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In Stanley EC2, it is an open question whether $\sum b_nx^{n^3}$ can satisfy an ADE. Stanley remarks that if this is true then it leads to a "completely unexpected result about representing integers as sum of cubes."

QUESTIONS.

(a) What completely unexpected result is he referring to?

(b) Does it have to do with the constructive nature of being able to write an integer as a sum of cubes? By Waring-Hilbert, we know it can be done with 9 cubes, but perhaps not constructively.

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  • $\begingroup$ Summoning @RichardStanley ... $\endgroup$
    – Ilya Bogdanov
    Commented Oct 19, 2016 at 21:17
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    $\begingroup$ $\vartheta_{11}(z, \tau) = i \sum_{n=-\infty}^\infty (-1)^n e^{2i\pi z\left(n+1/2\right)}e^{i\pi \tau\left(n + 1/2\right)^2}$ is related to the Weierstrass function by $\wp_\tau(z) = -(\log \vartheta_{11}(z;\tau))'' + c$, satisfying the differential equation $(\wp_\tau'(z))^2 = 4(\wp_\tau(z))^3-g_{2,\tau}\wp_\tau(z)-g_{3,\tau}$. So I guess it is about finding the same kind of relation for $\sum_n b_n e^{i \pi z n^3}$ $\endgroup$
    – reuns
    Commented Oct 19, 2016 at 21:56
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    $\begingroup$ @IlyaBogdanov unfortunately (or maybe fortunately!) summoning like this does not work on MO -- the '@' sign triggers a message only to those already involved with the question. $\endgroup$
    – Suvrit
    Commented Oct 19, 2016 at 22:32
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    $\begingroup$ (The reference is the solution to Ex. 6.63.c, BTW.) If $f(x) = \sum_{n} x^{n^3}$ would satisfy an algebraic differential equation, then (I think?) so would its powers $f^k$. In particular, the coefficients $a_{n} = [x^n] f^2(x) = \# \{ (a,b): a^3+b^3 = n \}$ would satisfy a certain recurrence relation. My take is that Stanley refers to the fact that there's no known recurrence relation satisfied by $\{ a_n\}$. $\endgroup$
    – Ofir Gorodetsky
    Commented Oct 19, 2016 at 22:39
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    $\begingroup$ @IlyaBogdanov: your summons is answered! The comment of Ofir is essentially what I had in mind. There is no nice theory concerning the number $f_k(n)$ of representations of $n$ as a sum of $k$ cubes analogous to sums of squares. If $\sum b_nx^{n^3}$ satistified an ADE then we would get a recurrence involving these numbers, which would be a big surprise to number theorists. $\endgroup$
    – Richard Stanley
    Commented Oct 20, 2016 at 17:31

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