{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:11:17Z","timestamp":1760242277701,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2017,3,31]],"date-time":"2017-03-31T00:00:00Z","timestamp":1490918400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The paper devises a family of leptokurtic bell-shaped distributions which is based on the hyperbolic secant raised to a positive power, and bridges the Laplace and Gaussian laws on asymptotic arguments. Moment and cumulant generating functions are then derived and represented in terms of polygamma functions. The behaviour of shape parameters, namely kurtosis and entropy, is investigated. In addition, Gram\u2013Charlier-type (GCT) expansions, based on the aforementioned distributions and their orthogonal polynomials, are specified, and an operational criterion is provided to meet modelling requirements in a possibly severe kurtosis and skewness environment. The role played by entropy within the kurtosis ranges of GCT expansions is also examined.<\/jats:p>","DOI":"10.3390\/e19040149","type":"journal-article","created":{"date-parts":[[2017,3,31]],"date-time":"2017-03-31T10:35:49Z","timestamp":1490956549000},"page":"149","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Distribution Family Bridging the Gaussian and the Laplace Laws, Gram\u2013Charlier Expansions, Kurtosis Behaviour, and Entropy Features"],"prefix":"10.3390","volume":"19","author":[{"given":"Mario","family":"Faliva","sequence":"first","affiliation":[{"name":"Dipartimento di Discipline matematiche, Finanza matematica ed Econometria, Universit\u00e0 Cattolica del Sacro Cuore, Largo Gemelli 1, 20123 Milano, Italy"}]},{"given":"Maria","family":"Zoia","sequence":"additional","affiliation":[{"name":"Dipartimento di Discipline matematiche, Finanza matematica ed Econometria, Universit\u00e0 Cattolica del Sacro Cuore, Largo Gemelli 1, 20123 Milano, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2017,3,31]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Benes, V., and Step\u00e1n, J. 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Distribution Theory.","key":"ref_5"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1080\/03610920802696596","article-title":"Tailoring the Gaussian law for excess kurtosis and skewness by Hermite polynomials","volume":"39","author":"Zoia","year":"2010","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1007\/s00362-014-0633-3","article-title":"The role of orthogonal polynomials in adjusting hyperbolic secant and logistic distributions to analyse financial asset returns","volume":"56","author":"Bagnato","year":"2015","journal-title":"Stat. Pap."},{"unstructured":"R\u00e9nyi, A. (1970). Probability Theory, North-Holland.","key":"ref_8"},{"unstructured":"Johnson, N.L., Kotz, S., and Balakrisnam, N. (1995). Continuous Univariate Distributions, Wiley. [2nd ed.].","key":"ref_9"},{"unstructured":"Chihara, T.S. (1978). An Introduction to Orthogonal Polynomials, Gordon & Breach.","key":"ref_10"},{"unstructured":"Szeg\u00f6, G. (1939). Orthogonal Polynomials, American Mathematical Society.","key":"ref_11"},{"doi-asserted-by":"crossref","unstructured":"Gelfand, I.M., Kapranov, M.M., and Zelevinsky, A.V. (1994). Discriminants, Resultants and Multidimensional Determinats, Birkhauser.","key":"ref_12","DOI":"10.1007\/978-0-8176-4771-1"},{"unstructured":"Erd\u00e9lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. (1953). Higher Transcendental Functions, Mc. Graw Hill.","key":"ref_13"},{"unstructured":"Ledermann, W. (1980). Handbook of Applicable Mathematics, Wiley.","key":"ref_14"},{"unstructured":"Abramovitz, M., and Stegun, I.A. (1965). Handbook of Mathematical Functions, Dover.","key":"ref_15"},{"unstructured":"Askey, R.A., and Roy, R. (2009). NIST Handbook of Mathematical Functions, NIST and Cambridge University Press.","key":"ref_16"},{"doi-asserted-by":"crossref","unstructured":"Copson, E.T. (1965). Asymptotic Expansions, Cambridge University Press.","key":"ref_17","DOI":"10.1017\/CBO9780511526121"},{"unstructured":"Erd\u00e9lyi, A. (1965). Asymptotic Expansions, Dover.","key":"ref_18"},{"unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (1980). Table of Integrals, Series and Product, Academic Press.","key":"ref_19"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/4\/149\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:31:44Z","timestamp":1760207504000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/4\/149"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,31]]},"references-count":19,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,4]]}},"alternative-id":["e19040149"],"URL":"https:\/\/doi.org\/10.3390\/e19040149","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2017,3,31]]}}}