Pochhammer symbol

In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer ,

$(x)_n\,$

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

$(x)_n=x(x+1)(x+2)\cdots(x+n-1)$

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. Note that the falling factorial can be written as a binomial coefficient:

$\frac{(x)_n}{n!} = {x \choose n}$

and thus a large number of identities on the binomial coefficients carry over to the Pochhammer symbols.

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 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

Δ(x) k = k (x) k - 1

and

D x k = k x k - 1

(where D denotes differentiation with respect to x). The study of similarities of this type is known as umbral calculus. The general theory covering such relations, including the Pochhammer polynomials, is given by the theory of polynomial sequences of binomial type and by Sheffer sequences.

Notation

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.