}^{n}{\begin{Bmatrix}n\\k\end{Bmatrix}}(x)_{k}\\&=\sum _{k=0}^{n}{\begin{Bmatrix}n\\k\end{Bmatrix}}(-1)^{n-k}x^{(k)}.\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/cbd08f8ee6ab95185deb0a315abc430ee519710c)
The falling and rising factorials are related to one another through the Lah numbers
:[9]![{\displaystyle {\begin{aligned}(x)_{n}&=\sum _{k=0}^{n}L(n,k)x^{(k)}\\x^{(n)}&=\sum _{k=0}^{n}L(n,k)(-1)^{n-k}(x)_{k}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/23ccaf37a95259acb186f9a7a186d82148210fc1)
Since the falling factorials are a basis for the polynomial ring, one can express the product of two of them as a linear combination of falling factorials:[10]
![{\displaystyle (x)_{m}(x)_{n}=\sum _{k=0}^{m}{\binom {m}{k}}{\binom {n}{k}}k!\cdot (x)_{m+n-k}\ .}](//wikimedia.org/api/rest_v1/media/math/render/svg/3904d6421c586b9acb571540b0f95182b785743c)
The coefficients
are called connection coefficients, and have a combinatorial interpretation as the number of ways to identify (or "glue together") k elements each from a set of size m and a set of size n.
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)},\quad {\text{for }}n\geq i.}](//wikimedia.org/api/rest_v1/media/math/render/svg/3d28d4ab199bec9dc6e1cf96fda2867cca55b1b6)
Additionally, we can expand generalized exponent laws and negative rising and falling powers through the following identities:[11](p 52)
![{\displaystyle {\begin{aligned}(x)_{m+n}&=(x)_{m}(x-m)_{n}=(x)_{n}(x-n)_{m}\\[6pt]x^{(m+n)}&=x^{(m)}(x+m)^{(n)}=x^{(n)}(x+n)^{(m)}\\[6pt]x^{(-n)}&={\frac {\Gamma (x-n)}{\Gamma (x)}}={\frac {(x-n-1)!}{(x-1)!}}={\frac {1}{(x-n)^{(n)}}}={\frac {1}{(x-1)_{n}}}={\frac {1}{(x-1)(x-2)\cdots (x-n)}}\\[6pt](x)_{-n}&={\frac {\Gamma (x+1)}{\Gamma (x+n+1)}}={\frac {x!}{(x+n)!}}={\frac {1}{(x+n)_{n}}}={\frac {1}{(x+1)^{(n)}}}={\frac {1}{(x+1)(x+2)\cdots (x+n)}}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/bd3efcab373ab20c11ed840a79a942066bda8e4c)
Finally, duplication and multiplication formulas for the falling and rising factorials provide the next relations:
![{\displaystyle {\begin{aligned}(x)_{k+mn}&=x^{(k)}m^{mn}\prod _{j=0}^{m-1}\left({\frac {x-k-j}{m}}\right)_{n}\,,&{\text{for }}m&\in \mathbb {N} \\[6pt]x^{(k+mn)}&=x^{(k)}m^{mn}\prod _{j=0}^{m-1}\left({\frac {x+k+j}{m}}\right)^{(n)},&{\text{for }}m&\in \mathbb {N} \\[6pt](ax+b)^{(n)}&=x^{n}\prod _{j=0}^{n-1}\left(a+{\frac {b+j}{x}}\right),&{\text{for }}x&\in \mathbb {Z} ^{+}\\[6pt](2x)^{(2n)}&=2^{2n}x^{(n)}\left(x+{\frac {1}{2}}\right)^{(n)}.\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/47fee6455aa4324bfb34466e431e9c1dc43c009b)
The falling factorial occurs in a formula which represents polynomials using the forward difference operator
and which is formally similar to Taylor's theorem:
![{\displaystyle f(x)=\sum _{n=0}^{\infty }{\frac {\Delta ^{n}f(0)}{n!}}(x)_{n}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/1461dd9f9470cb0a3ce24bc8a6c33181920b5162)
In this formula and in many other places, the falling factorial
in the calculus of finite differences plays the role of
in differential calculus. Note for instance the similarity of
to
.
A similar result holds for the rising factorial and the backward difference operator.
The study of analogies of this type is known as umbral calculus. A general theory covering such relations, including the falling and rising factorial functions, is given by the theory of polynomial sequences of binomial type and Sheffer sequences. Falling and rising factorials are Sheffer sequences of binomial type, as shown by the relations:
![{\displaystyle {\begin{aligned}(a+b)_{n}&=\sum _{j=0}^{n}{\binom {n}{j}}(a)_{n-j}(b)_{j}\\[6pt](a+b)^{(n)}&=\sum _{j=0}^{n}{\binom {n}{j}}a^{(n-j)}b^{(j)}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/ebb8afc540a08c40d7ed93301b09c269cfb31681)
where the coefficients are the same as those in the binomial theorem.
Similarly, the generating function of Pochhammer polynomials then amounts to the umbral exponential,
![{\displaystyle \sum _{n=0}^{\infty }(x)_{n}{\frac {t^{n}}{n!}}=\left(1+t\right)^{x},}](//wikimedia.org/api/rest_v1/media/math/render/svg/55394fbe0938ac2b60f7ba9514756cf3794872ae)
since
![{\displaystyle \operatorname {\Delta } _{x}\left(1+t\right)^{x}=t\cdot \left(1+t\right)^{x}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/be5e08b2c03b8596ac915060e0f02f3b88af3aa2)
The Pochhammer symbol has a generalized version called the generalized Pochhammer symbol, used in multivariate analysis. There is also a q-analogue, the q-Pochhammer symbol.
For any fixed arithmetic function
and symbolic parameters x, 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)}](//wikimedia.org/api/rest_v1/media/math/render/svg/5eae268966ccc0f1811c44ec9a61cd261abd6308)
may be studied from the point of view of the classes of generalized Stirling numbers of the first kind defined by the following coefficients of the powers of x in the expansions of (x)n,f,t and then by the next corresponding triangular recurrence relation:
![{\displaystyle {\begin{aligned}\left[{\begin{matrix}n\\k\end{matrix}}\right]_{f,t}&=\left[x^{k-1}\right](x)_{n,f,t}\\&=f(n-1)t^{1-n}\left[{\begin{matrix}n-1\\k\end{matrix}}\right]_{f,t}+\left[{\begin{matrix}n-1\\k-1\end{matrix}}\right]_{f,t}+\delta _{n,0}\delta _{k,0}.\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/57404d462daa4b98d900770ab2a24c0823d0ded4)
These coefficients satisfy a number of analogous properties to those for the Stirling numbers of the first kind as well as recurrence relations and functional equations related to the f-harmonic numbers,[12]
![{\displaystyle F_{n}^{(r)}(t):=\sum _{k\leq n}{\frac {t^{k}}{f(k)^{r}}}\,.}](//wikimedia.org/api/rest_v1/media/math/render/svg/7be56aa402ec6266f2e6e978cf52b820db3d1162)
Here the parts are distinct; for example, when x = n = 2, the (2)(2) = 6 partitions are
,
,
,
,
, and
, where − denotes an empty part.
Harris; Hirst; Mossinghoff (2008). Combinatorics and Graph Theory. Springer. ch. 2. ISBN 978-0-387-79710-6.
Slater, Lucy J. (1966). Generalized Hypergeometric Functions. Cambridge University Press. Appendix I. MR 0201688. — Gives a useful list of formulas for manipulating the rising factorial in (x)n notation.
Feller, William. An Introduction to Probability Theory and Its Applications. Vol. 1. Ch. 2.
Schmidt, Maxie D. (2018). "Combinatorial identities for generalized Stirling numbers expanding f-factorial functions and the f-harmonic numbers". Journal of Integer Sequences. 21 (2) 18.2.7. arXiv:1611.04708v2. MR 3779776.