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# Pochhammer symbol

In mathematics, the Pochhammer symbol

$\left(x\right)_\left\{n\right\},$

introduced by Leo August Pochhammer, represents either the rising or the falling factorial. Unfortunately there is no standard convention about which sort of factorial it represents.

The Pochhammer symbol (x)n is used in the theory of special functions (in particular the hypergeometric function) for the rising sequential product, sometimes called the "rising factorial" or "upper factorial".

$\left(x\right)_n=x\left(x+1\right)\left(x+2\right)cdots\left(x+n-1\right)=frac\left\{Gamma\left(x+n\right)\right\}\left\{Gamma\left(x\right)\right\}= frac\left\{\left(x+n-1\right)!\right\}\left\{\left(x-1\right)!\right\},$

but it is used in combinatorics to represent the falling sequential product (or "falling factorial" or "lower factorial")

$\left(x\right)_\left\{n\right\}=x\left(x-1\right)\left(x-2\right)cdots\left(x-n+1\right)=frac\left\{x!\right\}\left\{\left(x-n\right)!\right\}.,$

To distinguish the two, the notations $x^\left\{\left(n\right)\right\}$ and $\left(x\right)_\left\{n\right\}$ are sometimes used in combinatorics to denote the rising and falling sequential products, respectively. They are related by a difference in sign:

$\left(-x\right)^\left\{\left(n\right)\right\} = \left(-1\right)^n \left(x\right)_\left\{n\right\},,$

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

The two are related to the genuine factorial function by the formula:

$1^\left\{\left(n\right)\right\}=\left(n\right)_\left\{n\right\}=n!.,$

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.

## Properties

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

The rising and falling sequential products (sometimes improperly called "factorials") can be used to express a binomial coefficient:

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

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

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

as can the falling sequential product:

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

Rising and falling sequential products obey an equation similar to the binomial theorem:

$\left(a + b\right)^\left\{\left(n\right)\right\} = sum_^n \left\{n choose j\right\} \left(a\right)^\left\{\left(n-j\right)\right\}\left(b\right)^\left\{\left(j\right)\right\}$

$\left(a + b\right)_n = sum_^n \left\{n choose j\right\} \left(a\right)_\left\{n-j\right\}\left(b\right)_\left\{j\right\}$

where the coefficients are the same as the ones in the binomial expansion.

A rising sequential product can be expressed as a falling sequential product that starts from the other end: a(n) = (a + n − 1)n.

## 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 sequential product:

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

and for the falling sequential product:

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

Other notations for the falling sequential product include P(xn), xPn, Px,n, or xPn. (See permutation and combination). An alternate notation for the rising sequential product x(n) is the less common (x)+n. When the notation (x)+n is used for the rising product, the notation (x)n is typically used for the ordinary falling product to avoid confusion.

Another notation of the falling sequential product using a function is:

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

where −h is the decrement and k is the number of terms. The rising sequential product is written:

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

## Relation to umbral calculus

The falling sequential product 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 sequential product (x)k in the calculus of finite differences plays the role of xk in differential calculus. Note for instance the similarity of

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

and

$D x^k = k x^\left\{k-1\right\},$

(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.

## Connection coefficients

Since the falling sequential products are a basis of the polynomial ring, we can re-express the product of two of them as a linear combination of falling sequential products:

$x^\left\{underline\left\{m\right\}\right\} x^\left\{underline\left\{n\right\}\right\} = sum_\left\{k=0\right\}^\left\{m\right\} \left\{m choose k\right\} \left\{n choose k\right\} k!, x^\left\{underline\left\{m+n-k\right\}\right\}.$

The connection coefficients have a combinatorial interpretation as the number of ways to identify (or glue together) k elements from a set of size m and a set of size n.

## Notes

• Pochhammer actually used (x)n to denote the binomial coefficient .

## References

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