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](http://en.wikipedia.org/wiki/Rogers_polynomials). 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](http://en.wikipedia.org/wiki/Ultraspherical_polynomials)).


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 $x^k$,
$$
D\_q(x^k) = [k]\_qk^{k-1}.
$$



$q$-differential equation for the Rodgers polynomials?
Yes.


See <http://arxiv.org/pdf/0704.3123v1.pdf> and references therein. 


This is known since long, I guess Koelink's paper was one of the first on the argument...


<http://www.jstor.org/discover/10.2307/2161278?uid=3738296&uid=2129&uid=2&uid=70&uid=4&sid=21102660095777>


Finally let me mention that the famous work of Koekoek and Swarttouw which summarizes a lot of such informations was finally made into a book:


Hypergeometric Orthogonal Polynomials and Their q-Analogues
Koekoek, Lesky, Swarttouw 
Springer Monographs in Math 2010.


