## On complete monotonicity of certain special functions

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- by Ruiming Zhang PDF
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**146**(2018), 2049-2062 Request permission

## Abstract:

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.## References

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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: ruimingzhang@yahoo.com
- 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: https://doi.org/10.1090/proc/13878
- MathSciNet review: 3767356