# 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:
• ${\ displaystyle {\ frac {(x, n)} {(x, m)}} = (x + m, nm); \ quad n> m}$
• ${\ displaystyle {\ frac {(x, n)} {(x, m)}} = {\ frac {1} {(x + m, mn)}}; \ quad m> n}$
• 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.