The Dirichlet-type Laplace transforms

Volume 15, Issue 3, pp 225--239 http://dx.doi.org/10.22436/jnsa.015.03.05

Publication Date: April 16, 2022 Submission Date: January 08, 2022 Revision Date: January 26, 2022 Accteptance Date: March 10, 2022

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Authors

D. Caratelli - Electromagnetics Group, Department of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. S. Pinelas - Academia Militar-Departamento de Ciências Exatas e Engenharias, Av. Conde Castro Guimarães, 2720-113, Amadora, Portugal. P. E. Ricci - Sezione di Matematica, International Telematic University UniNettuno, 39 Corso Vittorio Emanuele II, I-00186 Rome, Italy.

Abstract

We show that it is possible to define extensions of the Laplace transform that use a general Dirichlet series as a kernel. These transforms, denoted by DLTs, further generalize those, considered in previous papers, in which the kernels were related to Laguerre-type exponentials or Bell polynomials. Computational techniques, exploiting expansions in Laguerre polynomials, and using Tricomi's method, have been considered. Since it turns out that the transforms considered are obtained as linear combinations of ordinary Laplace transforms, it is also possible to define an approximation of the relevant inverse transforms. Numerical experiments, performed with the algebra program Mathematica, show that the introduced technique is fast and efficient.

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Keywords

• General Dirichlet series
• Laguerre-type exponentials
• Bell polynomials
• Laplace transform

MSC

•  30B50
•  44A10
•  05A15
•  05A40
•  11B83

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