Complete Monotonicity of the Srivastava-Tomovski Function and Its Laplace-Wright Realization: Necessary and Sufficient Conditions
Abstract
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, Eα,βγ,κ(−x) is completely monotone on (0,∞) if and only if α≤κ and κβ≥αγ. 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↦ tκ of a power-biased Wright distribution; at α=κ it becomes a powered beta law for β>γ and the atom Γ(γ)−1δ1 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.
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BibTeX
@misc{cuenca2026complete,
title = {{Complete Monotonicity of the Srivastava-Tomovski Function and Its Laplace-Wright Realization: Necessary and Sufficient Conditions}},
author = {Rey R. Cuenca and Welfredo R. Patungan and Ruzzel Ragas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.08198},
eprint = {2608.08198},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.08198},
}
RIS
TY - UNPB
TI - Complete Monotonicity of the Srivastava-Tomovski Function and Its Laplace-Wright Realization: Necessary and Sufficient Conditions
AU - Rey R. Cuenca
AU - Welfredo R. Patungan
AU - Ruzzel Ragas
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2608.08198
UR - https://arxiv.org/abs/2608.08198
AB - 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,∞) if and only if α≤κ and κβ≥αγ. 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↦ 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)^−1δ_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.
ER -