Operational Versus Umbral Methods and the Borel Transform

Dattoli, G.; Palma, E. Di; Sabia, E.; Górska, K.; Horzela, A.; Penson, K. A.

doi:10.1007/s40819-017-0315-7

Cited by 9 publications

(15 citation statements)

References 45 publications

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“…Compared to the umbral calculus literature, we find the analogous equations (see e.g., [20][21][22])…”

Section: Examples and Relation To Traditional Umbral Calculusmentioning

confidence: 53%

“…Section 4), and finally several auxiliary results and examples (cf. Section 2) that illustrate our novel umbral calculus-type integral transform techniques, and which highlight a certain resemblance with earlier ideas as expressed in [22,23] (referred to therein as the "principle" of permanence of formal properties). The results we have obtained appear to be promising and seem to offer new and interesting developments within the framework of the theory of special functions and of the relevant applications.…”

Section: Discussionmentioning

confidence: 72%

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Operational Methods in the Study of Sobolev-Jacobi Polynomials

Behr¹,

Dattoli

Duchamp

et al. 2019

Mathematics

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Inspired by ideas from umbral calculus and based on the two types of integrals occurring in the defining equations for the gamma and the reciprocal gamma functions, respectively, we develop a multi-variate version of umbral calculus and of the so-called umbral image technique. Besides providing a class of new formulae for generalized hypergeometric functions and an implementation of series manipulations for computing lacunary generating functions, our main application of these techniques is the study of Sobolev-Jacobi polynomials. Motivated by applications to theoretical chemistry, we moreover present a deep link between generalized normal-ordering techniques introduced by Gurappa and Panigrahi, two-variable Hermite polynomials and our integral-based series transforms. Notably, we thus calculate all K-tuple L-shifted lacunary exponential generating functions for a certain family of Sobolev-Jacobi (SJ) polynomials explicitly.

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“…Compared to the umbral calculus literature, we find the analogous equations (see e.g., [20][21][22])…”

Section: Examples and Relation To Traditional Umbral Calculusmentioning

confidence: 53%

Section: Discussionmentioning

confidence: 72%

Operational Methods in the Study of Sobolev-Jacobi Polynomials

Behr¹,

Dattoli

Duchamp

et al. 2019

Mathematics

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“…Although we are borrowing terms, such as the vacuum, from Physics, the legitimacy of the above procedure has been justified by the use of methods based on the Borel transform [7]. The Gaussian integral identity…”

Section: Theoremmentioning

confidence: 99%

“…In deriving the previous identity, we have again taken the freedom, supposed as valid in [1,4,5] and then justified in [7], of treating the umbral operator as a standard algebraic quantity; such a point of view has been shown to be the leitmotiv underlying the umbral heuristic proof of the RMT outlined in [12].…”

Section: The Umbral Version Of the Trigonometric Functionsmentioning

confidence: 99%

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From Circular to Bessel Functions: A Transition through the Umbral Method

Dattoli¹,

Palma²,

Licciardi

et al. 2017

Fractal Fract

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A common environment in which to place Bessel and circular functions is envisaged. We show, by the use of operational methods, that the Gaussian provides the umbral image of these functions. We emphasize the role of the spherical Bessel functions and a family of associated auxiliary polynomials, as transition elements between these families of functions. The consequences of this point of view and the relevant impact on the study of the properties of special functions is carefully discussed.

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