Since I lost my paper notes in a domestic conflagration in Japan some ten years ago, I've occasionally tried to recall one particular author who wrote in the 1900s about Lyndon words / strings, or such (perhaps I'm misusing the qualifier Lyndon, but see edit at bottom)--strings composed of two symbols, say $V$ and $U$--and their normal reordering via the commutator relation $[V,U] = VU -UV = 1$ such that all the $U$s are to the left of all the $V$s. This is, of course, connected to the normal re-ordering found in quantum mechanics (operator disentanglement), algebraic combinatorics, special function theory, and Graves-Pincherle-Lie theory. One of the most commonly depicted examples is, with $V = D = \tfrac{d}{dx}$ and $U = x$,
$$(xD)^n = ST2_n(:xD:),$$
where $:xD:^n = x^n D^n$ and $ST2_n(x)$ are the Stirling polynomials of the second kind / Touchard / Bell / exponential polynomials.
For example,
$[D,x] = Dx-xD = 1$, so $Dx = 1 + xD$
and
$(xD)^2 = xDxD = (xDx)D = (x (1+xD))D = xD + x^2D^2 = ST2_1(:xD:).$
Then, avoiding explicit iterative use of the commutator (but the following can be reduced to such because it defines differentiation),
$$(xD)^3 = xDxDxD = xD(xD)^2 = xD(xD +x^2D^2) $$
$$= x (D + xD^2 + 2 xD^2 + x^2D^2) = xD + 3x^2D^2 + x^3D^3 = :xD:^1 + 3:xD:^2 + :xD:^3$$
$$ = ST2_3(:xD:).$$
More generally,
$$(RL)^n = ST2_n(:RL:),$$
is true for any set of lowering and raising ops $L$ and $R$ acting on a space spanned by $\psi_n$ such that
$L \psi_n = n \psi_{n-1}$ and $R \psi_n = \psi_{n+1}.$
Another classic example, investigated by the likes of Stokes, Al-Salam, and Chatterjea, is
$$(RLR)^n = R^nL^nR^n = R^n :LR:^n = R^n Lah_n(:RL:) =n! R^n Lag^{(-1)}_n(-:RL:),$$
where $Lah_n(x)$ are the Lah polynomials and $Lag_n^{(-1)}(x)$ are the fundamental Laguerre polynomials of order -1. Conjugate this appropriately and you get a sequence used by Getzler in characterizing the Virasoro algebra.
Another: As Erdelyi, Gelfand, and Shilov knew, reorder the binomial $\binom{RL + \alpha + \beta}{\alpha}$ and you get the confluent hypergeometric functions.
This illustrates the ultimate importance of the re-ordering allowed by the commutator to the Sheffer polynomial calculus / finite operator calculus (see this MO-Q) and related special functions ($(x+D)^n$, Hermite) and Weyl-Lie group theory, in which the Stirling polynomials of the first and second kinds and their refined versions the cycle index polynomials of the symmetric groups OEIS A036039 and the Faa di Bruno composition polynomials of A036040 play special roles.
All this ultimately issues from the commutator definition of derivation, so, at the very least from a historical viewpoint, it would be appropriate to acknowledge the work of the developers.
In "Recent Developments in Combinatorial Aspects of Normal Ordering", Matthias Schork (see also this MO-Q) goes into some history and has a fairly extensive set of references listing several clans of collaborators that have published on this topic; however, he does leave out some important contributors such as Al-Salaam, Chatterjea, and Stokes (see. e.g., this MO-A and OEIS A132440) and Feinsilver.
My question:
Other than the researchers listed in Schork's survey, who else has published since, say between 1950-2000, specifically on the notion of re-ordering of strings of symbols explicitly using the commutator?
(Of course if you hadn't already been aware of the above elaboration, you probably have not come across the author and his papers that I've forgotten, but if you are interested in the combinatorics of string perhaps you have. In addition, if you've been particularly impressed by similar papers on the topic but not lying exactly within the scope of my criteria, I'd still be interested, e.g., see this MO-Q.)
Edit Mar 2, 2024:
From "The origins of combinatorics on words" by Jean Berstel, Dominique Perrin (2007),
Lyndon words give rise to commutators through a process of iterated dichotomy, using the notion of standard factorization. For example, the Lyndon word aababb with standard factorization (a)(ababb) gives rise to the commutator [a,[[a, b],[[a, b], b]]].
