Completely monotonic functions associated with the gamma function and its q-analogues
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.
scientific article; zbMATH DE number 3946629