{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T09:07:20Z","timestamp":1747213640890,"version":"3.40.5"},"reference-count":15,"publisher":"Erdal Karapinar","issue":"1","funder":[{"name":"No support by our institutions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2023,1,12]]},"abstract":"<jats:p xml:lang=\"en\">Bell's polynomials have been used in many different fields, ranging from number theory to operators theory. In this article we show a method to compute the Laplace Transform (LT) of nested analytic functions.  To this aim, we provide a table of the first few values of the complete Bell's polynomials, which are then used to evaluate the LT of composite exponential functions. Furthermore a code for approximating the Laplace Transform of general analytic composite functions is created and presented. A graphical verification of the proposed technique is illustrated in the last section.<\/jats:p>","DOI":"10.31197\/atnaa.1187617","type":"journal-article","created":{"date-parts":[[2023,1,12]],"date-time":"2023-01-12T07:55:52Z","timestamp":1673510152000},"page":"162-177","source":"Crossref","is-referenced-by-count":1,"title":["Laplace Transform of nested analytic functions via Bell\u2019s polynomials"],"prefix":"10.31197","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7899-3087","authenticated-orcid":true,"given":"Paolo Emilio","family":"R\u0130CC\u0130","sequence":"first","affiliation":[{"name":"UniNettuno Telematic University"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0969-884X","authenticated-orcid":true,"given":"Diego","family":"CARATELL\u0130","sequence":"additional","affiliation":[{"name":"Eindhoven University of Technology"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0984-0159","authenticated-orcid":true,"given":"Sandra","family":"P\u0130NELAS","sequence":"additional","affiliation":[{"name":"Academia Militar"}]}],"member":"15935","published-online":{"date-parts":[[2023,3,31]]},"reference":[{"key":"ref1","doi-asserted-by":"crossref","unstructured":"[1] Bell, E.T. Exponential polynomials. Annals of Mathematics 1934, 35, 258\u2013277.","DOI":"10.2307\/1968431"},{"key":"ref2","doi-asserted-by":"crossref","unstructured":"[2] Beerends, R.J., Ter Morsche, H.G., Van Den Berg, J.C., Van De Vrie, E.M., Fourier and\r\nLaplace Transforms. Cambridge Univ. Press, Cambridge, 2003.","DOI":"10.1017\/CBO9780511806834"},{"key":"ref3","unstructured":"[3] Caratelli, D. Cesarano, C., Ricci, P.E. Computation of the Bell-Laplace transforms.\r\nDolomites Res. Notes Approx. 2021 14, 74\u201391."},{"key":"ref4","unstructured":"[4] Cassisa, C., Ricci, P.E. Orthogonal invariants and the Bell polynomials. Rend. Mat. Appl.\r\n2000 (Ser. 7) 20, 293\u2013303."},{"key":"ref5","doi-asserted-by":"crossref","unstructured":"[5] Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions. D. 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Integer Seq., 2021 24, Article 21.8.615"},{"key":"ref10","doi-asserted-by":"crossref","unstructured":"[10] Qi, F., Niu, D-W., Lim, D., Yao, Y-H., Special values of the Bell polynomials of the second\r\nkind for some sequences and functions. J. Math. Anal. Appl., 2020 491 (2), 124382.","DOI":"10.1016\/j.jmaa.2020.124382"},{"key":"ref11","doi-asserted-by":"crossref","unstructured":"[11] Ricci, P.E. Bell polynomials and generalized Laplace transforms. Integral Transforms\r\nSpec. Funct., (2022); doi.org\/10.1080\/10652469.2022.2059077.","DOI":"10.1080\/10652469.2022.2059077"},{"key":"ref12","unstructured":"[12] Riordan, J. An Introduction to Combinatorial Analysis. J. Wiley & Sons, Chichester,\r\n1958."},{"key":"ref13","unstructured":"[13] Robert, D. Invariants orthogonaux pour certaines classes d\u2019operateurs. Annales Math\u00b4em.\r\npures appl. 1973 52, 81\u2013114."},{"key":"ref14","doi-asserted-by":"crossref","unstructured":"[14] Roman, S.M., The Fa`a di Bruno Formula. Amer. Math. Monthly 87 (1980), 805\u2013809.","DOI":"10.1080\/00029890.1980.11995156"},{"key":"ref15","doi-asserted-by":"crossref","unstructured":"[15] Roman, S.M., Rota, G.C. The umbral calculus. Advanced in Math. 1978 27, 95\u2013188.","DOI":"10.1016\/0001-8708(78)90087-7"}],"container-title":["Advances in the Theory of Nonlinear Analysis and its Application"],"original-title":[],"deposited":{"date-parts":[[2023,8,9]],"date-time":"2023-08-09T13:40:47Z","timestamp":1691588447000},"score":1,"resource":{"primary":{"URL":"http:\/\/dergipark.org.tr\/en\/doi\/10.31197\/atnaa.1187617"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,31]]},"references-count":15,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,3,31]]}},"URL":"https:\/\/doi.org\/10.31197\/atnaa.1187617","relation":{},"ISSN":["2587-2648"],"issn-type":[{"type":"electronic","value":"2587-2648"}],"subject":[],"published":{"date-parts":[[2023,3,31]]}}}