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Lagrange inversion gives the Catalan coefficients of the inverse of $x - x^{2}$

Example

The compositional inverse $w$ of $x - x^{2}$ in $Q ⟦ x ⟧$ is

$w = x + x^{2} + 2 x^{3} + 5 x^{4} + 14 x^{5} + 42 x^{6} + \dots,$

and for $n \geq 1$ ,

$[x^{n}] w = \frac{1}{n} (\binom{2 n - 2}{n - 1}) .$

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

If $K$ contains $Q$ , $ϕ \in K ⟦ u ⟧$ has nonzero constant term, $w \in x K ⟦ x ⟧$ is the unique solution of $w = x ϕ (w)$ , $H \in K ⟦ u ⟧$ , and $n \geq 1$ , then $[x^{n}] H (w) = \frac{1}{n} [u^{n - 1}] H^{'} (u) ϕ (u)^{n}$ (Lagrange–Bürmann inversion extracts coefficients of a compositional inverse and of functions of it).

[F2]

In a commutative $Q$ -algebra, for $u \in x R ⟦ x ⟧$ and $c \in R$ , formal binomial powers satisfy $(1 + u)^{c} = \sum_{n \geq 0} c (c - 1) \dots (c - n + 1) u^{n} / n!$ (Formal $\exp$ and $\log$ are inverse homomorphisms and formal binomial powers obey the expected addition laws).

[F3]

For a commutative ring $R$ and $f \in x R ⟦ x ⟧$ , there is a unique $g \in x R ⟦ x ⟧$ with $f \circ g = x = g \circ f$ exactly when $[x] f$ is a unit (A zero-constant formal series has a compositional inverse exactly when its linear coefficient is a unit).

Verification

technique · put the inverse equation in Lagrange form

1.1

The equation $x = w - w^{2}$ is equivalent to $w = x (1 - w)^{- 1}$ . Lagrange inversion with $ϕ (u) = (1 - u)^{- 1}$ and $H (u) = u$ gives $[x^{n}] w = \frac{1}{n} [u^{n - 1}] (1 - u)^{- n}$ .

givenF1

2.1

Apply the generalized-binomial formula with exponent $- n$ and argument $- u$ . The coefficient of $u^{n - 1}$ is $(- 1)^{n - 1} (- n) (- n - 1) \dots (- 2 n + 2) / (n - 1)! = (\binom{2 n - 2}{n - 1})$ . At $n = 1, \dots, 6$ this yields $1, 1, 2, 5, 14, 42$ .

step 1.1givenalgebraF2

3.1

Direct substitution of these coefficients gives $w - w^{2} \equiv x (m o d x^{7})$ , and uniqueness of the compositional inverse confirms the displayed initial segment.

step 2.1givenF3∎

Depends on

Used by

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Sources

Benjamin Sambale, An Invitation to Formal Power Series (standard reference, not scraped)

Lagrange inversion gives the Catalan coefficients of the inverse of x−x2x−x2