The Lagrange Reversion Theorem and the Lagrange Inversion Formula

From Wikipedia on the LRT, with

$v(x,y) = x + y \; h(v(x,y))$ ,

$v(x,y) = x + \sum_{ n > 0} \frac {y^n}{n!} \; D_x^{n-1} h^n(x)$ .

Letting $x = 0$ and $w(y) = v (0,y)$ ,

$w (y) = y \; h (w(y))$ and $h(y)=y/w^{(-1)}(y)$ , giving

$w(y) = \sum_{n > 0} \frac {y^n}{n!} \; D_x^{n-1} h^n(x) \; |_{x=0} \;$ ,

the Lagrange inversion formula about the origin, whose expansion in the Taylor series coefficients of $h(x)$ is discussed in OEIS A248927. See also A134685. For connections to free probability, free cumulants and moments, Appell sequences, noncrossing partitions, and other combinatorics, see A134264.

Let $v^{(-1)}(x,y)$ be the inverse of $v(x,y)$ w.r.t. to $x$ . Then

$x = v^{(-1)}(x,y) + y \; h(x)$ , or

$v^{(-1)}(x,y) = x - y \; h(x)$ , and

$v(x,y) = x + \sum_{ n > 0} \frac {y^n}{n!} \; D_x^{n-1} h^n(x) = x + \sum_{ n > 0} \frac {1}{n!} \; D_x^{n-1} [x-v^{(-1)}(x,y) ]^n$ .

The solution for the inverse of this last type is also presented in the post on the inviscid Burgers’ equation and the post Generators, Inversion, and Matrix, Binomial, and Integral Transforms.

The Laplace transform (LPT) argument in Appendix II of the Generators pdf can be briefly extended to derive the last form of the LRT above. (See below.)

Comparing $v (x,t)$ with $A(x,t)$ on page 2 of the Burgers’ equation pdf, we see that the the factor $D^{n-1} \frac {h^n(x)}{n!}$ conjoined with $y^n =t^n$ is equivalent to a summation over the $t^n$ terms of all the partition polynomials (unsigned) in the expansion of $A(x,t)$ on page 2 . Each partition polynomial is associated with a refined face polynomial for a Stasheff associahedron (cf. MO-Q: … enumerative geometry and nonlinear waves?), or its dual, and the coefficient of the $t^n$ term of the polynomial is a weighting of the (m-n+1)-dimensional face of the m-dimensional associahedron. For example, the $t$ terms flag the full associahedron and the $t^2$ terms the facets, or the next lower dimensional faces, the (m-1)-dimensional faces of the m-dimensional associahedron. The summation for a given $t^n$ then is a summation over the “column” space for the partition polynomials, representing a constant dimensional difference from the top-dimension of the polytopes.

The lead to the connection between the OEIS entries and the LRT was Terry Tao’s post Another Problem about Power Series.

For comparison, in Tao’s notation the last equation here becomes

$G(z)= v(z,y)-z = \sum_{ n > 0} \frac {y^n}{n!} \; D_z^{n-1} h^n(z) = \sum_{ n > 0} \frac {1}{n!} \; D_z^{n-1} [z-v^{(-1)}(z,y) ]^n$
$=\sum_{ n > 0} \frac {1}{n!} \; D_z^{n-1} F^n(z)$ ,

so the expansions in A248927 apply here as well with evaluation at $z$ rather than at the origin.

A formal derivation of the LRT:

Let

$f(x,t) = x -t \; F(x)= x - t \sum_{n>1} \; a_n \frac{x^n}{n!}$

and

$f^{-1}(x,t)= x + \sum_{n>1} \; b_n \frac{x^n}{n!}$

be its compositional inverse in $x$ . Then formally the Borel-Laplace transform gives, for a suitable class of functions,

$1 + \sum_{n>1} \; b_n \; z^{n-1} = \int_0^\infty \frac{1}{z} \; \exp(-\frac {u}{z}) \; D_u(f^{-1}(u,t)) \; du = \int_0^\infty \frac{1}{z} \; \exp(-\frac {f(u,t)}{z}) \; du$

$= \int_0^\infty \frac{1}{z} \; \exp(-\frac {u-tF (u)}{z}) \; du = \sum_{n \geq 0} (\frac{t}{z})^n \int_0^\infty \frac{1}{z} \; \exp(-\frac {u}{z}) \; \frac{F^n(u)}{n!} \; du$ .

Now inspecting the first expression in the chain of equalities shows that the terms $b_n z^{n-1}$ need to be multiplied by $z/n!$ to obtain the terms of the formal Taylor series for $f^{-1}(z,t)$ , but this amounts to replacing $z$ by $1/p$ , multiplying by $1/p^2$ , and taking the inverse Laplace transform, giving

$f^{-1}(z,t) = z \; + \; LPT^{-1}_{p \to z} \left [\sum_{n > 0} \frac{t^n}{n!} \; p^{n-1} LPT_{z \to p} \left [ F^n(z) \right] \right] = z \;+ \; \sum_{ n > 0} \frac {t^n}{n!} \; D_z^{n-1} F^n(z)$ .

Do a sanity check with $F (x)= x^m$ .

Related stuff:

1) “Formal groups, power systems, and Adams operators” by Buchstaber and Novikov

2) “The Hirzebruch criterion for the Todd class” earlier posting

3) “The methods of algebraic topology from the viewpont of cobordism theory” by Novikov

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The Lagrange Reversion Theorem and the Lagrange Inversion Formula

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