{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T00:12:17Z","timestamp":1773187937212,"version":"3.50.1"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"2","license":[{"start":{"date-parts":[[2023,12,1]],"date-time":"2023-12-01T00:00:00Z","timestamp":1701388800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["CEECIND\/00640\/2017"],"award-info":[{"award-number":["CEECIND\/00640\/2017"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Trans. Amer. Math. Soc."],"abstract":"<p>\n                    We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series obtained via Appell series of the first kind and Thomae\u2019s relationships for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Subscript 3 Baseline upper F 2 left-parenthesis 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\"\/>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:msub>\n                              <mml:mi>F<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{}_3F_2(1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Using a self-decomposability argument, the identities are applied to derive complete monotonicity properties for quotients of confluent hypergeometric functions having a doubling character. By means of probability, we also obtain a simple proof of Tur\u00e1n\u2019s inequality for the parabolic cylinder function and the confluent hypergeometric function of the second kind. The case of Mill\u2019s ratio is discussed in detail.\n                  <\/p>","DOI":"10.1090\/tran\/8748","type":"journal-article","created":{"date-parts":[[2022,6,8]],"date-time":"2022-06-08T10:04:43Z","timestamp":1654682683000},"page":"855-890","source":"Crossref","is-referenced-by-count":2,"special_numbering":"1065","title":["Convolution of beta prime distribution"],"prefix":"10.1090","volume":"376","author":[{"given":"Rui","family":"Ferreira","sequence":"first","affiliation":[]},{"given":"Thomas","family":"Simon","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,12,1]]},"reference":[{"key":"1","unstructured":"P. Appell and J. Kamp\u00e9 de F\u00e9riet, Fonctions hyperg\u00e9ometriques et hypersph\u00e9riques, Gauthier-Villars, Paris, 1926."},{"issue":"1-2","key":"2","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1017\/S0308210500020412","article-title":"Associated Laguerre and Hermite polynomials","volume":"96","author":"Askey, Richard","year":"1984","journal-title":"Proc. Roy. Soc. Edinburgh Sect. A","ISSN":"https:\/\/id.crossref.org\/issn\/0308-2105","issn-type":"print"},{"key":"3","series-title":"Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics","isbn-type":"print","volume-title":"Continuous univariate distributions. 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