An important property of (any extension of) the umbral calculus is that it has its own generalization of Lagrange's inversion formula (as follows from the closed forms for basic polynomials [67, Theorem 4]). Thus we find many papers in which new forms of the Lagrange's inversion formula is derived using umbral calculus [4,5,17,41,45,46,52,78,115,124,131].

In [58], the umbral calculus is generalized to symmetric
functions. When
counting enriched functions (functions, injections, reluctant
functions, dispositions, etc.) from **N** to **X**, we can
assign a weight to each function according to its fiber structure.
The total number of such functions is a symmetric function
of degree **n** where **n=|N|**.
The elementary and complete symmetric functions are (up to a multiple
of ) good examples of such sequences. They obey their own sort of
binomial theorem

The generating functions of is directly related to that of
its underlying species. By specializing all **x** variables to 1, we
return to the study to polynomials.

Another rich field of application is linear recurrences and lattice path counting. Here we should first of all mention the work of Niederhausen [72,71,70,74,76,77], but there other papers as well in this direction [90,91,104,135,136].

There are also papers on graph theory and umbral calculus, in particular on chromatic polynomials [85,89].

General papers on the combinatorial properties of polynomials of binomial type are [36,54,53,93,120].

Coalgebraic aspects of umbral calculus are treated in [13,14,24,50,66,68,83].

sandro@win.tue.nl / loeb@labri.u-bordeaux.fr