CAOP

Continuous Big q-Hermite Polynomials

Definition

The Continuous Big q-Hermite polynomials are defined as

\[ \begin{align} H_n(x;a|q) &= a^{-n} \sum_{k=0}^\infty \frac{(q^{-n};q)_k (a e^{i \theta};q)_k (a e^{-i \theta};q)_k}{(q;q)_k} q^k\\ &= a^{-n} {}_{3}\phi_{2}\!\left(\left. {q^{-n}, a e^{i \theta}, a e^{-i\theta} \atop 0, 0} \; \right| q ; q \right),\quad x=\cos(\theta) \end{align} \]

q-Recurrence Equation

Parameters

Variables

\(n\)

\(x\)

\(q\)

Parameters

\(a\) \(a > 1\)

factor (use Maple-style input)

   q-hypergeometric term in \(n\)