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Some comments on Rota's Umbral Calculus [PDF]
Journal of Mathematical Analysis and Applications, 1980 AbstractRota's Umbral Calculus is put in the context of general Fourier analysis. Also, some shortcuts in the proofs are illustrated and a new characterization of sequences of binomial type is given. Finally it is shown that there are few (classical) orthogonal polynomials of binomial type.
Zeilberger, Doron
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An application of the umbral calculus [PDF]
Journal of Mathematical Analysis and Applications, 1990 AbstractThe partial difference equation r(i, j) = r(i, j − 1) + r(i − 1, j) + r(i − 1, J + 1), where r(i, j) are defined for integer numbers i and j, i ⩾ 0, by the conditions r(0, j) = 1 for all j and r(i, −1) = 0 for i ⩾ 1 is solved. For i ⩾ 0 and j ⩾ 0 a combinatorial meaning of numbers r(i, j) is given.
Razpet, Marko
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An infinite dimensional umbral calculus [PDF]
Journal of Functional Analysis, 2019 The aim of this paper is to develop foundations of umbral calculus on the space $\mathcal D'$ of distributions on $\mathbb R^d$, which leads to a general theory of Sheffer polynomial sequences on $\mathcal D'$. We define a sequence of monic polynomials on $\mathcal D'$, a polynomial sequence of binomial type, and a Sheffer sequence.
Dmitri Finkelshtein +3 more
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Some Identities of Fully Degenerate r-Dowling Polynomials Arising from λ-Umbral Calculus [PDF]
MathematicsThis paper introduces fully Dowling polynomials of the first and second kinds, which are degenerate versions of the ordinary Dowling polynomials. Then, several important identities for these degenerate polynomials are derived.
Xiaoxue Li, Siqi Dong, Yuankui Ma
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Umbral Calculus and New Trigonometries [PDF]
, 2023 The present work proposes an overview of the Umbral calculus and of its wide range of applicability. The aim is to provide new tools, embedding umbral, symbolic and operatorial methods to be exploited in pure and applied Mathematics, in the research ...
Silvia Licciardi
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Representations by degenerate Daehee polynomials
Open Mathematics, 2022 In this paper, we consider the problem of representing any polynomial in terms of the degenerate Daehee polynomials and more generally of the higher-order degenerate Daehee polynomials.
Kim Taekyun +3 more
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Degenerate Bell polynomials associated with umbral calculus
Journal of Inequalities and Applications, 2020 Carlitz initiated a study of degenerate Bernoulli and Euler numbers and polynomials which is the pioneering work on degenerate versions of special numbers and polynomials.
Taekyun Kim +4 more
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Formal Calculus and Umbral Calculus [PDF]
The Electronic Journal of Combinatorics, 2010 We use the viewpoint of the formal calculus underlying vertex operator algebra theory to study certain aspects of the classical umbral calculus. We begin by calculating the exponential generating function of the higher derivatives of a composite function, following a very short proof which naturally arose as a motivating computation related to a ...
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Degenerate Lah–Bell polynomials arising from degenerate Sheffer sequences
Advances in Difference Equations, 2020 Umbral calculus is one of the important methods for obtaining the symmetric identities for the degenerate version of special numbers and polynomials. Recently, Kim–Kim (J. Math. Anal. Appl.
Hye Kyung Kim
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