The umbral calculus of  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)  and then went on to generalize this to arbitrary polynomials . 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 . Roman remarks  that Ward back in 1936 attempted to construct an umbral calculus for generalized Appell polynomials . 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 , 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
However, a generalization of this by Kurbanov and Maksimov  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.