Let
$$N(x) = 1 + \sum_{k \ge 1} N_k(h_1,h_2,...,h_k) \;x^k$$
$$ = 1 + h_1 x + (h_1^2 + h_2) x^2 + (h_1^3 + 3h_1h_2 + h_3)x^3 + (h_1^4 + 6 h_2 h_1^2 + 4 h_3 h_1 + 2 h_2^2 + h_4) x^4 + \cdots$$
be the o.g.f. of the Kreweras-Voiculescu polynomials that can be characterized as the formal compositional inverse $f^{(-1)}(x) = xN(x)$ about the origin $x=0$ of the invertible function / formal power series $f$ where the infinite set of indeterminates $[h]$ with $h_0=1$ are the coefficients of the shifted reciprocal of $f$; that is,
$$h(x) = 1 + \sum_{k \geq 1} h_k \; x^k = \frac{x}{f(x)} = \frac{x}{x+ \sum_{k \geq 1} a_k \;x^{k+1}}.$$
Then
$$f(x)h(x) =x,$$
and composing with $f^{(-1)}$ gives
$$xh(f^{(-1)}(x)) = f^{(-1)}(x),$$
implying the unique solution
$$xh(xN(x)) = xN(x).$$
This is sufficient with some fairly straightforward algebraic arguments to prove an iterative generator for the K-V polynomials for $n \geq 1$ is
$(IGEN)$
$$T_n((xh)^{(n-1)}(x)) = T_n(xN(x))$$
where $T_n$ is the truncation op whose action on a power series $p(x) = \sum_{k\geq0} b_k \;x^k$ gives $T_n(p(x)) = \sum_{k=0}^n b_k \;x^k$ and where, e.g., $(xh)^{(3)}(x)$ represents the iteration $xh(xh(xh(x)))$ with $(xh)^{(0)}(x) =x$.
Request:
I'm looking for references (particularly journal articles, books, or personal postings freely accessible on the Net) giving proofs for the iterative formula IGEN.
Significance of the polynomials:
The K-V polynomials flag and enumerate distinct noncrossing partitions and give the free moments in terms of the free cumulants in free probability theory. Obviously they are often useful in mathematical / physical analyses involving compositional inverses and are also of broad use in combinatorics characterizing parking functions, trees, Dyck lattice paths, cluster complexes, and other combinatorial geometric constructs.
(Peripheral question: What are the earliest occurrences of these polynomials in the literature between Newton and Kreweras? An explicit formula for the numerical coefficients for each monomial summand of the K_V polynomials, the set $[N]$, is given in this MO-Q in item 2c.)
