Pochhammer symbol

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In mathematics, the Pochhammer symbol

$(x)_n,\,$

introduced by Leo August Pochhammer, is used in the theory of special functions to represent the "rising factorial" or "upper factorial"

$(x)^{(n)}=x(x+1)(x+2)\cdots(x+n-1)=\frac{(x+n-1)!}{(x-1)!}$

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

$(x)_n=x(x-1)(x-2)\cdots(x-n+1)=\frac{x!}{(x-n)!}.$

To distinguish the two, the notations $x (n)$ and $(x) n$ are commonly used to denote the rising and falling factorials, respectively. They are related by a difference in sign:

$(-x)^{(n)} = (-1)^n (x)_n, \,$

where the left-hand side is a rising factorial and the right-hand side is a falling factorial. This notation will be used below.

1 Properties
2 Alternate notations
3 Relation to umbral calculus
4 Note

[edit] Properties

The empty products x⁽⁰⁾ and (x)₀ are defined to be 1 in both cases.

The rising and falling factorials can be expressed in terms of a binomial coefficient:

$\frac{x^{(n)}}{n!} = {x+n-1 \choose n} \quad\mbox{and}\quad \frac{(x)_n}{n!} = {x \choose n}.$

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

It follows from these expressions that the product of n consecutive integers is divisible by n!. Furthermore, the product of four consecutive integers is a perfect square minus one.

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

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

as can the falling factorial:

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

[edit] Alternate notations

Another, less common notation was introduced by Ronald L. Graham, Donald E. Knuth and Oren Patashnik in their book Concrete Mathematics. They define, for the rising factorial:

$x^{\overline{n}}=\frac{(x+n-1)!}{(x-1)!}$

and for the falling factorial:

$x^{\underline{n}}=\frac{x!}{(x-n)!}$

Other notations for the falling factorial include P(x, n), ^xP_n, P_x,n, or _xP_n. (See permutation and combination).

Another notation of the falling factorial using a function 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 terms. The raising factorial is written:

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

In combinatorial usage, the falling factorial is commonly denoted $(x) n$ and the rising factorial is denoted $x (n)$ (Comtet 1974, p. 6; Roman 1984, p. 5; Hardy 1999, p. 101), whereas in the calculus of finite differences and the theory of special functions, the falling factorial is denoted $x (n)$ and the rising factorial is denoted $(x) n$ (Roman 1984, p. 5; Abramowitz and Stegun 1972, p. 256; Spanier 1987). Extreme caution is therefore needed in interpreting the meanings of the notations $(x) n$ and $x (n)$ .

[edit] 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

$\Delta x^{\underline{k}} = k x^{\underline{k-1}}\,$

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.