- Q-Pochhammer symbol
In
mathematics , in the area ofcombinatorics , a q-Pochhammer symbol, also called a q-shifted factorial, is aq-analog of the commonPochhammer symbol . It is defined as:
The q-Pochhammer symbol is a major building block in the construction of q-analogs; for instance, in the theory of
basic hypergeometric series , it plays the role that the ordinary Pochhammer symbol plays in the theory ofhypergeometric series .Unlike the ordinary Pochhammer symbol, the q-Pochhammer symbol can be extended to an infinite product:
:
This is an
analytic function of "q" in the interior of theunit disk , and can also be considered as aformal power series in "q". The special case:
is known as
Euler's function , and is important incombinatorics ,number theory , and the theory ofmodular forms .A q-series is a series in which the coefficients are functions of "q", typically depending on "q" via q-Pochhammer symbols.
Identities
The finite product can be expressed in terms of the infinite product:
:
which extends the definition to negative integers "n". Thus, for nonnegative "n", one has
:
and
:
The "q"-Pochhammer symbol is the subject of a number of q-series identities, particularly the infinite series expansions
:
and
:,
which are both special cases of the
q-binomial theorem ::
Combinatorial interpretation
The "q"-Pochhammer symbol is closely related to the enumerative combinatorics of partitions. The coefficient of in:is the number of partitions of "m" into at most "n" parts.
Since, by conjugation of partitions, this is the same as the number of partitions of "m" into parts of size at most "n", by identification of generating series we obtain the identity:
:as in the above section.
We also have that the coefficient of in:is the number of partitions of "m" into "n" or "n"-1 distinct parts.
By removing a triangular partition with "n-1" parts from such a partition, we are left with an arbitrary partition with at most "n" parts. This gives a weight-preserving bijection between the set of partitions into "n" or "n"-1 distinct parts and the set of pairs consisting of a triangular partition having "n"-1 parts and a partition with at most "n" parts. By identifying generating series, this leads to the identity:
:also described in the above section.
The "q"-binomial theorem itself can also be handled by a slightly more involved combinatorial argument of a similar flavour.
Multiple arguments convention
Since identities involving "q"-Pochhammer symbols so frequently involve products of many symbols, the standard convention is to write a product as a single symbol of multiple arguments:
:
Relationship to the "q"-bracket and the "q"-binomial
Noticing that
:
we define the "q"-analog of "n", also known as the "q"-bracket or "q"-number of "n" to be
:
From this one can define the "q"-analog of the
factorial , the "q"-factorial, as:
Again, one recovers the usual factorial by taking the limit as "q" approaches 1.
From the "q"-factorials, one can move on to define the "q"-binomial coefficients, also known as
Gaussian coefficient s, Gaussian polynomials, orGaussian binomial s::
One can check that
:
One also obtains a q-analog of the
Gamma function , called the q-Gamma function, and defined as:
Note that
:
and
:
This converges to the usual Gamma function as "q" approaches 1 from inside the unit disc.
ee also
*
Basic hypergeometric series
*Q-derivative
*Q-theta function
*Elliptic gamma function
* Jacobi theta functionReferences
* George Gasper and Mizan Rahman, "Basic Hypergeometric Series, 2nd Edition", (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0-521-83357-4.
* Roelof Koekoek and Rene F. Swarttouw, " [http://fa.its.tudelft.nl/~koekoek/askey/ The Askey scheme of orthogonal polynomials and its q-analogues] ", section 0.2.External links
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