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

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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.

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