beyond differentially algebraic power series

Question

In a recent question, we learned about the existence of functions that do not satisfy any algebraic differential equation.

One nice property of such equations is that there is a good way to enumerate a basis: we can produce the stream of "monomials" $\left(\prod_i D^{\lambda_i-1}f(x)\right)_\lambda$, where $D$ is the differentiation operator and $\lambda=(\lambda_1,\lambda_2,\dots)$ runs over the integer partitions in lexicographic order: $$ 1, f(x), f(x)^2, f'(x), f(x)^3, f(x)f'(x), f''(x), f(x)^4, f(x)^2f'(x),\dots $$

I'm wondering: is there a "natural" class of equations, more general than ADEs, that has a similar basis. (Natural meaning: equations that specify many functions occurring in "nature")? Or, alternatively, just another class of equations.

I realise this is vague, please bear with me...

edit: I should add that I'm aware of "algebraic recurrences" (i.e., shift instead of differentiation) and "Mahler-type functional equations" (i.e., $f(x^{k+1})$ instead of $f(x)^{(k)}$).

Martin Klazar mentions that a few interesting sequences (eg. the ordinary generating function for Bell numbers) satisfy functional equations of the form $$p_1(x)f(x)=p_2(x)+p_3(x)f(\frac{x}{1-x}),$$ with polynomials p1, p2, p3 (and concludes that they are not differentially algebraic), but I'm not sure how common such equations are.

edit: the motivation for this question comes from the desire of being able to guess a formula (or recurrence, differential or functional equation) for a given sequence (of numbers or polynomials, etc.), as pioneered be GFUN, see also Section 7 in my preprint with Waldek Hebisch.

For example, given the first few (say 100) terms of the sequence, we compute it's (truncated) generating function $f_1 := f(x)$, and also $f_2 := f(x)^2, f_3 := f'(x), f_4 := f(x)^3, f_5 := f(x)f'(x), f_6 := f''(x), f_7 := f(x)^4, f_8 := f(x)^2f'(x), \dots$. We fix the maximal degree, say $N$ of the coefficient polynomials $p_1, p_2, \dots, p_m$, and then try to solve the linear system of equations obtained by equating coefficients in $$ ord(p_1 f_1 + \dots + p_m f_m)\geq\sigma $$ for large sigma. If we get a solution, and the given sequence is somehow naturally defined, chances are good that the equation holds for all terms of the sequence.

Answer 1

I am not quite sure whether the question is about a "natural" graded algebra which is infinitely generated, or finitely generated algebras are fine as well. Because there are nice examples of algebras of multiple zeta values, but also of multiple polylogarithms and of finite multiple harmonic sums, as well as the algebra of classical modular forms. The latter gives rise to a certain structure which is presumably richer than the algebra of differential monomials, and I could probably try to explain this point.

The Eisenstein series $E_2=1-24\sum_{n=1}\sigma_1(n)q^n$, $E_4=1+240\sum_{n=1}\sigma_3(n)q^n$ and $E_6=1-504\sum_{n=1}\sigma_5(n)q^n$, where $\sigma_k(n)=\sum_{d\mid n}d^k$, generate a differentially stable ring over $\mathbb Q$ with respect to the differentiation $D=q\dfrac{d}{dq}$ (a result usually attributed to Ramanujan). The weights $2,4,6$ are assigned to $E_2,E_4,E_6$ respectively, and $D$ increases the weight by $2$. The graded ring $\mathbb Q[E_2,E_4,E_6]$ possesses an additional structure coming from the functional equations for replacing $q$ by $q^k$ where $k$ is a positive integer, although it's very hard to write down the structure explicitly. Let me call the corresponding scale operators (substitutions $q^k$ for $q$) $D_k$. They do not change weights.

The counterpart consists of the infinite family $F_{2m+1}(q)=\sum_{n=1}^\infty\sigma_{2m}(n)q^n$, $m=0,1,2,\dots$, which are known to be linearly independent over $\mathbb Q$ and even over the field of meromorphic functions on $\mathbb C$. We can formally assign the weight $2m+1$ to each $F_{2m+1}(q)$, although there could be reasons to normalize them in a way used for the Eisenstein series. Again, the differential operator $D$ increases weights by 2, and the open problem here is to show that the $F_{2m+1}(q)$ are all algebraically differentially independent over $\mathbb C$ (or $\mathbb Q$). An expanded version of the problem is to show that the ring of all $D$- and $D_k$-monomials have no nontrivial relations at all. In a sense this includes both the algebraic differential structure from the problem, as well as all kind of Mahler-type equations. If one restricts to considering $D$- and $D_2$-monomials (or $D_k$ monomials for a finite set of $k$'s), the corresponding set of monomials of finite weight will be finite.