Srivastava-Tomovski函数的完全单调性及其Laplace-Wright实现
Complete Monotonicity of the Srivastava-Tomovski Function and Its Laplace-Wright Realization: Necessary and Sufficient Conditions
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中文总结 AI 辅助
该研究探讨Srivastava-Tomovski函数的完全单调性，分离其广义Wright级数与Laplace-Wright实现，推导参数条件、测度性质及特例结果，为相关函数的分析提供理论支撑。
中文摘要 AI 辅助
针对正参数，我们研究负轴Srivastava-Tomovski函数的完全单调性，同时将其广义Wright级数与正轴Laplace-Wright实现分离。该级数仅当Δ=1+α−κ>0时为整函数；在该区域内，当且仅当α≤κ且κβ≥αγ时，该函数完全单调。对于0<α<κ，Laplace-Wright实现延伸过有限半径和零半径阶段，并遵循相同的特征。其归一化内部测度为先验律：当a=α/κ、μ=β−aγ时，它是基于Ferreira-Simon Wright密度的Wang幂偏Wright律在映射t↦t^κ下的推前。这些结果提供了内部正性、可允许参数区域、矩以及β/原子端点。我们将这些律与四参数实现的解析阶段关联，并对排除区域使用有限符号Laplace唯一性和矩支撑论证。特例κ=1整合了已知的Prabhakar充分性和负值结果与先验支撑坍缩机制。每个可允许测度的质量为1/Γ(β)。
英文摘要
We determine necessary and sufficient conditions for complete monotonicity of the Srivastava--Tomovski extension, a generalized Mittag--Leffler kernel arising in fractional calculus, while separating the defining generalized-Wright series from its positive-axis realization when the series is not entire. For positive parameters, let \(Δ=1+α-κ\). In the entire regime \(Δ>0\), \(E_{α,β}^{γ,κ}(-x)\) is completely monotone on \((0,\infty)\) if and only if \(α\leqκ\) and \(κβ\geqαγ\). For arbitrary positive parameters, the corresponding positive-axis realization satisfies the same criterion; when \(0<α<κ\), it is defined by a second-kind Wright kernel and remains meaningful through the finite-radius and zero-radius phases of the defining series. We also identify the Bernstein measure in every admissible case. In the interior region it is, after normalization, the pushforward under \(t\mapsto t^κ\) of a power-biased Wright distribution; at \(α=κ\) it becomes a powered beta law for \(β>γ\) and the atom \(Γ(γ)^{-1}δ_1\) for \(β=γ\). Necessity in the excluded regions follows from a Mellin-transform zero, uniqueness of finite signed Laplace transforms, and moment-support asymptotics. The Prabhakar criterion is recovered by setting \(κ=1\).
发表机构
- School of Statistics, University of the Philippines（菲律宾大学统计学院）
- Department of Mathematics and Statistics, Mindanao State University–Iligan Institute of Technology（棉兰老国立大学-伊利甘理工学院数学与统计系）
- School of Mathematics and Statistics, The University of Sydney（悉尼大学数学与统计学院）
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