- Askey Wilson
- q-Racah
- Continuous Dual q-Hahn
- Continuous q-Hahn
- Big q-Jacobi
- q-Hahn
- Dual q-Hahn
- Al Salam Chihara
- q-Meixner Pollaczek
- Continuous q-Jacobi
- Big q-Laguerre
- Little q-Jacobi
- q-Meixner
- Quantum q-Krawtchouk
- q-Krawtchouk
- Affine q-Krawtchouk
- Dual q-Krawtchouk
- Continuous Big q-Hermite
- Continuous q-Laguerre
- Little q-Laguerre / Wall
- q-Laguerre
- Alternative q-Charlier
- q-Charlier
- Al-Salam-Carlitz I
- Al-Salam-Carlitz II
- Continuous q-Hermite
- Stieltjes-Wigert
- Discrete q-Hermite I
- Discrete q-Hermite II
Discrete q-Hermite I Polynomials
Definition
The Discrete q-Hermite I polynomials are defined as
\[
\begin{align}
h_n(x;q) &= q^{n \choose 2} \sum_{k=0}^\infty \frac{(q^{-n};q)_k (x^{-1};q)_k}{(q;q)_k} \left(-qx\right)^k\\
&= q^{n \choose 2} {}_{2}\phi_{1}\!\left(\left. {q^{-n}, x^{-1} \atop 0} \; \right| q ; -q x \right)
\end{align}
\]