Page 521 - Advanced Linear Algebra

The Umbral Calculus 505

A Final Remark
Unfortunately, space does not permit a detailed discussion of examples of
Sheffer sequences nor the application of the umbral calculus to various classical
problems. In [105], one can find a discussion of the following polynomial
sequences:

The lower factorial polynomials and Stirling numbers
The exponential polynomials and Dobinski's formula
The Gould polynomials
The central factorial polynomials
The Abel polynomials
The Mittag-Leffler polynomials
The Bessel polynomials
The Bell polynomials
The Hermite polynomials
The Bernoulli polynomials and the Euler–MacLaurin expansion
The Euler polynomials
The Laguerre polynomials
The Bernoulli polynomials of the second kind
The Poisson–Charlier polynomials
The actuarial polynomials
The Meixner polynomials of the first and second kinds
The Pidduck polynomials
The Narumi polynomials
The Boole polynomials
The Peters polynomials
The squared Hermite polynomials
The Stirling polynomials
The Mahler polynomials
The Mott polynomials

and more. In [105], we also find a discussion of how the umbral calculus can be
used to approach the following types of problems:

The connection constants problem
Duplication formulas
The Lagrange inversion formula
Cross sequences
Steffensen sequences
Operational formulas
Inverse relations
Sheffer sequence solutions to recurrence relations
Binomial convolution