q-Pochhammer symbol

The finite product can be expressed in terms of the infinite product:

$${\displaystyle (a;q)_{n}={\frac {(a;q)_{\infty }}{(aq^{n};q)_{\infty }}},}$$

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

$${\displaystyle (a;q)_{-n}={\frac {1}{(aq^{-n};q)_{n}}}=\prod _{k=1}^{n}{\frac {1}{(1-a/q^{k})}}}$$

and

$${\displaystyle (a;q)_{-n}={\frac {(-q/a)^{n}q^{n(n-1)/2}}{(q/a;q)_{n}}}.}$$

Alternatively,

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

which is useful for some of the generating functions of partition functions.

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

$${\displaystyle (x;q)_{\infty }=\sum _{n=0}^{\infty }{\frac {(-1)^{n}q^{n(n-1)/2}}{(q;q)_{n}}}x^{n}}$$

and

$${\displaystyle {\frac {1}{(x;q)_{\infty }}}=\sum _{n=0}^{\infty }{\frac {x^{n}}{(q;q)_{n}}},}$$

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

$${\displaystyle {\frac {(ax;q)_{\infty }}{(x;q)_{\infty }}}=\sum _{n=0}^{\infty }{\frac {(a;q)_{n}}{(q;q)_{n}}}x^{n}.}$$

Fridrikh Karpelevich found the following identity (see Olshanetsky and Rogov (1995) for the proof):

$${\displaystyle {\frac {(q;q)_{\infty }}{(z;q)_{\infty }}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}q^{n(n+1)/2}}{(q;q)_{n}(1-zq^{n})}},\ |z|<1.}$$

The q-Pochhammer symbol is closely related to the enumerative combinatorics of partitions. The coefficient of ${\displaystyle q^{m}a^{n}}$ in

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

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

$${\displaystyle (a;q)_{\infty }^{-1}=\sum _{k=0}^{\infty }\left(\prod _{j=1}^{k}{\frac {1}{1-q^{j}}}\right)a^{k}=\sum _{k=0}^{\infty }{\frac {a^{k}}{(q;q)_{k}}}}$$

as in the above section.

We also have that the coefficient of ${\displaystyle q^{m}a^{n}}$ in

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

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

$${\displaystyle (-a;q)_{\infty }=\prod _{k=0}^{\infty }(1+aq^{k})=\sum _{k=0}^{\infty }\left(q^{k \choose 2}\prod _{j=1}^{k}{\frac {1}{1-q^{j}}}\right)a^{k}=\sum _{k=0}^{\infty }{\frac {q^{k \choose 2}}{(q;q)_{k}}}a^{k}}$$

also described in the above section. The reciprocal of the function ${\displaystyle (q)_{\infty }:=(q;q)_{\infty }}$ similarly arises as the generating function for the partition function, ${\displaystyle p(n)}$, which is also expanded by the second two q-series expansions given below:[1]

$${\displaystyle {\frac {1}{(q;q)_{\infty }}}=\sum _{n\geq 0}p(n)q^{n}=\sum _{n\geq 0}{\frac {q^{n}}{(q;q)_{n}}}=\sum _{n\geq 0}{\frac {q^{n^{2}}}{(q;q)_{n}^{2}}}.}$$

The q-binomial theorem itself can also be handled by a slightly more involved combinatorial argument of a similar flavor (see also the expansions given in the next subsection).

Similarly,

$${\displaystyle (q;q)_{\infty }=1-\sum _{n\geq 0}q^{n+1}(q;q)_{n}=\sum _{n\geq 0}q^{\frac {n(n+1)}{2}}{\frac {(-1)^{n}}{(q;q)_{n}}}.}$$

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:

$${\displaystyle (a_{1},a_{2},\ldots ,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots (a_{m};q)_{n}.}$$

A q-series is a series in which the coefficients are functions of q, typically expressions of ${\displaystyle (a;q)_{n}}$.[2] Early results are due to Euler, Gauss, and Cauchy. The systematic study begins with Eduard Heine (1843).[3]

The q-analog of n, also known as the q-bracket or q-number of n, is defined to be

$${\displaystyle [n]_{q}={\frac {1-q^{n}}{1-q}}.}$$

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

$${\displaystyle [n]!_{q}=\prod _{k=1}^{n}[k]_{q}=[1]_{q}\cdot [2]_{q}\cdots [n-1]_{q}\cdot [n]_{q}.}$$

This can be rewritten in many equivalent ways, including as ${\displaystyle {\frac {1-q}{1-q}}{\frac {1-q^{2}}{1-q}}\cdots {\frac {1-q^{n-1}}{1-q}}{\frac {1-q^{n}}{1-q}}}$, ${\displaystyle 1\cdot (1+q)\cdots (1+q+\cdots +q^{n-2})\cdot (1+q+\cdots +q^{n-1})}$, and ${\displaystyle {\frac {(q;q)_{n}}{(1-q)^{n}}}.}$

These numbers are analogues in the sense that

$${\displaystyle \lim _{q\rightarrow 1}[n]_{q}=n,}$$

and so also

$${\displaystyle \lim _{q\rightarrow 1}[n]!_{q}=n!.}$$

The limit value n! counts permutations of an n-element set S. Equivalently, it counts the number of sequences of nested sets ${\displaystyle E_{1}\subset E_{2}\subset \cdots \subset E_{n}=S}$ such that ${\displaystyle E_{i}}$ contains exactly i elements.[4] By comparison, when q is a prime power and V is an n-dimensional vector space over the field with q elements, the q-analogue ${\displaystyle [n]!_{q}}$ is the number of complete flags in V, that is, it is the number of sequences ${\displaystyle V_{1}\subset V_{2}\subset \cdots \subset V_{n}=V}$ of subspaces such that ${\displaystyle V_{i}}$ has dimension i.[4] The preceding considerations suggest that one can regard a sequence of nested sets as a flag over a conjectural field with one element.

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

$${\displaystyle \prod _{k=1}^{n}[-k]_{q}={\frac {(-1)^{n}\,[n]!_{q}}{q^{n(n+1)/2}}}}$$

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

$${\displaystyle {\begin{bmatrix}n\\k\end{bmatrix}}_{q}={\frac {[n]!_{q}}{[n-k]!_{q}[k]!_{q}}},}$$

where it is easy to see that the triangle of these coefficients is symmetric in the sense that ${\displaystyle {\begin{bmatrix}n\\m\end{bmatrix}}_{q}={\begin{bmatrix}n\\n-m\end{bmatrix}}_{q}}$ for all ${\displaystyle 0\leq m\leq n}$.

One can check that

$${\displaystyle {\begin{aligned}{\begin{bmatrix}n+1\\k\end{bmatrix}}_{q}&={\begin{bmatrix}n\\k\end{bmatrix}}_{q}+q^{n-k+1}{\begin{bmatrix}n\\k-1\end{bmatrix}}_{q}\\&={\begin{bmatrix}n\\k-1\end{bmatrix}}_{q}+q^{k}{\begin{bmatrix}n\\k\end{bmatrix}}_{q}.\end{aligned}}}$$

One can also see from the previous recurrence relations that the next variants of the ${\displaystyle q}$-binomial theorem are expanded in terms of these coefficients as follows:[5]

$${\displaystyle {\begin{aligned}(z;q)_{n}&=\sum _{j=0}^{n}{\begin{bmatrix}n\\j\end{bmatrix}}_{q}(-z)^{j}q^{\binom {j}{2}}=(1-z)(1-qz)\cdots (1-zq^{n-1})\\(-q;q)_{n}&=\sum _{j=0}^{n}{\begin{bmatrix}n\\j\end{bmatrix}}_{q^{2}}q^{j}\\(q;q^{2})_{n}&=\sum _{j=0}^{2n}{\begin{bmatrix}2n\\j\end{bmatrix}}_{q}(-1)^{j}\\{\frac {1}{(z;q)_{m+1}}}&=\sum _{n\geq 0}{\begin{bmatrix}n+m\\n\end{bmatrix}}_{q}z^{n}.\end{aligned}}}$$

One may further define the q-multinomial coefficients

$${\displaystyle {\begin{bmatrix}n\\k_{1},\ldots ,k_{m}\end{bmatrix}}_{q}={\frac {[n]!_{q}}{[k_{1}]!_{q}\cdots [k_{m}]!_{q}}},}$$

where the arguments ${\displaystyle k_{1},\ldots ,k_{m}}$ are nonnegative integers that satisfy ${\displaystyle \sum _{i=1}^{m}k_{i}=n}$. The coefficient above counts the number of flags ${\displaystyle V_{1}\subset \dots \subset V_{m}}$ of subspaces in an n-dimensional vector space over the field with q elements such that ${\displaystyle \dim V_{i}=\sum _{j=1}^{i}k_{j}}$.

The limit ${\displaystyle q\to 1}$ gives the usual multinomial coefficient ${\displaystyle {n \choose k_{1},\dots ,k_{m}}}$, which counts words in n different symbols ${\displaystyle \{s_{1},\dots ,s_{m}\}}$ such that each ${\displaystyle s_{i}}$ appears ${\displaystyle k_{i}}$ times.

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

$${\displaystyle \Gamma _{q}(x)={\frac {(1-q)^{1-x}(q;q)_{\infty }}{(q^{x};q)_{\infty }}}}$$

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

$${\displaystyle \Gamma _{q}(x+1)=[x]_{q}\Gamma _{q}(x)}$$

for any x and

$${\displaystyle \Gamma _{q}(n+1)=[n]!_{q}}$$

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.

• Basic hypergeometric series
• Elliptic gamma function
• Jacobi theta function
• Lambert series
• Pentagonal number theorem
• q-derivative
• q-theta function
• q-Vandermonde identity
• Rogers–Ramanujan identities
• Rogers–Ramanujan continued fraction

• 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.
• Exton, H. (1983), q-Hypergeometric Functions and Applications, New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914, ISBN 0470274530, ISBN 978-0470274538
• M.A. Olshanetsky and V.B.K. Rogov (1995), The Modified q-Bessel Functions and the q-Bessel-Macdonald Functions, arXiv:q-alg/9509013.

• Weisstein, Eric W. "q-Analog". MathWorld.
• Weisstein, Eric W. "q-Bracket". MathWorld.
• Weisstein, Eric W. "q-Factorial". MathWorld.
• Weisstein, Eric W. "q-Series". MathWorld.
• Weisstein, Eric W. "q-Binomial Coefficient". MathWorld.