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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On complete monotonicity of certain special functions
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by Ruiming Zhang PDF
Proc. Amer. Math. Soc. 146 (2018), 2049-2062 Request permission


Given an entire function $f(z)$ that has only negative zeros, we shall prove that all the functions of type $f^{(m)}(x)/f^{(n)}(x),\ m>n$ are completely monotonic. Examples of this type are given for Laguerre polynomials, ultraspherical polynomials, Jacobi polynomials, Stieltjes-Wigert polynomials, $q$-Laguerre polynomials, Askey-Wilson polynomials, Ramanujan function, $q$-exponential functions, $q$-Bessel functions, Euler’s gamma function, Airy function, modified Bessel functions of the first and the second kind, and the confluent basic hypergeometric series.
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Additional Information
  • Ruiming Zhang
  • Affiliation: College of Science, Northwest A&F University, Yangling, Shaanxi 712100, People’s Republic of China.
  • MR Author ID: 257230
  • Email:
  • Received by editor(s): June 27, 2017
  • Received by editor(s) in revised form: July 2, 2017
  • Published electronically: December 11, 2017
  • Additional Notes: This research was supported by National Natural Science Foundation of China, grant No. 11371294.
  • Communicated by: Mourad Ismail
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 146 (2018), 2049-2062
  • MSC (2010): Primary 33D15; Secondary 33C45, 33C10, 33E20
  • DOI:
  • MathSciNet review: 3767356