Completely monotonic functions associated with the gamma function and its q-analogues
type: Item
instance of: scholarly article
title: Completely monotonic functions associated with the gamma function and its q-analogues
zbMATH Open document ID: 0589.33001
DOI: 10.1016/0022-247X(86)90042-9
author: Mourad E. H. Ismail
author: Lee Lorch
author: Martin E. Muldoon
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: The property of complete monotonicity has recently been observed to be of importance with respect to certain functions occurring in the theory of elasticity, probability and numerical analysis [see, for example, \textit{W. A. Day}, Proc. Cambridge Philos. Soc. 67, 503-508 (1970; Zbl 0202.253)]. In this important paper, two general theorems on monotonicity are employed to establish this property for a number of combinations of gamma functions and their basic analogues. For example, if \(0<t\leq e^{- \gamma}\), \(\gamma\) being Euler's constant, \(\delta >0\), and let \(\{t^{x^{\delta}}/\Gamma (x^{\delta}+1)\}=e^{-h(x)}\). Then, for \(0<\delta \leq\), \(h(x)>0\), h'(x) is completely monotonic on (0,\(\infty)\). For \(\delta =1\), \(h(x)>0\), \(h'(x)>0\) and h''(x) is completely monotonic on (0,\(\infty).\) In conclusion, the main results of this study are applied to some cases of infinitely divisible probability distribution functions.
reviewed by: H. Exton
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 60E07
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 33E99
Mathematics Subject Classification ID: 74D99
zbMATH DE Number: 3946629
zbMATH Keywords: q-analogues of gamma functions
zbMATH Keywords: monotonicity
MaRDI profile type: Publication
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full work available at URL: 0022-247x(86)90042-9
OpenAlex ID: W2047004726
description: scientific article; zbMATH DE number 3946629