Abstract
We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at −∞, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le Roy function. The main ingredient for the proof is a probabilistic representation of these functions in terms of the stable subordinator.
Highlights
IntroductionZn, z ∈ C, where the parameters are such that α, m > 0 and l > −1/α. It can be viewed as a generalization of the one- or two-parameter Mittag–Leffler function since, with standard notations, Eα,0 (z)
The Kilbas-Saigo function is a three-parameter entire function with the convergent series representationEα,m,l(z) = 1 + ∑ n≥1 n ∏ k=1Γ(1 + α((k − 1)m + l)) Γ(1 + α((k − 1)m + l + 1))zn, z ∈ C, where the parameters are such that α, m > 0 and l > −1/α
We provide the exact asymptotics of the fractional Weibull and Fréchet densities at both ends of their support and we give a series of probabilistic factorizations
Summary
Zn, z ∈ C, where the parameters are such that α, m > 0 and l > −1/α. It can be viewed as a generalization of the one- or two-parameter Mittag–Leffler function since, with standard notations, Eα,0 (z). For the negative half-line and α ∈ In the former case, the behavior is in cα,m x−(1+1/m) with a non-trivial constant cα,m obtained from the connection with the fractional Fréchet distribution and given in terms of the double Gamma function—see Proposition 7 and Remark 8 (c) below. Introduced in [6] in the context of analytic continuation, a couple of years before the Mittag–Leffler function, the Le Roy function has been much less studied It was shown in [3] that this function encodes for α ∈ [0, 1] a Gumbel distribution of fractional type, as the moment generating function of the perpetuity of the α−stable subordinator. We have gathered in Appendix A all the needed facts and formulæ on this double Gamma function, whose connection with the Kilbas-Saigo function has probably a broader focus than the content of the present paper (we leave this topic open to further research)
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