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?).