q-Number

📐 Definition

For $q \neq 1$ and $x \in \mathbb{C}$ , the q-number is

[x]_q = \frac{1 - q^{x}}{1 - q}

Domain and Codomain

Defined for $|q|<1$ (or more generally $q>0$ , $q\ne 1$ ) and all complex $x$ (with a branch choice for $q^x$ when $x$ is non-integer). Output is complex-valued and analytic in $x$ away from branch issues.

⚙️ Key Properties

Recovers the identity in the classical limit:

\lim_{q \to 1} [x]_q = x

Additivity Rule

[x+y]_q = [x]_q + q^{x}[y]_q

Integer Case (Geometric Sum)

For integers $n\ge 1$ ,

[n]_q = 1 + q + \cdots + q^{n-1}

🎯 Special Cases and Limits

$x$	$[x]_q$
$0$	$0$
$1$	$1$

For integer $n$ , $[n]_q$ is the q-analog of $n$ used throughout basic hypergeometric series and q-calculus.

q-exponentials and q-zeta functions are built from the same q-calculus primitives. Many q-analogs (q-factorials, q-binomial coefficients) can be expressed in terms of q-numbers.

Usage in Oakfield

Oakfield implements Jackson q-numbers as part of its q-special-function toolkit:

Special function core provides sim_q_number (real) and sim_q_number_complex plus “safe” variants with diagnostics/fallback hooks.
Lua API exposure: q-numbers are available to scripts (useful for experimenting with q-series models and parameter sweeps).
q-numbers are also used internally by the q-series implementations (e.g. q-zeta) that build on $[n+a]_q$ denominators.

Historical Foundations

📜 q-Calculus

q-numbers are a core building block of q-calculus, where geometric progressions replace linear increments and many classical identities admit q-deformed analogues.

🌍 Modern Perspective

They serve as the “q-version” of a scalar, feeding into q-derivatives, q-factorials, and basic hypergeometric series.

📚 References

Gasper & Rahman, Basic Hypergeometric Series
Andrews, Askey, Roy, Special Functions