The q-Rodrigues conjecture for Askey–Wilson polynomials

Let $D_{q,t}^{n}$ denote the $n$-fold $q$-derivative with respect to $t$, let $(q)_i$ and $(at,bt,ct)_n$ denote the corresponding $q$-shifted factorials, and let $\psi_i(x|a,b,c,q)$ be the polynomial family used in the generating function. Let $AW_n(x|a,b,c,t,q)$ denote the $n$th Askey–Wilson polynomial.

q-Rodrigues conjecture. The identity

$$D_{q,t}^{n}\left( \sum_{i\geq 0}\frac{t^{i}}{(q)_{i}}\psi _{i}\left( x|a,b,c,q\right) \right) = \frac{(1-q)^{n}AW_{n} \left( x|a,b,c,t,q\right) }{(at,bt,ct)_{n}}\sum_{i\geq 0}\frac{t^{i}}{(q)_{i} }\psi _{i}\left( x|a,b,c,q\right)$$

should hold. This extends the analogous $q$-Rodrigues-type identities established earlier in the paper for the preceding polynomial families; its validity for the Askey–Wilson generating function is posed as an open problem.