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Modeling in Mathematics pp 43–52Cite as

Hermite Calculus

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Part of the book series: Atlantis Transactions in Geometry ((ATLANTIS,volume 2))

Abstract

We develop a new method of umbral nature to treat blocks of Hermite and of Hermite like polynomials as independent algebraic quantities. The Calculus we propose allows the formulation of a number of “practical rules” yielding significant simplifications in computational problems involving integrals and partial differential equations as well. The procedure we adopt is particularly useful to enter more deeply in the algebraic structure of Hermite polynomials. It provides indeed a tool allowing a generalization of the recently introduced geometrical point of view to the interplay between ordinary monomials and Hermite polynomials.

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Notes

  1. 1.

    We have used the umbral notation in a rather inaccurate way, without specifying on which space the operators are acting. For further comments and an appropriate discussion of the formal content, see the second of Ref. [2].

  2. 2.

    The subscript \(\left( \gamma ,-\beta \right) \) has been omitted because the identity holds for \(\hat{h}\) operators with the same basis, hereafter it will be included whenever necessary.

References

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Author information

Authors and Affiliations

  1. ENEA - Frascati Research Center, Via Enrico Fermi 45, 00044, Frascati, Rome, Italy

    Giuseppe Dattoli & Silvia Licciardi

  2. Department of Methods and Mathematic Models for Applied Sciences, University of Rome, La Sapienza, Via A. Scarpa, 14, 00161, Rome, Italy

    Bruna Germano & Maria Renata Martinelli

  3. Department of Mathematics, University of Catania, Via Santa Sofia, 64, 95125, Catania, Italy

    Silvia Licciardi

Authors
  1. Giuseppe Dattoli
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  2. Bruna Germano
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  3. Silvia Licciardi
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  4. Maria Renata Martinelli
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Corresponding author

Correspondence to Giuseppe Dattoli .

Editor information

Editors and Affiliations

  1. Bio-Engineering Sciences, University of Antwerp, Antwerp, Belgium

    Johan Gielis

  2. International Telematic University UniNettuno, Roma, Italy

    Paolo Emilio Ricci

  3. Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University, Tbilisi, Georgia

    Ilia Tavkhelidze

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Dattoli, G., Germano, B., Licciardi, S., Martinelli, M.R. (2017). Hermite Calculus. In: Gielis, J., Ricci , P., Tavkhelidze, I. (eds) Modeling in Mathematics . Atlantis Transactions in Geometry, vol 2. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-261-8_4

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