Pochhammer Icon
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 .
definition
The Pochhammer symbol is defined using the gamma function :
From the functional equation of the gamma function it then follows
- .
So you have an identity
with the increasing factorial .
properties
- The Pochhammer symbol is a meromorphic function .
- Is can be represented as a polynomial in . These have a common zero at .
- Relationship between coefficients of different signs :
- Division rule:
- Special values:
q pounding hammer icon
The - Pochhammer symbol is the - analogue of the Pochhammer symbol and plays a role in combinatorics for - analogues of classic formulas, where, stimulated by the border crossing
- ,
das - analogue of natural numbers above
is defined.
The - Pochhammer symbol is defined by the formal power series in the variable :
With
- .
They are also called - series and as abbreviated, e.g. B. .
It can also be expanded into an infinite product:
The special case
is Euler's product, which plays a role in the theory of partition function .
Individual evidence
- ↑ L. Pochhammer: About the differential equation of the more general hypergeometric series with two finite singular points . Journal for pure and applied mathematics, Volume 102, pp. 76-159, 1888; especially pp. 80-81. Pochhammer uses the term for the binomial coefficient, for the falling factorial and for the increasing factorial.
- ↑ Eric W. Weisstein: Pochhammer symbol. In: MathWorld . Retrieved February 9, 2019 .
- ↑ Eric W. Weisstein: q -Pochhammer symbol. In: MathWorld . Retrieved February 9, 2019 .
- ↑ Eric W. Weisstein: q -Analog. In: MathWorld . Retrieved February 9, 2019 .
- ↑ Euler's partition product. Also Euler function in English , but this term is ambiguous.