Some properties of the Kilbas-Saigo function

Boudabsa, Lotfi; Simon, Thomas

doi:10.48550/arxiv.2012.05666

Cited by 2 publications

(8 citation statements)

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“…This theorem extends in its part (a) the aforementioned characterizations of [3,12]. In its parts (b) and (c), it improves on all the results of [7] and solves the numerical conjecture stated therein that for every γ = n an integer, every α ≤ 1/n and β ≥ (n + 1)/2n, the CM property holds for…”

Section: Introductionsupporting

confidence: 76%

“…reasoning exactly as in the proof of Theorem 2.3 in [3] with Carlson's theorem -see e.g. Section 5.81 in [13] -implies the identification…”

Section: Proof Of the Theoremmentioning

confidence: 61%

“…It has been shown in [12] that x → E α,β (−x) is completely monotone (CM) on (0, ∞) iff α ∈ [0, 1] and β ≥ α. It was observed in [3] -see Proposition 5.3 therein -that x → F (γ) 1,1 (−x) is CM on (0, ∞) iff γ ∈ [0, 1]. In the case when γ is an integer, the problem was recently studied in [7],…”

Section: Introductionmentioning

confidence: 99%

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Remark on a Mittag-Leffler function of Le Roy type

Simon¹

2020

Preprint

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Section: Introductionsupporting

confidence: 76%

“…reasoning exactly as in the proof of Theorem 2.3 in [3] with Carlson's theorem -see e.g. Section 5.81 in [13] -implies the identification…”

Section: Proof Of the Theoremmentioning

confidence: 61%

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Remark on a Mittag-Leffler function of Le Roy type

Simon¹

2020

Preprint

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“…It is a generalization of the classical Mittag-Leffler function that is proposed by Kilbas and Saigo [141], (see also [4], p. 48, and Section 5.2 in [129], pp. [128][129][130][131][132][133][134][135][136][137][138][139][140][141][142][143][144][145][146][147] that is defined by the series…”

Section: Gf Distributions With Kilbas-saigo Functionmentioning

confidence: 99%

“…An asymptotic formula for Function (378) can be derived by using the results of Boudabsa, Simon, and Vallois in the works [142,143]. The following three cases should be considered:…”

Section: Gf Distributions With Kilbas-saigo Functionmentioning

confidence: 99%

Nonlocal Probability Theory: General Fractional Calculus Approach

Tarasov

2022

Mathematics

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Nonlocal generalization of the standard (classical) probability theory of a continuous distribution on a positive semi-axis is proposed. An approach to the formulation of a nonlocal generalization of the standard probability theory based on the use of the general fractional calculus in the Luchko form is proposed. Some basic concepts of the nonlocal probability theory are proposed, including nonlocal (general fractional) generalizations of probability density, cumulative distribution functions, probability, average values, and characteristic functions. Nonlocality is described by the pairs of Sonin kernels that belong to the Luchko set. Properties of the general fractional probability density function and the general fractional cumulative distribution function are described. The truncated GF probability density function, truncated GF cumulative distribution function, and truncated GF average values are defined. Examples of the general fractional (GF) probability distributions, the corresponding probability density functions, and cumulative distribution functions are described. Nonlocal (general fractional) distributions are described, including generalizations of uniform, degenerate, and exponential type distributions; distributions with the Mittag-Leffler, power law, Prabhakar, Kilbas–Saigo functions; and distributions that are described as convolutions of the operator kernels and standard probability density.

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Some properties of the Kilbas-Saigo function

Cited by 2 publications

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Remark on a Mittag-Leffler function of Le Roy type

Remark on a Mittag-Leffler function of Le Roy type

Nonlocal Probability Theory: General Fractional Calculus Approach

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