# Differential Equations of Infinite Order

As a physicist I was playing with some QM problem and stumbled upon an ordinary differential equation of infinite order (coefficients are polynomials) that could be cast in the form:

$\sum_{n=0}^{\infty}(a_{n0}+a_{n1}x + a_{n2}x^2 + ... + a_{np}x^p)u^{(n)}(x) = 0$

After a little bit of searching I came across papers by H.T.Davis and L.Carleson that deal with variations of this problem (those papers are from the 30s or 40s). I was wondering whether there are newer results and where I could find them (maybe a more general theory?).

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Would you mind linking to those papers you say you found? –  Ｊ. Ｍ. Apr 17 '11 at 17:47
Even if Lennart Carleson (born 1928) is undoubtedly a genius, I have difficulties believing that he wrote papers in the 30s and 40s. –  t.b. Apr 17 '11 at 17:55
Sorry, the paper by Carleson is obviously from 1953. The papers I found by Davis: Differential Equations of Infinite Order with Constant Coefficients The Laplace Differential Equation of Infinite Order –  Jim Apr 17 '11 at 20:12