# Nascent formal group law

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).

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?

• Hazewinkel link is to "On formal groups. The functional equation lemma and some of its applications." – Tom Copeland Feb 16 at 21:02