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 :

${\displaystyle (x,n)\equiv {\frac {\Gamma (x+n)}{\Gamma (x)}}}$

From the functional equation of the gamma function it then follows

${\displaystyle (x,n)\equiv x(x+1)\dotsm (x+n-1)}$.

So you have an identity

${\displaystyle (x,n)=x^{\overline {n}}}$

with the increasing factorial .

properties

Function graphs of the first four Pochhammer symbols

• The Pochhammer symbol is a meromorphic function .
• Is can be represented as a polynomial in . These have a common zero at . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle (x,n)}$ ${\displaystyle x}$ ${\displaystyle x=0}$
• Relationship between coefficients of different signs :

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

• Division rule:
  • ${\displaystyle {\frac {(x,n)}{(x,m)}}=(x+m,n-m);\quad n>m}$
  • ${\displaystyle {\frac {(x,n)}{(x,m)}}={\frac {1}{(x+m,m-n)}};\quad m>n}$
• Special values:
  • ${\displaystyle (1,n)=n!}$
  • ${\displaystyle ({\tfrac {1}{2}},n)=2^{-n}(2n-1)!!}$
  • ${\displaystyle (0,0)=1}$

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 ${\displaystyle q}$ ${\displaystyle q}$ ${\displaystyle q}$

${\displaystyle \lim _{q\rightarrow 1}{\frac {1-q^{n}}{1-q}}=n}$,

das - analogue of natural numbers above ${\displaystyle q}$

${\displaystyle [n]_{q}={\frac {1-q^{n}}{1-q}}=1+q+q^{2}+\dotsb +q^{n-1}}$

is defined.

The - Pochhammer symbol is defined by the formal power series in the variable : ${\displaystyle q}$ ${\displaystyle q}$

${\displaystyle (a;q)_{n}=\prod _{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\dotsm (1-aq^{n-1})}$

With

${\displaystyle (a;q)_{0}=1}$.

They are also called - series and as abbreviated, e.g. B. . ${\displaystyle q}$ ${\displaystyle (a;q)_{n}}$ ${\displaystyle (a)_{n}}$ ${\displaystyle (q;q)_{n}=(q)_{n}=\prod _{k=1}^{n}(1-q^{k})=(1-q)(1-q^{2})\dotsm (1-q^{n})}$

It can also be expanded into an infinite product:

${\displaystyle (a;q)_{\infty }=\prod _{k=0}^{\infty }(1-aq^{k})}$

The special case

${\displaystyle \phi (q)=(q;q)_{\infty }=\prod _{k=1}^{\infty }(1-q^{k})}$

is Euler's product, which plays a role in the theory of partition function .

Individual evidence

1. ↑ 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. ${\displaystyle (x)_{n}}$ ${\displaystyle [x]_{n}}$ ${\displaystyle [x]_{n}^{+}}$
2. ↑ Eric W. Weisstein: Pochhammer symbol. In: MathWorld . Retrieved February 9, 2019 . 
3. ↑ Eric W. Weisstein: q -Pochhammer symbol. In: MathWorld . Retrieved February 9, 2019 . 
4. ↑ Eric W. Weisstein: q -Analog. In: MathWorld . Retrieved February 9, 2019 . 
5. ↑ Euler's partition product. Also Euler function in English , but this term is ambiguous.