If $D_q$ is the Askey Wilson operator, then I have been pondering about sequence of monic polynomials $\{ P_n \}$ satisfying $D_q (P_n) = \gamma_n P_{n-1}$

To be honest, this is only Case One of a certain structure equation mentioned by Al-Salam and Ismail who can be considered as “giants” in this area of Mathematics called Special Functions.  It can be shown,  by comparison of coefficients, that $\gamma_n = [n]_q$
where here $[n]_q$ is the so-called q-numbers by Medem and Marcellán  (other two giants in this area of Mathematics) given by

$[n]_q = \frac { q^{n/2} - q^{-n/2} } { q^{1/2} - q^{-1/2} } = \frac{q^n-1}{q-1} q^{(1-n)/2}$

so these the q-numbers are related to $[n]$ of Hahn by

$[n] = [n]_q q^{(n-1)/2 }$

We have that

$D_q (T_n) = [n]_q U_{n-1}$

where $T_n$ is a degree n Chebyshev polynomial of the first type

and  $U_n$ is a degree n Chebyshev polynomial of the second type

From the book of Ismail (referrence [1] below), on page 135 Theorem 13.1.8, we have

one would think that the above (STR) is enjoyed by a polynomial “closely related” to $H_n(x|q)$, the degree n continuous (Rogers) q-Hermite polynomial.

But after intense computation, it turns out that the MOPS  $\{ P_n \}$ that
satisfy the STR above also satisfy the three-term recurrence relations given by

So which MOPS is that?  Is it “closely related” to the continuous q-Hermite polynomial?  I do not know the answer to both the questions.  I do not even know a simpler form of $C_n$.  All I know is, for my polynomials to be Rogers q-Hermite, all the $B_n$‘s must be zeros!

[1]  Mourad E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, (ENCYCLOPEDIA OF MATHEMATICS), Cambridge University Press.

$(D_q f) (x) = \frac{\delta_q f (x)}{\delta_q x}$ where $x=\cos \theta$ and $\delta_q g(e^{i \theta}) = g(q^{1/2}e^{i \theta}) -g(q^{-{1/2} e^{i \theta})$