symmetry S S Article Tricomi’s Method for the Laplace Transform and Orthogonal Polynomials Paolo Emilio Ricci 1, * , Diego Caratelli 2,3 and Francesco Mainardi 4   Citation: Ricci, P.E.; Caratelli, D.; Mainardi, F. Tricomi’s Method for the Laplace Transform and Orthogonal Polynomials. Symmetry 2021, 13, 589. https://doi.org/10.3390/sym13040589 Academic Editor: Dorian Popa Received: 10 March 2021 Accepted: 1 April 2021 Published: 2 April 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Section of Mathematics, International Telematic University UniNettuno, Corso Vittorio Emanuele II 39, 00186 Rome, Italy 2 Department of Research and Development, The Antenna Company, High Tech Campus 29, 5656 AE Eindhoven, The Netherlands; d.caratelli@tue.nl 3 Department of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands 4 Department of Physics and Astronomy, University of Bologna, Via Irnerio 46, 40126 Bologna, Italy; francesco.mainardi@bo.infn.it * Correspondence: p.ricci@uninettunouniversity.net or paoloemilioricci@gmail.com Abstract: Tricomi’s method for computing a set of inverse Laplace transforms in terms of Laguerre polynomials is revisited. By using the more recent results about the inversion and the connection coef- ficients for the series of orthogonal polynomials, we find the possibility to extend the Tricomi method to more general series expansions. Some examples showing the effectiveness of the considered procedure are shown. Keywords: resolvent; Laplace transform; Laguerre polynomials; connection coefficients 1. Introduction In three notes presented to the Accademia dei Lincei in 1935 [1,2], Francesco G. Tricomi introduced a method for the computation of the Laplace transform of functions that can be developed in a series of Laguerre polynomials. Precisely, he proved the following proposition: Proposition 1. If the analytic function F(s) is regular at infinity and we can find a real number h such that it can be represented with a series of the form F(s)= 1 s + h ∞ ∑ n=0 a n  s + h - 1 s + h  n , (1) then it is the Laplace transform of the sum of the series of Laguerre polynomials f (t)= e -ht ∞ ∑ n=0 a n L n (t) , (2) which is absolutely and uniformly convergent for t > 0. In particular, for h = 0, that is, avoiding the shift that follows from a basic rule of the Laplace transform, Tricomi found the functions pair ( F(s), f (t)): F(s)= 1 s ∞ ∑ n=0 a n  s - 1 s  n ↔ f (t)= ∞ ∑ n=0 a n L n (t) . (3) Symmetry 2021, 13, 589. https://doi.org/10.3390/sym13040589 https://www.mdpi.com/journal/symmetry