Reference request for q-numbers?

Question

Let $q$ be an element of a field $k$ (possibly $\mathbb{C}$), different from $-1$ and $1$. We have $$[n]=\frac{q^n-q^{-n}}{q-q^{-1}}=q^{n-1}+q^{n-2}+\dots+q^{-n+1}$$

Where $n$ is a natural number.

I know some basic facts about this type "number"s when $n$ is natural. I am wondering about the case when $n$ is not a natural number (or the more general case like what we have in the case of factorials, I mean the gamma function)? Could anyone possibly direct me to a reference that talks about the general case.

Is there a specific name for this "number"?

*ps feel free to tag the right keyword

Answer 1

You might find this book useful: "Quantum Calculus" by Victor Kac and Pokman Cheung, Springer, 2002.

http://www.springer.com/gp/book/9780387953410

Answer 2

These are q-numbers. Rev. F. H. Jackson and E. Heine made them into more useable forms, ie Bessel, Hypergeometric, and so on function. The q-numbers are naturally defined by \begin{align} [n]_{q} = \frac{1-q^{n}}{1-q} \end{align}

q-numbers, Wiki

q-gamma integrals by Alberto De Sole , Victor G. Kac

Many many more papers are readily found. G. E. Andrews and a few others have books packed with q-functions

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The q-numbers defined by \begin{align} [m] = \frac{q^{m} - q^{-m}}{q - q^{-1}} \end{align} are used in symmetric algebra structures of quantum mechanics.