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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. 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.

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    $\begingroup$ Note that my question here is not restricted to algebraic differential equations. $\endgroup$
    – Unknown
    Commented Dec 26, 2010 at 14:07
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    $\begingroup$ $x=f$ is the unique solution to $x'=(f'/f)x, x(0)=f(0)$ if $f(0)\ne 0$. This is of course stupid, but it shows shows that you need to make your question a bit more precise. $\endgroup$
    – Felipe Voloch
    Commented Dec 26, 2010 at 14:53
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    $\begingroup$ Thanks. I reworked it exactly as you explained(even with the same variable t) before commenting but wondered why you chose to do so with the $x$'s and $f$'s in that cognitively dissonant way. $\endgroup$
    – Unknown
    Commented Dec 26, 2010 at 17:42
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    $\begingroup$ Maybe an algebraist thinks $x$ is an unknown to be solved for, while an analyst thinks $x$ is the independent variable of a function. $\endgroup$
    – Gerald Edgar
    Commented Dec 27, 2010 at 1:23
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    $\begingroup$ If you consider invertible functions as in oeis.org/A145271, then the autonomous diff. eqn. in that entry applies to characterize f. $\endgroup$
    – Tom Copeland
    Commented Feb 1, 2012 at 9:37

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I think there are more power series than there are differential equations you can write down, in the same sense that there are more real numbers than there are names for real numbers.

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  • $\begingroup$ @Gerry: not really: for every (formal!) power series $f$ there is an ODE $u'(x)=f(x)$, so there are at least as many ODEs as series. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Dec 26, 2010 at 15:44
  • $\begingroup$ @Mariano, yes, but: if you propose to write the right side out as an infinite power series, then you can't actually write it down, while if you just call the right side $f$, then eventually you run out of names for power series, without running out of power series. $\endgroup$
    – Gerry Myerson
    Commented Dec 26, 2010 at 16:00
  • $\begingroup$ The question asks "can we find, or prove that it exists"... $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Dec 26, 2010 at 16:05
  • $\begingroup$ @Mariano, so, if the question had been whether for any real number $\alpha$ we can find, or prove there exists, a linear polynomial that characterizes it, you would say, yes, $x-\alpha$, and I would say that's cheating, because you generally can't write $\alpha$. I take your point; but I suspect OP had in mind an equation that didn't make such blatant use of the given power series, but rather used only, say, rational numbers and common functions. So I think it's up to OP to clarify intent. $\endgroup$
    – Gerry Myerson
    Commented Dec 27, 2010 at 2:40
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If the problem as it stands has been solved, then Wikipedia would not hint that the problem of determining a non-algebraic differential equation characterizing the Gamma function is open. So, my current bet is that nothing is known about characterizations of power series in terms of DE's beyond the theory of algebraic DE's.

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