In this question we manage to show the existence of a closed form for arbitrary $e^{a(x) \frac{d}{dx} + b(x)I}$ as a single term of the form $k_1(x) f(k_2(x))$ where $k_1, k_2$ obey an interesting formula.
I now want to turn my attention to $e^{\frac{d^2}{dx^2}}$ which (thanks to Sri Tata for pointing it out) can be thought of as a one dimensional heat kernel. Is there any formula involving finitely many pairs of functions $q_1(x), p_1(x) , ... q_n(x), p_n(x)$ and finitely many non-negative integers $c_1 ... c_n$ such that
$$ e^{\frac{d^2}{dx^2}} = q_1(x)f^{(c_1)}(p_1(x)) + ... + q_n(x)f^{(c_n)}(p_n(x))$$
Where the $(c_i)$ refer to derivatives.
I would suspect the answer is "no" since heat kernels have been known for a long time so we would have found a closed form by now. But I'm not even sure what kinds of tools are required to prove a result like this.
Would something like Differential Algebra or Picard Vessiot Theory work in this setting?
