The Rodgers polynomials $C_{\alpha,q}$ are a particular family of well-known $q$-hypergeometric function. For example, a description can be found here on Wikipedia. For the special case of $q=1$, we get back the famous ultra-spherical polynomials $C_{\alpha}$ (a description of which can also be found on Wikipedia).

Now the ultra-spherical polynomials are completely described by the differential equation $$ (1-x^2)y'' - (2\alpha +1)xy' +n(n+2\alpha)y = 0. $$ What I would like to know is, can one naively replace this by the corresponding $q$-differential equation (ie integers to $q$-integers and derivatives to $q$-derivatives) and get a $q$-differential equation characterization of the Rodgers polynomials? If so, what is a good reference for this?

I should that by $q$-derivative I mean the one which acts as $$ D_q(f) = \frac{f(qx)- f(q^{-1}x)}{qx - q^{-1}x}, $$ or for the simple case of $x^k$, $$ D_q(x^k) = [k]_qk^{k-1}. $$