q-Digamma Function

📐 Definition

For $0 < q < 1$ , the q-Gamma function is

\Gamma_q(z) = (1-q)^{1-z} \frac{(q;q)_\infty}{(q^{z}; q)_\infty}, \qquad (a;q)_\infty = \prod_{k=0}^{\infty} (1 - a q^{k})

The q-digamma function is defined as the logarithmic derivative

\psi_q(z) = \frac{d}{dz} \log \Gamma_q(z)

Domain and Codomain

Meromorphic on $\mathbb{C}$ with simple poles at $z = 0, -1, -2, \dots$ . Maps to $\mathbb{C}$ .

⚙️ Key Properties

Series representation (absolutely convergent for $0<q<1$ ):

\psi_q(z) = -\log(1-q) + \log q \sum_{k=0}^{\infty} \frac{q^{k+z}}{1 - q^{k+z}}

Recurrence:

\psi_q(z+1) = \psi_q(z) - \frac{q^{z}\log q}{1 - q^{z}}

🎯 Special Cases and Limits

Classical limit: as $q \to 1^-$ , $\psi_q(z) \to \psi(z)$ (the classical digamma).
At positive integers $n$ , $\psi_q(n)$ can be expressed in terms of q-harmonic-type sums and $\log(1-q)$ .

The limit $q\to 1$ recovers the classical digamma. q-polygamma functions arise from higher derivatives in $z$ . Many identities connect $\Gamma_q$ and $\psi_q$ to basic hypergeometric series and q-exponentials.

Usage in Oakfield

Oakfield includes q-digamma as part of its q-analog special-function suite:

Special function core implements sim_q_digamma and complex sim_q_digamma_complex, with “safe” variants returning diagnostic reports and supporting fallback handlers.
Lua API exposure: available as sim.qdigamma(x_or_complex, q) for q-series exploration and validation (currently not a built-in warp profile on its own).

Historical Foundations

📜 q-Gamma and q-Polygamma Families

q-analogs of gamma and digamma functions emerged from the development of q-calculus and basic hypergeometric series, providing deformations of classical special functions with geometric progression structure.

🌍 Modern Perspective

q-digamma functions are standard tools in q-series analysis and appear in deformed statistical and spectral models.

📚 References

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