This book presents a novel approach to umbral calculus, which uses only elementary linear algebra (matrix calculus) based on the observation that there is an isomorphism between Sheffer polynomials and Riordan matrices, and that Sheffer polynomials can be expressed in terms of determinants. Additionally, applications to linear interpolation and operator approximation theory are presented in many settings related to various families of polynomials.

This book presents a novel approach to umbral calculus, which uses only elementary linear algebra (matrix calculus) based on the observation that there is an isomorphism between Sheffer polynomials and Riordan matrices, and that Sheffer polynomials can be expressed in terms of determinants. Additionally, applications to linear interpolation and operator approximation theory are presented in many settings related to various families of polynomials.

Geared toward upper-level undergraduates and graduate students, this elementary introduction to classical umbral calculus requires only an acquaintance with the basic notions of algebra and a bit of applied mathematics (such as differential equations) to help put the theory in mathematical perspective. The text focuses on classical umbral calculus, which dates back to the 1850s and continues to receive the attention of modern mathematicians. Subjects include Sheffer sequences and operators and their adjoints, with numerous examples of associated and other sequences. Related topics encompass the connection constants problem and duplication formulas, the Lagrange inversion formula, operational formulas, inverse relations, and binomial convolution. The final chapter offers a glimpse of the newer and less well-established forms of umbral calculus.

The theory of binomial enumeration is variously called the calculus of finite differences or the Umbral calculus. This theory studies the analogies between various sequences of polynomials pn and the powers sequence xn. The subscript n in pn was thought of as the shadow ("Umbra" means "Shadow" in Latin, whence the name umbral calculus) of the subscript n in xn, and many parallels were discovered between such sequences.At the very outset a brief explanation of the term modern umbral calculus is given. A large part of applied analysis is concerned with the study of certain sequences of special polynomials. Some of the most important of these sequences are associated with the names of Jacobi, Gegenbauer, Legendre, Chebyshev, Bessel, Laguerre, Hermite and Bernoulli. All of these sequences and many more, fall into a special class. Boas and Buck, in their work on polynomial expansions of analytic functions, used the term sequence of generalized Appell type for members of this class.

The Umbral Calculus

Geared toward upper-level undergraduates and graduate students, this elementary introduction to classical umbral calculus requires only an acquaintance with the basic notions of algebra and a bit of applied mathematics (such as differential equations) to help put the theory in mathematical perspective. The text focuses on classical umbral calculus, which dates back to the 1850s and continues to receive the attention of modern mathematicians. Subjects include Sheffer sequences and operators and their adjoints, with numerous examples of associated and other sequences. Related topics encompass the connection constants problem and duplication formulas, the Lagrange inversion formula, operational formulas, inverse relations, and binomial convolution. The final chapter offers a glimpse of the newer and less well-established forms of umbral calculus.

This book covers different aspects of umbral calculus and of its more recent developments. It discusses the technical details in depth, including its relevant applications. The book has therefore manyfold scopes to introduce a mathematical tool, not widespread known as it should be; to present a complete account of the relevant capabilities through the use of different examples of applications; to provide a formal bridge between different fields of research in pure and applied.