The umbral calculus of [107] is restricted to the class of Sheffer polynomials.
It was therefore natural to extend the umbral calculus to larger classes
of polynomials. Viskov first extended the umbral calculus to
so-called generalized Appell polynomials (or Boas-Buck polynomials) [129] and
then went on to generalize this to arbitrary polynomials [130]. The extension to
generalized Appell
polynomials makes it possible to apply umbral calculus to **q**-analysis
[1,18,17,42,43,51,94] or important classes of orthogonal
polynomials like the Jacobi polynomials [101]. Roman remarks [100]
that Ward back in 1936 attempted to construct an umbral
calculus for generalized Appell polynomials [132]. Other interesting papers in
this direction are [11,23,62].

An extension of the umbral calculus to certain classes of entire function can be found in [33,34].

Another extension of the umbral calculus is to allow several variables [3,10,28,47,64,80,92,95,112,133,134]. However, all these extensions suffer from the same drawback, viz. they are basis dependent. Currently, an attempt is being undertaken by Rota, Loeb and Di Bucchianico to construct a basis-free umbral calculus in finite and infinite dimensions.

Roman [97,98,99,100] developed a version to the umbral calculus for inverse formal power series of negative degree. Most theorems of umbral calculus have their analog in this context. In particular, any shift-invariant operator of degree 1 (delta operator) has a special sequence associated with it---satisfying a type of binomial theorem. Nevertheless, despite its philosophical connections, this theory remained completely distinct from Rota's theory treating polynomials.

Later, in [60], a theory was discovered which generalized
simultaneously Roman and Rota's umbral calculi by embedding them in a
logarithmic algebra containing both positive and negative powers of **x**, and
logarithms. A subsequent generalization [57,59] extends this algebra
to a field which includes not only **x** and but also the iterated
logarithms, all of whom may be raised to any real power.
Sequence of polynomials are then replaced with sequence of
asymptotic series where the degree **a** is a real and
the level is
a sequence of reals. Rota's theory is the restriction to level
and degree . Roman's theory is the restriction to
level and degree . Thus, the difficulty
in uniting Roman and Rota's theories was essentially that they lied on
different levels of some larger yet unknown algebra. Other papers in this direction are
[49,79,102,103].

Rota's operator approach to the calculus of finite difference can be
thought of as a systematic study of shift-invariant operators on the
algebra of polynomials. The expansion theorem [67, Theorem 2]
states that all
shift-invariant operators can be written as formal power series in the
derivative **D**. If is a
shift-invariant operator, then

where .

However, a generalization of this by Kurbanov and Maksimov [55] to
arbitrary linear operators has received surprisingly little attention.
Any linear operator can be
expanded as a formal power series in **X** and **D** where **X** is the
operator of multiplication by **x**. More generally, let
**B** be any linear operator which reduces the degree of nonzero
polynomials by one. (By convention, .) Thus, **B** might
be not only the derivative or any delta operator, but also the
**q**-derivative, the divided difference operator, etc.
Then can be expanded in terms of **x** and **B**:

Extensions of umbral calculus to symmetric functions have already been mentioned.

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