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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Convolution of beta prime distribution
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by Rui A. C. Ferreira and Thomas Simon HTML | PDF
Trans. Amer. Math. Soc. 376 (2023), 855-890 Request permission


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’s relationships for ${}_3F_2(1)$. 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 Turán’s inequality for the parabolic cylinder function and the confluent hypergeometric function of the second kind. The case of Mill’s ratio is discussed in detail.
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Additional Information
  • Rui A. C. Ferreira
  • Affiliation: Grupo Física-Matemática, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal
  • MR Author ID: 839886
  • Email:
  • Thomas Simon
  • Affiliation: Université de Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, 59000 Lille, France
  • MR Author ID: 640288
  • Email:
  • Received by editor(s): June 29, 2021
  • Received by editor(s) in revised form: February 22, 2022
  • Published electronically: December 1, 2022
  • Additional Notes: The first author was supported by the “Fundação para a Ciência e a Tecnologia (FCT)” through the program “Stimulus of Scientific Employment, Individual Support-2017 Call” with reference CEECIND/00640/2017
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 855-890
  • MSC (2020): Primary 26A48, 33C15, 33C20, 33C45, 33C65, 60E07, 60E15, 62E15
  • DOI:
  • MathSciNet review: 4531664