I was wondering whether for any convergent real power series (or a Frobenius series) we can find (or prove that there exists) a corresponding differential equation that characterizes it. I am aware of [Hölder's theorem][1] (I do not know whether the gamma function has a power series representation). So, in effect I am looking for results in these lines but, of course , for real analytic functions. (Generally, my question reads: Can real analytic functions be characterized by differential equations?)

P.S.: I feel these statements are rather vague. But I am eager to hear your comments/answers. [1]: http://en.wikipedia.org/wiki/H%25C3%25B6lder%2527s_theorem