Combinatorial q-Analogues

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Abstract

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

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

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

(- a; q)_{n} = \prod_{k = 0}^{n - 1} (1 + a q^{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{aligned} (a; q)_{\infty} & = \prod_{k = 0}^{\infty} (1 - a q^{k}) & = \sum_{n = 0}^{\infty} \frac{(- a)^{n} q^{n (n - 1) / 2}}{(1 - q) \dots (1 - q^{n})} \\ \frac{1}{(a; q)_{\infty}} & = \prod_{k = 0}^{\infty} \frac{1}{1 - a q^{k}} & = \sum_{n = 0}^{\infty} \frac{a^{n}}{(1 - q) \dots (1 - q^{n})} \end{aligned}

License

BSD License

Topics

Mathematics/Number theory

Session Combinatorial_Q_Analogues

Used by

Theta Functions

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