Generators, Inversion, and Matrix, Binomial, and Integral Transforms is a belated set of notes (pdf) on a derivation of a generating function for the row polynomials of OEIS-A111999 from its relation to the compositional inversion (a Lagrange inversion formula, LIF) presented in A133932 of invertible functions represented umbrally as logarithmic series . The results show that A111999 is a natural reduction of A133932.

Along the way, more general results are given involving the relations among Borel-Laplace transforms; compositional inversion in general; binomial transforms of rows, columns, and diagonals of matrices; infinitesimal generators; and the generating functions and reversals of binomial and Appell Sheffer polynomials, in particular the cycle index polynomials of the symmetric groups, or partition polynomials of the refined Stirling numbers of the first kind A036039.

A table has been added in Appendix III to illustrate how the analysis applies to the two other complementary LIFs A134685 , based on the refined Stirling numbers of the second kind A036040 (a refinement of the Bell / Touchard / exponential polynomials A008277), and A133437, based on the refined Lah numbers A130561 (a refinement of the Lah polynomials A008297, A105278, normalized Laguerre polynomials of order -1).

**Errata**:

Equation at top of page 5 should have 1/n rather than 1/n! in the series expanson.

### Like this:

Like Loading...

Pingback: The Lagrange Reversion Theorem and the Lagrange Inversion Formula | Shadows of Simplicity