q-Pochhammer symbol

In mathematics, in the area of combinatorics, a q-Pochhammer symbol, also called a q-shifted factorial, is a q-analog of the common Pochhammer symbol. It is defined as

\[ (a;q)_n = \prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1}) \]

with

(a;q)₀ = 1

by definition. 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 of generalized hypergeometric series.

Unlike the ordinary Pochhammer symbol, the q-Pochhammer symbol can be extended to an infinite product:

\[ (a;q)_\infty = \prod_{k=0}^{\infty} (1-aq^k). \]

This is an analytic function of q in the interior of the unit disk, and can also be considered as a formal power series in q. The special case

\[ \phi(q) = (q;q)_\infty=\prod_{k=1}^\infty (1-q^k) \]

is known as Euler's function, and is important in combinatorics, number theory, and the theory of modular 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:

\[ (a;q)_n = \frac{(a;q)_\infty} {(aq^n;q)_\infty}, \]

which extends the definition to negative integers n. Thus, for nonnegative n, one has

\[ (a;q)_{-n} = \frac{1}{(aq^{-n};q)_n}=\prod_{k=1}^n \frac{1}{(1-a/q^k)} \]

and

\[ (a;q)_{-n} = \frac{(-q/a)^n q^{n(n-1)/2}} {(q/a;q)_n}. \]

The q-Pochhammer symbol is the subject of a number of q-series identities, particularly the infinite series expansions

\[ (x;q)_\infty = \sum_{n=0}^\infty \frac{(-1)^n q^{n(n-1)/2}}{(q;q)_n} x^n \]

and

\[ \frac{1}{(x;q)_\infty}=\sum_{n=0}^\infty \frac{x^n}{(q;q)_n}, \]

which are both special cases of the q-binomial theorem:

\[ \frac{(ax;q)_\infty}{(x;q)_\infty} = \sum_{n=0}^\infty \frac{(a;q)_n}{(q;q)_n} x^n. \]

Combinatorial interpretation

The q-Pochhammer symbol is closely related to the enumerative combinatorics of partitions. The coefficient of qman in

\[ (a;q)_\infty^{-1} = \prod_{k=0}^{\infty} (1-aq^k)^{-1} \]

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:

\[ (a;q)_\infty^{-1} = \sum_{k=0}^\infty \left(\prod_{j=1}^k \frac{1}{1-q^j} \right) a^k = \sum_{k=0}^\infty \frac{a^k}{(q;q)_k} \]

as in the above section.

We also have that the coefficient of qman in

\[ (-a;q)_\infty = \prod_{k=0}^{\infty} (1+aq^k) \]

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:

\[ (-a;q)_\infty = \prod_{k=0}^\infty (1+aq^k) = \sum_{k=0}^\infty \left(q^{k\choose 2} \prod_{j=1}^k \frac{1}{1-q^j}\right) a^k = \sum_{k=0}^\infty \frac{q^{k\choose 2}}{(q;q)_k} a^k \]

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:

\[ (a_1,a_2,\ldots,a_m;q)_n = (a_1;q)_n (a_2;q)_n \ldots (a_m;q)_n. \]

Relationship to other q-functions

Noticing that

\[ \lim_{q\rightarrow 1}\frac{1-q^n}{1-q}=n, \]

we define the q-analog of n, also known as the q-bracket or q-number of n to be

\[ [n]_q=\frac{1-q^n}{1-q}. \]

From this one can define the q-analog of the factorial, the q-factorial, as

\[ \big[n]_q! =\prod_{k=1}^n [k]_q
= [1]_q [2]_q \cdots [n-1]_q [n]_q
=\frac{1-q}{1-q} \frac{1-q^2}{1-q} \cdots \frac{1-q^{n-1}}{1-q} \frac{1-q^n}{1-q}
=1(1+q)\cdots (1+q+\cdots + q^{n-2}) (1+q+\cdots + q^{n-1})
=\frac{(q;q)_n}{(1-q)^n}. \]

Again, one recovers the usual factorial by taking the limit as q approaches 1. This can be interpreted as the number of flags in an n-dimensional vector space over the field with q elements, and taking the limit as q goes to 1 yields the interpretation of an ordering on a set as a flag in a vector space over the field with one element.

A product of negative integer q-brackets can be expressed in terms of the q-factorial as:

\[ \prod_{k=1}^n [-k]_q = \frac{(-1)^n\,[n]_q!}{q^{n(n+1)/2}} \]

From the q-factorials, one can move on to define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients:

\[ \begin{bmatrix} n\\ k \end{bmatrix}_q = \frac{[n]_q!}{[n-k]_q! [k]_q!}. \]

One can check that

\[ \begin{bmatrix} n+1\\ k \end{bmatrix}_q = \begin{bmatrix} n\\ k \end{bmatrix}_q + q^{n-k+1} \begin{bmatrix} n\\ k-1 \end{bmatrix}_q. \]

One also obtains a q-analog of the Gamma function, called the q-gamma function, and defined as

\[ \Gamma_q(x)=\frac{(1-q)^{1-x} (q;q)_\infty}{(q^x;q)_\infty} \]

This converges to the usual Gamma function as q approaches 1 from inside the unit disc.. Note that

\[\Gamma_q(x+1)=[x]_q\Gamma_q(x)\, \]

for any x and

\[ \Gamma_q(n+1)=[n]_q!\frac{}{}. \]

for non-negative integer values of n. Alternatively, this may be taken as an extension of the q-factorial function to the real number system.
See also

Basic hypergeometric series
Pochhammer symbol
Q-derivative
Q-theta function
Elliptic gamma function
Jacobi theta function

References

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, The Askey scheme of orthogonal polynomials and its q-analogues, section 0.2.