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.