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$\DeclareMathOperator\FGL{FGL}$The formal group law (cf. Wikipedia, Ex. 1.6 of nLab, Hazewinkel) derived from an analytic function or formal series $f(x) = x + a_2 x^2 + a_3 x^3 + ...$ and its formal compositional inverse, perhaps derived from Lagrange inversion, $f^{(-1)}(x)$ is defined by

$$\FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)]$$.

I'm interested in earlier investigations of this formula and the closely related formula

$$F(x,t)=\exp[t \cdot g(x)d/dx]x = f[f^{-1}(x)+t],$$

where $g(x) = [df^{(-1)}/dx]^{-1}.$

Both occasionally pop up in MO (see Q1, Q2, O3).

(Edit 9/19/22: In "Coefficient rings of formal group laws", Buchstaber and Ustinov give a nice presentation of some history of FGLs and their applications in algebraic geometry, algebraic topology, and mathematical physics.)

So far the earliest presentations I know of are

  1. by Abel for the $\FGL(x,y)$ in 1826 (cf. "From Abel's heritage: Transcendental objects in algebraic geometry and their algebraization" by F. Catanese, p. 6, Thm. 2.1, and "The Work of Niels Henrik Abel" by Houzel, p. 24, Eqn. 5.)

  2. by Abel for $F(x,t)$ in 1826 (cf. Abel equation, 1826).

  3. by Charles Graves for $F(x,t)$ in 1853 in "A generalization of the symbolic statement of Taylor's theorem" in the Proceedings of the Royal Irish Academy, Vol. 5, (1850-1853), p. 285-287 (cf. p. 13 in The Theory of Linear Operators ... , Principia Press, 1936, by Harold T. Davis and MO-Q4).

Aware of any earlier presentations than Abel's or Graves?

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  • $\begingroup$ Hazewinkel link is to "On formal groups. The functional equation lemma and some of its applications." $\endgroup$
    – Tom Copeland
    Commented Feb 16, 2020 at 21:02
  • $\begingroup$ Inselberg in "Superpositions of nonlinear operators. I. Strong superpositions and linearizability" on p. 502 states, "Abel in 1826 <ref below> was the first to show that if $F(x, y)$ is an abelian group operation on the reals and satisfies certain other conditions, then there exist a one-to-one function, $f$, of one variable such that $F(x, y) = f^{ -1}( f (x) + f(y))$." $\endgroup$
    – Tom Copeland
    Commented Mar 18, 2021 at 19:56
  • $\begingroup$ N. H. Abel, Recherche de fonctions de deux quantites variable independantes x et y, telles que $f(x, y)$, qui ont la propriM que $f(z,f(x, y))$ est une fonction symttrique de $z, x$ et $y$, J. Reine Angew. Math. 1 (1826), 11-15 [Oeuvres completes, I, pp. 61-65, Christiania 1881]. $\endgroup$
    – Tom Copeland
    Commented Mar 18, 2021 at 19:57

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