Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial,[1] falling sequential product, or lower factorial) is defined as the polynomial

\( {\displaystyle (x)_{n}=x^{\underline {n}}=x(x-1)(x-2)\cdots (x-n+1)=\prod _{k=1}^{n}(x-k+1)=\prod _{k=0}^{n-1}(x-k).} \)

The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial,[1] rising sequential product, or upper factorial) is defined as

\( {\displaystyle x^{(n)}=x^{\overline {n}}=x(x+1)(x+2)\cdots (x+n-1)=\prod _{k=1}^{n}(x+k-1)=\prod _{k=0}^{n-1}(x+k).} \)

The value of each is taken to be 1 (an empty product) when n = 0. These symbols are collectively called factorial powers.[2]

The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (x)n, 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 (x)n with yet another meaning, namely to denote the binomial coefficient \( {\tbinom {x}{n}} \).[3]

In this article, the symbol (x)n is used to represent the falling factorial, and the symbol x(n) is used for the rising factorial. These conventions are used in combinatorics,[4] although Knuth's underline/overline notations \( {\displaystyle x^{\underline {n}},x^{\overline {n}}} \) 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 (x)n is used to represent the rising factorial.[6][7]

When x is a positive integer, (x)n gives the number of n-permutations of an x-element set, or equivalently the number of injective functions from a set of size n to a set of size x. Also, (x)n is "the number of ways to arrange n flags on x flagpoles" [8], where all flags must be used and each flagpole can have at most one flag. In this context, other notations like xPn and P(x, n) are also sometimes used.

Examples

The first few rising factorials are as follows:

\( {\displaystyle x^{(0)}=x^{\overline {0}}=1} \)
\( {\displaystyle x^{(1)}=x^{\overline {1}}=x} \)
\( {\displaystyle x^{(2)}=x^{\overline {2}}=x(x+1)=x^{2}+x} \)
\( {\displaystyle x^{(3)}=x^{\overline {3}}=x(x+1)(x+2)=x^{3}+3x^{2}+2x} \)
\( {\displaystyle x^{(4)}=x^{\overline {4}}=x(x+1)(x+2)(x+3)=x^{4}+6x^{3}+11x^{2}+6x} \)

The first few falling factorials are as follows:

\( {\displaystyle (x)_{0}=x^{\underline {0}}=1} \)
\( {\displaystyle (x)_{1}=x^{\underline {1}}=x} \)
\( {\displaystyle (x)_{2}=x^{\underline {2}}=x(x-1)=x^{2}-x} \)
\( {\displaystyle (x)_{3}=x^{\underline {3}}=x(x-1)(x-2)=x^{3}-3x^{2}+2x} \)
\( {\displaystyle (x)_{4}=x^{\underline {4}}=x(x-1)(x-2)(x-3)=x^{4}-6x^{3}+11x^{2}-6x} \)

The coefficients that appear in the expansions are Stirling numbers of the first kind.
Properties

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

\( {\displaystyle m^{(n)}={(m+n-1)}_{n}=(-1)^{n}(-m)_{n}} \)

\( {\displaystyle {(m)}_{n}={(m-n+1)}^{(n)}=(-1)^{n}(-m)^{(n)}} \)

The rising and falling factorials are directly related to the ordinary factorial:

\( {\displaystyle n!=1^{(n)}=(n)_{n}} \)

\( {\displaystyle (m)_{n}={\frac {m!}{(m-n)!}}} \)

\( {\displaystyle m^{(n)}={\frac {(m+n-1)!}{(m-1)!}}} \)

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

\( {\displaystyle {\frac {x^{(n)}}{n!}}={x+n-1 \choose n}\quad {\text{and}}\quad {\frac {(x)_{n}}{n!}}={x \choose n}.} \)

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 x can be taken to be, for example, a complex number, including negative integers, or a polynomial with complex coefficients, or any complex-valued function.

The rising factorial can be extended to real values of n using the gamma function provided x and x + n are real numbers that are not negative integers:

\( x^{(n)}={\frac {\Gamma (x+n)}{\Gamma (x)}}, \)

and so can the falling factorial:

\( (x)_{n}={\frac {\Gamma (x+1)}{\Gamma (x-n+1)}}. \)

If D denotes differentiation with respect to x, one has

\( {\displaystyle D^{n}(x^{a})=(a)_{n}\cdot x^{a-n}.} \)

The Pochhammer symbol is also integral to the definition of the hypergeometric function: The hypergeometric function is defined for |z| < 1 by the power series

\( {\displaystyle {}_{2}F_{1}(a,b;c;z)=\sum _{n=0}^{\infty }{a^{(n)}b^{(n)} \over c^{(n)}}{z^{n} \over n!}} \)

provided that c does not equal 0, −1, −2, ... . Note, however, that the hypergeometric function literature typically uses the notation \( {(a)}_{n} \) for rising factorials.

Relation to umbral calculus

The falling factorial occurs in a formula which represents polynomials using the forward difference operator Δ and which is formally similar to Taylor's theorem:

\( {\displaystyle f(x)=\sum _{n=0}^{\infty }\left[{\frac {\,\Delta ^{n}\!f(0)\,}{n!}}\right]\,(x)_{n}.} \)

In this formula and in many other places, the falling factorial (x)n in the calculus of finite differences plays the role of xn in differential calculus. Note for instance the similarity of \( {\displaystyle \Delta \!\left[\,(x)_{n}\,\right]=n\,(x)_{n-1}} \) to \( {\displaystyle {\tfrac {\operatorname {d} }{\operatorname {d} x}}\left[\,x^{n}\,\right]=n\,x^{n-1}} \).

A similar result holds for the rising factorial.

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. Rising and falling factorials are Sheffer sequences of binomial type, as shown by the relations:

\( {\displaystyle (a+b)^{(n)}=\sum _{j=0}^{n}{n \choose j}(a)^{(n-j)}(b)^{(j)}} \)

\( {\displaystyle (a+b)_{n}=\sum _{j=0}^{n}{n \choose j}(a)_{n-j}(b)_{j}} \)

where the coefficients are the same as the ones in the expansion of a power of a binomial (Chu–Vandermonde identity).

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

\( \sum _{n=0}^{\infty }(x)_{n}~{\frac {t^{n}}{n!}}=(1+t)^{x}~, \)

since

\( {\displaystyle \operatorname {\Delta } _{x}\left(1+t\right)^{x}=t\,\left(1+t\right)^{x}~.} \)

Connection coefficients and identities

The falling and rising factorials are related to one another through the Lah numbers:[9]

\( {\displaystyle {\begin{aligned}(x)_{n}&=\sum _{k=1}^{n}{\binom {n-1}{k-1}}{\frac {n!}{k!}}\times x^{(k)}\\&=(-1)^{n}(-x)^{(n)}=(x-n+1)^{(n)}\\x^{(n)}&=\sum _{k=0}^{n}{\binom {n}{k}}(n-1)_{n-k}\times (x)_{k}\\&=(-1)^{n}(-x)_{n}=(x+n-1)_{n}\end{aligned}}}. \)

The following formulas relate integral powers of a variable x through sums using the Stirling numbers of the second kind ( notated by curly brackets {n
k} ):[9]

\( {\displaystyle {\begin{aligned}x^{n}&=\sum _{k=0}^{n}\left\{{\begin{matrix}n\\k\end{matrix}}\right\}(x)_{k}\\&=\sum _{k=0}^{n}\left\{{\begin{matrix}n\\k\end{matrix}}\right\}(-1)^{n-k}x^{(k)}\end{aligned}}}. \)

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:

\( {\displaystyle (x)_{m}(x)_{n}=\sum _{k=0}^{m}{m \choose k}{n \choose k}k!\cdot (x)_{m+n-k}~.} \)

The coefficients \( {\displaystyle {m \choose k}{n \choose k}k!} \) 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

\( {\displaystyle {\frac {x^{(n)}}{x^{(i)}}}=(x+i)^{(n-i)},n\geq i~.} \)

Additionally, we can expand generalized exponent laws and negative rising and falling powers through the following identities:

\( {\displaystyle {\begin{aligned}(x)_{m+n}&=(x)_{m}(x-m)_{n}\\x^{(m+n)}&=x^{(m)}(x+m)^{(n)}\\x^{(-n)}&={\frac {1}{(x-n)^{(n)}}}={\frac {1}{(x-1)_{n}}}\\(x)_{-n}&={\frac {1}{(x+1)^{(n)}}}={\frac {1}{n!{\binom {x+n}{n}}}}={\frac {1}{(x+1)(x+2)\cdots (x+n)}}.\end{aligned}}} \)

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

\( {\displaystyle x^{(k+mn)}=x^{(k)}m^{mn}\prod _{j=0}^{m-1}\left({\frac {x+j+k}{m}}\right)^{(n)},m\in \mathbb {N} } \)
\( {\displaystyle (ax+b)^{(n)}=x^{n}\prod _{k=0}^{x-1}\left(a+{\frac {b+k}{x}}\right)^{(n/x)},x\in \mathbb {Z} ^{+}} \)
\( {\displaystyle (2x)^{(2n)}=2^{2n}x^{(n)}\left(x+{\frac {1}{2}}\right)^{(n)}.} \)

Alternate notations

An alternate notation for the rising factorial

\( {\displaystyle x^{\overline {m}}=\overbrace {x(x+1)\ldots (x+m-1)} ^{m{\text{ factors}}}\qquad {\text{for integer }}m\geq 0,} \)

and for the falling factorial

\( {\displaystyle x^{\underline {m}}=\overbrace {x(x-1)\ldots (x-m+1)} ^{m{\text{ factors}}}\qquad {\text{for integer }}m\geq 0~;} \)

goes back to A. Capelli (1893) and L. Toscano (1939), respectively.[2] Graham, Knuth, and Patashnik[10] propose to pronounce these expressions as "x to the m rising" and "x to the m falling", respectively.

Other notations for the falling factorial include P(x, n) , xPn , Px,n , or xPn . (See permutation and combination.)

An alternate notation for the rising factorial x(n) is the less common (x)+
n . When (x)+
n is used to denote the rising factorial, the notation (x)−
n is typically used for the ordinary falling factorial, to avoid confusion.[3]
Generalizations

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.

A generalization of the falling factorial in which a function is evaluated on a descending arithmetic sequence of integers and the values are multiplied is:

\( [f(x)]^{k/-h}=f(x)\cdot f(x-h)\cdot f(x-2h)\cdots f(x-(k-1)h), \)

where −h is the decrement and k is the number of factors. The corresponding generalization of the rising factorial is
\)
\( [f(x)]^{k/h}=f(x)\cdot f(x+h)\cdot f(x+2h)\cdots f(x+(k-1)h).

This notation unifies the rising and falling factorials, which are [x]k/1 and [x]k/−1, respectively.

For any fixed arithmetic function \( {\displaystyle f:\mathbb {N} \rightarrow \mathbb {C} } \) and symbolic parameters \( {\displaystyle x,t} \) , related generalized factorial products of the form

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

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 \({\displaystyle (x)_{n,f,t}} \) and then by the next corresponding triangular recurrence relation:

\( {\displaystyle {\begin{aligned}\left[{\begin{matrix}n\\k\end{matrix}}\right]_{f,t}&=[x^{k-1}](x)_{n,f,t}\\&=f(n-1)t^{1-n}\left[{\begin{matrix}n-1\\k\end{matrix}}\right]_{f,t}+\left[{\begin{matrix}n-1\\k-1\end{matrix}}\right]_{f,t}+\delta _{n,0}\delta _{k,0}.\end{aligned}}} \)

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, \( {\displaystyle F_{n}^{(r)}(t):=\sum _{k\leq n}{\frac {t^{k}}{f(k)^{r}}}}. \) [11]