Falling and rising factorials

In mathematics, the Pochhammer symbol

$(x)_{n}\,$

is used in the theory of special functions to represent the "rising factorial" or "upper factorial"

$(x)_{n}=x(x+1)(x+2)\cdots (x+n-1)$

and, confusingly, is used in combinatorics to represent the "falling factorial" or "lower factorial"

$(x)_{n}=x(x-1)(x-2)\cdots (x-n+1).$

The empty product (x)₀ is defined to be 1 in both cases.

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 of calculus. In this formula and in many other places, the falling factorial (x)_k in the calculus of finite differences plays the role of x^k in differential calculus. Note for instance the similarity of

$\Delta (x)_{k}=k(x)_{k-1}$

and

$Dx^{k}=kx^{k-1}$

(where D denotes differentiation with respect to x).

The notation was introduced by Leo August Pochhammer.

An alternative notation used by Ronald L. Graham, Donald E. Knuth and Oren Patashnik in their book Concrete Mathematics uses

$x^{\overline {n}}$

for the rising factorial and

$x^{\underline {n}}$

for the falling factorial.