Does there exist a framework for determining if a power series is "differentially algebraic"

Question

It is a well studied problem to take a function $f$ expressed (usually expressed as a solution to a differential equation w/ some initial conditions) and ask if it has an "elementary closed form". The Risch "algorithm" allows one to do this for just integration and generally the field of Differential Galois Theory allows one to look at the more general case of differential equations.

I wanted to consider a slightly higher-order problem: given a function as a power series, is there any polynomial differential equation of finite differential order $P(f,f',f'',....f^{(k)})$ which this $f$ satisfies, without necessarily finding such a polynomial differential equation?

My concrete motivation for the problem was if the theta function $\theta(z) = \sum_{n=0}^{\infty} z^{n^2}$ can be expressed as the solution to some polynomial Diff-Eq.

I suspect the answer is no, (in fact probably no for any lacunary series)

But I don't know of any theory/tools/framework for answering problems like this.

Has anyone seen something similar?

Update:

It seems people have thought about this before: An example of a series that is not differentially algebraic?

It also seems that consensus is that lacunary functions are transcendental.

Answer 1

I had not seen this post before, but although the differential equation may be complicated, the result is easy: any modular form on $\Gamma_0(4)$ (but this is true much more generally), here $\theta(\tau)$, is a solution of a third degree differential equation. A nonconstructive proof in this special case is as follows: the ring of quasi-modular forms (i.e., including $E_2$) of $\Gamma_0(4)$ is $\Bbb C[E_2,\theta,F_2]$ where $F_2=\sum_{n\ge1,\ n\text{ odd}}\sigma_1(n)q^n$, and this ring is stable by differentiation $D=qd/dq$ and of transcendence degree $3$, so for any $F$ in the ring, $F$, $DF$, $D^2F$, and $D^3F$ must be algebraically dependent.

Answer 2

For the lazy reader I have wrapped Richard Stanley's very helpful comment into an answer. Richard Stanley's Enumerative Combinatorics Volume 2 on Page 282 reviews Jacobis Result that

$$ y = \sum_{n=-\infty}^{\infty} x^{n^2}$$

Obeys $$ \left( y^2z_3 - 15yz_1 z_2 + 30 z_1^3 \right)^2 + 32(yz_2 -3z_1^2)^3 - y^{10} (yz_2 -3z_1^2)^2 = 0 $$

Where

$$ z_1 = xy' \\ z_2 = xy' + x^2 y'' \\ z_3 = xy' + 3x^2 y'' + x^3 y''' $$

Where Jacobi's Original paper is : K. G. J. Jacobi, J. Reine Angew. Math. ( = Crelle’s Journal.) 36 (1847), 97-112

I tried reading Jacobi's original paper but unfortunately it is in German which I cannot read and in google translate he does say something to the effect of "this identity cannot be verified without going to the 14th order..." suggesting that his proof which doesn't involve series level manipulations was at the time the ONLY way to verify the identity (it was physically too much effort to expand and combine terms at the series level to check it (although the fact he had some number like 14 to measure how hard it was might mean Jacobi did actually go ahead and check)) but using Mathematica today hopefully this can be verified in the naive way much more easily.