In the mathematical field of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product

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

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

${\displaystyle (x)_{n}=x(x+1)\dots (x+n-1)}$

$${\displaystyle \lim _{q\to 1}{\frac {(q^{x};q)_{n)){(1-q)^{n))}=(x)_{n}.}$$

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

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

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

Identities

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

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

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

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

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

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

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

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

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

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

Combinatorial interpretation

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

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

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

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

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

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

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

${\displaystyle (q)_{\infty }:=(q;q)_{\infty ))$

${\displaystyle p(n)}$

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

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,

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

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:

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

q-series

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]

Relationship to other q-functions

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

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

$${\displaystyle {\begin{aligned}\left[n\right]!_{q}&=\prod _{k=1}^{n}[k]_{q}=[1]_{q}\cdot [2]_{q}\cdots [n-1]_{q}\cdot [n]_{q}\\&={\frac {1-q}{1-q)){\frac {1-q^{2)){1-q))\cdots {\frac {1-q^{n-1)){1-q)){\frac {1-q^{n)){1-q))\\&=1\cdot (1+q)\cdots (1+q+\cdots +q^{n-2})\cdot (1+q+\cdots +q^{n-1})\\&={\frac {(q;q)_{n)){(1-q)^{n))}\\\end{aligned))}$$

These numbers are analogues in the sense that

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

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

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

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

${\displaystyle k_{1},\ldots ,k_{m))$

${\displaystyle \sum _{i=1}^{m}k_{i}=n}$

${\displaystyle V_{1}\subset \dots \subset V_{m))$

${\displaystyle \dim V_{i}=\sum _{j=1}^{i}k_{j))$

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

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

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

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