Operational Methods in the Study of Sobolev-Jacobi Polynomials

Behr, Nicolas; Dattoli, G.; Duchamp, Gérard; Licciardi, Silvia; Penson, K. A.

doi:10.3390/math7020124

Cited by 7 publications

(11 citation statements)

References 36 publications

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“…A more complex set of examples is provided by the following expression quoted from [15] for the generalized hypergeometric functions, illustrating further the utility of the umbral image type formalism:…”

Section: (A2)mentioning

confidence: 99%

“…The recently introduced reformulation [15] of the umbral calculus framework in terms of umbral image type techniques permits to understand the calculations that lead to (37) in a very direct manner: taking advantage of the identity (see Appendix A for further details)…”

Section: Laguerre Derivative Laguerre Exponential and Operator-orderingmentioning

confidence: 99%

“…For the readers' convenience, we briefly recall the central definition of the umbral image type technique as introduced in [15]:…”

Section: Appendix a The Umbral Image Type Technique Of [15]mentioning

confidence: 99%

“…Definition A1 (Definition 2 of [15]). Let A = {λ} U V X be an alphabet of formal variables, where denotes the operation of disjoint union, and where {λ}, U , V and X are four (disjoint) alphabets of auxiliary formal variables.…”

Section: Appendix a The Umbral Image Type Technique Of [15]mentioning

confidence: 99%

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Operator Ordering and Solution of Pseudo-Evolutionary Equations

Behr

Dattoli

Lattanzi

2019

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The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational nature, has been demonstrated to be a very promising tool in order to approach within a unifying context non-canonical evolution problems. This article covers the extension of this approach to the solution of pseudo-evolutionary equations. We will comment on the explicit formulation of the necessary techniques, which are based on certain timeand operator ordering tools. We will in particular demonstrate how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools can be profitably extended to obtain solutions of fractional differential equations. We apply the method to a number of examples, in which fractional calculus and a certain umbral image calculus play a role of central importance.

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Section: (A2)mentioning

confidence: 99%

Section: Laguerre Derivative Laguerre Exponential and Operator-orderingmentioning

confidence: 99%

“…For the readers' convenience, we briefly recall the central definition of the umbral image type technique as introduced in [15]:…”

Section: Appendix a The Umbral Image Type Technique Of [15]mentioning

confidence: 99%

Section: Appendix a The Umbral Image Type Technique Of [15]mentioning

confidence: 99%

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Operator Ordering and Solution of Pseudo-Evolutionary Equations

Behr

Dattoli

Lattanzi

2019

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“…Behr et.al. have discussed lacunary generation functions in the context of their rather comprehensive study of Sobolev-Jacobo polynomials (see [12]). And, recently, Kişi, Gümüş, and Savas studied A I -lacunary convergence and Cesàro summability with respect to lacunary sequences (see Reference [13]).…”

Section: Introductionmentioning

confidence: 99%

Taming the Natural Boundary of Centered Polygonal Lacunary Functions—Restriction to the Symmetry Angle Space

2020

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This work investigates centered polygonal lacunary functions restricted from the unit disk onto symmetry angle space which is defined by the symmetry angles of a given centered polygonal lacunary function. This restriction allows for one to consider only the p-sequences of the centered polygonal lacunary functions which are bounded, but not convergent, at the natural boundary. The periodicity of the p-sequences naturally gives rise to a convergent subsequence, which can be used as a grounds for decomposition of the restricted centered polygonal lacunary functions. A mapping of the unit disk to the sphere allows for the study of the line integrals of restricted centered polygonal that includes analytic progress towards closed form representations. Obvious closures of the domain obtained from the spherical map lead to four distinct topological spaces of the “broom topology” type.

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