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Generalizations of the umbral calculus

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.

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