Martin Klazar mentions that a few interesting sequences (eg. the ordinary generating function for Bell numbers) satisfy functional equations of the formwith 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 inord(p_1 f_1 + \dots + p_m f_m)\geq\sigmafor 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.

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 $f(x)^{(k)}$).

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...