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Operational Versus Umbral Methods and the Borel Transform

  • G. Dattoli
  • E. di Palma
  • E. Sabia
  • K. GórskaEmail author
  • A. Horzela
  • K. A. Penson
Original Paper
First Online:

Abstract

Integro-differential methods, currently exploited in calculus, provide an inexhaustible source of tools to be applied to a wide class of problems, involving the theory of special functions and other subjects. The use of integral transforms of the Borel type and the associated formalism is shown to be a very effective mean, constituting a solid bridge between umbral and operational methods. We merge these different points of view to obtain new and efficient analytical techniques for the derivation of integrals of special functions and the summation of associated generating functions as well.

Keywords

Integral transforms Borel transforms Operational calculus Umbral calculus 
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Notes

Acknowledgements

The authors express their sincere appreciation to Dr. D. Babusci for interesting and enlightening discussions on the topics treated in this paper. It is also a pleasure to recognize the interest and the encouragement of Prof. V. Strehl. K. G., A. H. and K. A. P. were supported by the PAN-CNRS program for the French-Polish collaboration. Moreover, K. G. thanks for the support from MNiSW, Warsaw (Poland), under “Iuventus Plus 2015–2016”, program no IP2014 013073.

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Copyright information

© Springer India Pvt. Ltd. 2017

Authors and Affiliations

  • G. Dattoli
    • 1
  • E. di Palma
    • 1
  • E. Sabia
    • 1
  • K. Górska
    • 2
    Email author
  • A. Horzela
    • 2
  • K. A. Penson
    • 3
  1. 1.ENEA—Centro Ricerche FrascatiFrascati (Roma)Italy
  2. 2.Division of Theoretical Physics, H. Niewodniczański Institute of Nuclear PhysicsPolish Academy of SciencesKrakówPoland
  3. 3.Laboratoire de Physique Théorique de la Matière Condensée, CNRS UMR 7600Sorbonne Universités, Université Pierre et Marie CurieParis Cedex 05France

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