Pochhammer-Symbol

The Pochhammer symbol is a special function that is used in combinatorics and in the theory of hypergeometric functions. The name goes back to Leo August Pochhammer.

Notation

For the icon that represents this function, different variants can be used:

In theory, the special functions of

Denotes the rising factorial

However, in combinatorics, the falling factorial is thus referred

To avoid confusion, is often used for the rising and the falling factorial for. Furthermore, there is a new notation for the rising or falling factorials, which was introduced by Ronald L. Graham, Donald E. Knuth, and Oren Patashnik in their book Concrete Mathematics. For the rising factorial write

And for the falling factorial

The faculties have covered in the calculus of finite differences characteristics similar potencies in the differential calculus, and can be used to calculate ranks.

Definition in terms of special functions

Pochhammer the icon is generally defined by the Gamma function:

Then for natural numbers

Properties

• The Pochhammer symbol is a meromorphic function
• , So may be represented as a polynomial in x. These have a common root.
• Relationship between coefficients of different sign

• Division rule

• Special values