Combinatorial q-Analogues

This entry defines the $q$-analogues of various combinatorial symbols, namely:

• The $q$-bracket $[n]_q = \frac{1-q^n}{1-q}$ for $n\in\mathbb{Z}$
• The $q$-factorial $[n]_q! = [1]_q [2]_q \cdots [n]_q$ for $n\in\mathbb{Z}$
• The $q$-binomial coefficients $\binom{n}{k}_{\!q} = \frac{[n]_q!}{[k]_q!\,[n-k]_q!}$ for $n,k\in\mathbb{N}$ (also known as Gaussian binomial coefficients or Gaussian polynomials)
• The infinite $q$-Pochhammer symbol $(a; q)_\infty = \prod_{n=0}^\infty\, (1 - aq^n)$
• Euler's $\phi$ function $\phi(q) = (q; q)_\infty$
• The finite $q$-Pochhammer symbol $(a; q)_n = (a; q)_\infty / (aq^n; q)_\infty$ for $n\in\mathbb{Z}$

Proofs for many basic properties are provided, notably for the $q$-binomial theorem:

\[(-a; q)_n = \prod_{k=0}^{n-1} (1 + aq^n) = \sum_{k=0}^n \binom{n}{k}_{\!\!q} a^k q^{k(k-1)/2} \]

Additionally, two identities of Euler are formalised that give power series expansions for $(a; q)_\infty$ and $1/(a; q)_\infty$ in powers of $a$:

\[ \begin{alignat*}{3} (a; q)_\infty &{}= \prod_{k=0}^\infty (1 - aq^k) &{}= \sum_{n=0}^\infty \frac{(-a)^n q^{n(n-1)/2}}{(1-q) \cdots (1-q^n)}\\ \frac{1}{(a; q)_\infty} &{}= \prod_{k=0}^\infty \frac{1}{1 - aq^k} &{}= \sum_{n=0}^\infty \frac{a^n}{(1-q)\cdots(1-q^n)} \end{alignat*} \]