{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:46:00Z","timestamp":1760060760533,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,15]],"date-time":"2025-09-15T00:00:00Z","timestamp":1757894400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Using the Laplace transform and the Gamma function, we obtain the Laplace-type transform, with the property of mapping a function to a functional sequence, which cannot be realized by the Laplace transform. In addition, we construct a backward difference as a generalization of the backward difference operator \u2207. By connecting it to the Laplace-type transform, we deduce a method for solving difference equations and, relying on classical orthogonal polynomials, for obtaining combinatorial identities. A table of some elementary functions and their images is at the end of the text.<\/jats:p>","DOI":"10.3390\/axioms14090697","type":"journal-article","created":{"date-parts":[[2025,9,16]],"date-time":"2025-09-16T07:33:02Z","timestamp":1758007982000},"page":"697","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Laplace-Type Transform and Its Applications"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3766-8579","authenticated-orcid":false,"given":"Slobodan B.","family":"Tri\u010dkovi\u0107","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Ni\u0161, 18000 Ni\u0161, Serbia"}]},{"given":"Miomir S.","family":"Stankovi\u0107","sequence":"additional","affiliation":[{"name":"Mathematical Institute of the Serbian Academy of Sciences and Arts, 11000 Belgrade, Serbia"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"739","DOI":"10.1016\/j.indag.2013.08.001","article-title":"Uniform Asymptotic Methods for Integrals","volume":"24","author":"Temme","year":"2013","journal-title":"Indag. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1112\/plms\/s2-17.1.116","article-title":"The harmonic functions associated with the parabolic cylinder","volume":"2","author":"Watson","year":"1918","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"114897","DOI":"10.1016\/j.cam.2022.114897","article-title":"A convergent and asymptotic Laplace method for integrals","volume":"422","author":"Pagola","year":"2023","journal-title":"J. Comp. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Shiff, J. (1999). The Laplace Transform: Theory and Applications, Springer.","DOI":"10.1007\/978-0-387-22757-3"},{"key":"ref_5","unstructured":"Suetin, P.K. (2005). Classical Orthogonal Polynomials, Nauka."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Mitrinovi\u0107, D., and Ke\u010dki\u0107, J. (1984). The Cauchy Method of Residues\u2014Theory and Applications, D. Reidel Publishing Company.","DOI":"10.1007\/978-94-015-6872-2"},{"key":"ref_7","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_8","unstructured":"Kamke, E. (1959). Differentialgleichungen. L\u00f6sungsmethoden und L\u00f6sungen, Akademische Verlagsgesellschaft."},{"key":"ref_9","unstructured":"Krasnov, M., Kiselev, A., and Makarenko, G. (1971). Problems and Exercises in Integral Equations, Mir Publishers."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5437","DOI":"10.1016\/j.jfranklin.2014.09.007","article-title":"Laplace type integral expressions for a certain three-parameter family of generalized Mittag-Leffler functions with applications involving complete monotonicity","volume":"351","author":"Tomovski","year":"2014","journal-title":"J. Frankl. Inst."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Koepf, W., Kim, I., and Rathie, A.K. (2019). On a New Class of Laplace-Type Integrals Involving Generalized Hypergeometric Functions. Axioms, 8.","DOI":"10.3390\/axioms8030087"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/9\/697\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:46:00Z","timestamp":1760035560000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/9\/697"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,15]]},"references-count":11,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["axioms14090697"],"URL":"https:\/\/doi.org\/10.3390\/axioms14090697","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,9,15]]}}}