CAOP

Discrete q-Hermite I Polynomials

Definition

The Discrete q-Hermite I polynomials are defined as

\[ \begin{align} h_n(x;q) &= q^{n \choose 2} \sum_{k=0}^\infty \frac{(q^{-n};q)_k (x^{-1};q)_k}{(q;q)_k} \left(-qx\right)^k\\ &= q^{n \choose 2} {}_{2}\phi_{1}\!\left(\left. {q^{-n}, x^{-1} \atop 0} \; \right| q ; -q x \right) \end{align} \]

q-Differential Equation

q-Recurrence Equation

Parameters

Variables

\(n\)

\(x\)

\(q\)

factor (use Maple-style input)

   q-hypergeometric term in \(n\) and q-hyperexponential term in \(x\) required