Pochhammer symbol

In mathematics, the Pochhammer symbol

<math>(x)_n\,</math>

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

<math>(x)_n=x(x+1)(x+2)\cdots(x+n-1)</math>

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

<math>(x)_n=x(x-1)(x-2)\cdots(x-n+1).</math>

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

<math>\Delta (x)_k = k (x)_{k-1}</math>

and

<math>D x^k = k x^{k-1}</math>

(where D denotes differentiation with respect to x).