# Question on existence of a "uniquely" asymptotic series expansion for exact XC functional

It is known that the unknown exact XC functional is potentially discontinuous and thus might not be expressable as a Taylor series; even if it is continuous, the Taylor expansion is not guaranteed to converge anyway.

However, there exist asymptotic series that "uniquely" correspond to a specific function(al); for example, the formal pseudodifferential operator representation of the fractional Fourier transform(FRFT), $$e^{ia/8(d^2/dy^2-π^2y^2/2+π/4)}f(y)$$, whose MacLaurin expansion (w.r.t. a) does not generally converge even for Schwartz f, formally satisfies the unitarity property, the "derivative with respect to a" property, and the "reduction to the identity" (when $$a=0$$) property, whose combination uniquely defines the FRFT(see my answer on my post).

My question now follows- is any series expansion, that "uniquely" expresses the exact XC functional(even if it is non-convergent), existent; if it is, can it be expressed in quasi-closed form?

P.S. By "quasi-closed form', I mean as in (possibly divergent and/or asymptotic; see my post linked above) Taylor, Fourier, generalised Fourier(Hermite etc), and/or (psuedo)differential operator representations, and NOT "neural network-derived series".

• The sequence of functions that converges to the exact XC functional (mattermodeling.stackexchange.com/questions/8895/…) can be trivially converted into such a series. Suppose the sequence of functionals is F1, F2, F3, ..., then it can be converted to a series F1 + (F2-F1) + (F3-F2) + ... . But this is probably not the answer that you want, and you may want to exclude answers like this by modifying your question. Commented Jun 27, 2023 at 6:27
• Commented Jun 27, 2023 at 16:00
• @NikeDattani edited this post to make sure it does not answer my question as I (should have) intended. Commented Jun 28, 2023 at 2:44