q-Exponential

📐 Definition

For $|q| < 1$ , the q-Pochhammer symbol is $(a;q)_n = \prod_{k=0}^{n-1} (1 - a q^{k})$ with $(q;q)_0 = 1$ . The q-exponential is

e_q(x) = \sum_{n=0}^{\infty} \frac{x^n}{(q;q)_n}, \qquad (a;q)_n = \prod_{k=0}^{n-1} (1 - a q^{k})

Domain and Codomain

For fixed $|q|<1$ , the series converges for $|x|<1$ and defines an analytic function on the unit disk. The product form below provides a meromorphic continuation to $\mathbb{C}$ .

⚙️ Key Properties

Normalization

e_q(0) = 1

q-Derivative Eigenfunction

With the q-derivative $D_q f(x) = \frac{f(x) - f(qx)}{(1-q)x}$ ,

D_q\, e_q\!\big((1-q)ax\big) = a\, e_q\!\big((1-q)ax\big)

Product Form

For $|q|<1$ ,

e_q(x) = \frac{1}{(x;q)_\infty}, \qquad (x;q)_\infty = \prod_{k=0}^{\infty} (1-xq^k)

In particular, $e_q(x)$ is meromorphic in $x$ with poles at $x=q^{-k}$ for integers $k\ge 0$ .

🎯 Special Cases and Limits

Classical limit: as $q \to 1^-$ , $e_q((1-q)x) \to e^{x}$ .
$q=0$ : $e_0(x) = \frac{1}{1-x}$ (for $|x|<1$ ).

As $q\to 1$ , the q-exponential connects to the classical exponential and (via Euler’s formula) to the complex exponential. It is also linked to q-gamma and q-digamma through basic hypergeometric identities.

Usage in Oakfield

Oakfield does not currently expose a standalone “q-exponential” operator/function in the runtime API.

Related usage in the current codebase:

q-series implementations (q-number/q-zeta/q-digamma) rely on $q^x$ and related exponentials internally (implemented via exp(x * log(q)) in stable forms).
q-hyperexponential warp profile (analytic_warp QHYPEREXP) uses q-deformed hyperexponential helpers that depend on geometric q-progressions.

Historical Foundations

📜 Jackson q-Exponentials

Jackson introduced q-exponentials as core objects in q-calculus, serving as eigenfunctions of the q-derivative and as generating functions for q-series.

🌍 Modern Perspective

q-exponentials are standard primitives in basic hypergeometric function theory and q-deformed modeling.

📚 References

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