The umbral calculus

Roman, Steven; Rota, Gian‐Carlo

doi:10.1016/0001-8708(78)90087-7

Cited by 427 publications

(349 citation statements)

References 66 publications

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“…The set of all Sheffertype polynomial sequences {u n (x) = [t n ]d(t)e xh(t) } with an operation, "umbral composition" (cf. [25] and [27]), forms a group called the Sheffer group. [13] presents the isomorphism between the Riordan group and Sheffer group.…”

Section: ) · (D(t) H(t)) = (D(t) H(t)) · (1 T) = (D(t) H(t))mentioning

confidence: 99%

On an extension of Riordan array and its application in the construction of convolution-type and Abel-type identities

He¹,

Hsu²,

Ron³

2014

European Journal of Combinatorics

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Using the basic fact that any formal power series over the real or complex number field can always be expressed in terms of given polynomials {p n (t)}, where p n (t) is of degree n, we extend the ordinary Riordan array (resp. Riordan group) to a generalized Riordan array (resp. generalized Riordan group) associated with {p n (t)}. As new application of the latter, a rather general Vandermonde-type convolution formula and certain of its particular forms are presented. The construction of the Abel type identities using the generalized Riordan arrays is also discussed.

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Section: ) · (D(t) H(t)) = (D(t) H(t)) · (1 T) = (D(t) H(t))mentioning

confidence: 99%

On an extension of Riordan array and its application in the construction of convolution-type and Abel-type identities

He¹,

Hsu²,

Ron³

2014

European Journal of Combinatorics

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“…Following the notations and conventions described in [13], let P be the algebra of polynomials in the single variable x over the field C of complex numbers. Then the following formal power series:…”

Section: An Umbral-calculus Generalization Of the Addition Theoremsmentioning

confidence: 99%

Addition theorems for the Appell polynomials and the associated classes of polynomial expansions

Pintér

Srivástava

2012

Aequat. Math.

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n (x) and the generalized Euler polynomials E (α) n (x) of order α and degree n in x. The main purpose of this sequel to some of the aforecited investigations is to give several addition formulas for a general class of Appell sequences. The addition formulas, which are derived in this paper, involve not only the generalized Bernoulli polynomials B (α) n (x) and the generalized Euler polynomials E (α) n (x), but also the generalized Genocchi polynomials G (α) n (x), the Srivastava polynomials S N n (x), several general families of hypergeometric polynomials and such orthogonal polynomials as the Jacobi, Laguerre and Hermite polynomials. Some umbral-calculus generalizations of the addition formulas are also investigated.Mathematics Subject Classification (2010). Primary 11B68; Secondary 11B83, 33C45.

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“…The umbra ǫ is named augmentation as Roman and Rota first called it [24]. We will call the umbra u the unity umbra.…”

Section: The Classical Umbral Calculusmentioning

confidence: 99%

Umbral nature of the Poisson random variables

Nardo¹,

Senato²

2001

Algebraic Combinatorics and Computer Science

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Abstract. Extending the rigorous presentation of the "classical umbral calculus" [28], the so-called partition polynomials are interpreted with the aim to point out the umbral nature of the Poisson random variables. Among the new umbrae introduced, the main tool is the partition umbra that leads also to a simple expression of the functional composition of the exponential power series. Moreover a new short proof of the Lagrange inversion formula is given.

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The umbral calculus

Cited by 427 publications

References 66 publications

On an extension of Riordan array and its application in the construction of convolution-type and Abel-type identities

On an extension of Riordan array and its application in the construction of convolution-type and Abel-type identities

Addition theorems for the Appell polynomials and the associated classes of polynomial expansions

Umbral nature of the Poisson random variables

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