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Complete Monotonicity of the Srivastava-Tomovski Function and Its Laplace-Wright Realization

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Pith's one-line read This paper proves an exact if-and-only-if criterion for the Srivastava–Tomovski function to be completely monotone, and identifies the representing probability measure explicitly.

desk verdict A careful synthesis paper that delivers the first if-and-only-if complete-monotonicity classification for the four-parameter Srivastava-Tomovski function, with the main residual risk isolated in the Mellin identity for the second-kind Wright function with negative parameter. read the letter →

arxiv 2608.08198 v2 pith:35YZO2MK submitted 2026-08-08 math.ST math.PRstat.TH

Rey R. Cuenca , Welfredo R. Patungan , Ruzzel Ragas This is my paper · ORCID
classification math.STmath.PRstat.TH MSC 33E1226A4844A1033C60
keywords completemonotonicitySrivastava–TomovskifunctionPrabhakarWrightBernsteinmeasuregeneralizedseriesLaplacetransformMittag–Leffler
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper fixes exactly when the Srivastava–Tomovski function, a four-parameter Mittag–Leffler generalization, is completely monotone on the positive half-line: every derivative alternates in sign, or equivalently the function is the Laplace transform of a positive measure. For positive parameters $\alpha,\beta,\gamma,\kappa$, the negative-axis series is proved completely monotone if and only if $\alpha\le\kappa$ and $\kappa\beta\ge\alpha\gamma$, in the entire-series regime $\Delta=1+\alpha-\kappa>0$. A separately defined Laplace–Wright realization, which exists for all positive parameters, obeys the same characterization. In every admissible case the representing Bernstein measure is explicit: a strictly positive Wright density, a powered $\beta$ density, or a point mass, each with total mass $1/\Gamma(\beta)$. Because completely monotone functions are mixtures of exponential relaxations, the result identifies the exact parameter region in which this generalized relaxation kernel has a genuine positive spectral measure.

What carries the argument

The load-bearing object is the second-kind Wright function $W_{-a,\nu}$ and its Mellin identity (13): $\int_0^\infty t^{z-1}W_{-a,\nu}(-t)\,dt=\Gamma(z)/\Gamma(\nu+az)$ on the appropriate half-plane. The paper proves this identity for arbitrary real $\nu$ by combining the stretched-exponential decay of $W_{-a,\nu}$ with repeated integration by parts; this extension is what allows the kernel $K(u)=u^{\gamma/\kappa-1}W_{-\alpha/\kappa,\mu}(-u^{1/\kappa})/(\kappa\Gamma(\gamma))$ to be analyzed even when $\mu<0$. The sufficiency direction rides on the pushforward identity $\Gamma(\beta)K(u)\,du=(t\mapsto t^\kappa)_\# f_{a,\mu,\gamma-1}(t)\,dt$, showing the normalized interior measure is a power-biased Wright law. The necessity direction uses two auxiliary tools: a finite signed Laplace uniqueness lemma (a signed measure with zero Laplace transform is zero) and a moment-root lemma that identifies the upper endpoint of a measure's support as the limit of $m_n^{1/n}$.

What would settle it

Take $\alpha=1$, $\kappa=2$, $\gamma=2$, $\beta=1/2$, so $a=1/2$ and $\mu=-1/2$. The central Mellin identity (13) predicts $\int_0^\infty W_{-1/2,-1/2}(-t)\,dt=0$, since the right-hand side contains $\Gamma(0)$ in the denominator. Evaluating this integral by high-precision numerical quadrature, using the Wright series near zero and its stretched-exponential asymptotics at infinity, either confirms zero, as the theorem requires, or returns a nonzero value, which would refute the identity and the necessity argument.

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Extended reading notes

Core claim

The paper's central discovery is a complete positivity classification. Theorem 3.1 states that for $\alpha,\beta,\gamma,\kappa>0$ with $\Delta=1+\alpha-\kappa>0$, the negative-axis Srivastava–Tomovski series $E^{\gamma,\kappa}_{\alpha,\beta}(-x)$ is completely monotone on $(0,\infty)$ if and only if $\alpha\le\kappa$ and $\kappa\beta\ge\alpha\gamma$. Theorem 3.2 extends the same equivalence to the separately defined Laplace–Wright realization for arbitrary positive parameters, with strict complete monotonicity in every admissible case. The sufficiency direction identifies, up to normalization and the substitution $u=t^\kappa$, the integrating kernel with a power-biased Wright law, giving an explicit positive measure; the necessity direction splits into three exclusions: for $0<\alpha<\kappa$ and $\mu=\beta-\alpha\gamma/\kappa<0$, a zero in the Mellin transform forces the inverse kernel to change sign, so no positive measure can share its Laplace transform; for $\alpha=\kappa$ and $\beta<\gamma$, the moment roots force support in $[0,1]$ while the moments diverge; and for $\alpha>\kappa$, the moment roots collapse to zero, forcing support at a point while the first moment is positive. Every admissible Bernstein measure has total mass $1/\Gamma(\beta)$, so its normalization is a probability law.

Load-bearing premise

The load-bearing premise is the all-real Mellin identity for the second-kind Wright function: the integral $\int_0^\infty t^{z-1}W_{-a,\nu}(-t)\,dt$ must equal $\Gamma(z)/\Gamma(\nu+az)$ for every real $\nu$, including negative values, with no hidden exceptional case. If that identity failed for some negative $\nu$, the proof that the region $0<\alpha<\kappa$, $\beta<\alpha\gamma/\kappa$ is not completely monotone would collapse, and the necessity direction of the main theorem with it.

Editorial extensions

If this is right

  • For the Prabhakar specialization $\kappa=1$, the theorem becomes: $E^\gamma_{\alpha,\beta}(-x)$ is completely monotone if and only if $0<\alpha\le 1$ and $\beta\ge\alpha\gamma$, unifying earlier sufficiency and negative-value results.
  • In every admissible case the normalized Bernstein measure is a probability law, so the relaxation kernel can be evaluated or sampled by drawing from an explicit Wright, beta, or atomic law and exponentiating the draw.
  • Fractional-calculus operators whose kernels are built from this function inherit positivity, as Laplace transforms of positive measures, exactly in the parameter region $\alpha\le\kappa$ and $\kappa\beta\ge\alpha\gamma$.
  • The theorem separates the analytic transition controlled by $\Delta=1+\alpha-\kappa$ from the positivity transition controlled by $\alpha\le\kappa$ and $\mu\ge 0$; neither condition subsumes the other.
  • At the boundary $\alpha=\kappa$ the spectrum changes type, from a Wright density when $\mu\ge0$ to a compactly supported powered-beta law when $\beta>\gamma$, and finally to the point mass $e^{-x}/\Gamma(\gamma)$ when $\beta=\gamma$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same mechanism — a zero ordinate in a Mellin transform forcing a sign-changing inverse — may classify complete monotonicity for other generalized Wright or Fox–H kernels, not just this four-parameter family.
  • Because the realization exists and is smooth but non-analytic at zero when $\Delta<0$, this family offers explicit completely monotone functions that are not determined by a convergent Taylor series, which could be a testbed for numerical Laplace inversion.
  • The explicit probability structure suggests Bayesian or simulation interpretations: the normalized kernel is a prior on $(0,\infty)$ with tunable tail behavior, with the atomic endpoint as a degenerate limiting law.
  • One could test whether the same inequalities characterize fractional complete monotonicity or Bernstein-function properties of the realization, questions the paper does not address.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper studies complete monotonicity of the Srivastava-Tomovski function E^{γ,κ}_{α,β}(-x) on the positive half-line for positive parameters. It carefully distinguishes the historical entire series, which is entire only when Δ = 1 + α - κ > 0, from a separately defined positive-axis Laplace-Wright realization that is defined for arbitrary positive parameters. The main results, Theorems 3.1 and 3.2, assert that complete monotonicity holds if and only if α ≤ κ and κβ ≥ αγ, both for the entire series in its entire regime and for the realization on the full positive parameter space. Sufficiency is established by explicit Bernstein measures: a strictly positive Wright density in the interior 0 < α < κ with μ = β - αγ/κ ≥ 0, a powered beta density at α = κ and β > γ, and an atom at α = κ and β = γ. Necessity uses three separate mechanisms: a Mellin-zero sign-changing argument for 0 < α < κ with μ < 0 (Proposition 5.1), a moment-support argument for α = κ with β < γ (Proposition 5.2), and a moment-support collapse for α > κ (Proposition 5.3). The appendices provide the all-real Mellin identity for the second-kind Wright function, the convergence trichotomy, finite signed Laplace uniqueness, and the moment-support lemma. The Prabhakar specialization κ = 1 is derived as Corollary 6.1.

Significance. If correct, the paper settles the complete-monotonicity classification for the four-parameter Srivastava-Tomovski function, going beyond known sufficiency results for the three-parameter Prabhakar function. The characterization is sharp, and the Bernstein measures are explicitly identified, including the atomic endpoint. The paper is unusually transparent about which ingredients are new and which are prior results: the interior positive measure is exactly Wang's power-biased Wright law, and the beta and atomic endpoints are inherited from Ferreira-Simon and related constructions. The proof structure is rigorous: the load-bearing Mellin identity for an arbitrary real secondary parameter is proved in Appendix A, and the excluded-region arguments rely on standard uniqueness and moment-support lemmas proved in Appendix C. The paper also delivers a clean consolidation of the Prabhakar case. I find the derivations sound and the claims falsifiable; the main residual risk, the scope of the external Wright negative-ray theorem, is addressed by explicit citation and by a self-contained integration-by-parts extension. There is no circular reasoning and there are no fitted parameters.

minor comments (4)
  1. [Definition 2.3 and throughout] The notation for the series and for the realization should be made visually distinct; in the current typesetting the symbols E and E are easy to confuse, which matters because the two objects are not equal in the zero-radius phase.
  2. [Section 4, proof of Proposition 4.2] The text refers to 'Theorem 4.1' in the proof of Proposition 4.2, whereas the result is labeled Lemma 4.1 in the main text; the numbering should be harmonized between the main text and the appendices.
  3. [Appendix A, after Eq. (34)] Please state explicitly that the constants C and N in the bound (14) may depend on ν but that, for the finite family of shifted parameters ν+(j+1)a appearing in the integration-by-parts boundary terms, they can be chosen uniformly; this would remove a small potential ambiguity in the boundary-term argument.
  4. [Proposition 5.1] The proof that W_{-a,μ}(-t) is not identically zero is correct, but the reasoning would be easier to follow if the coefficient conditions for n=0 and n=1 were written out explicitly, since this step is essential for the sign-changing conclusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and rests on independent external theorems, with no fitted parameters or self-citation chain.

full rationale

The paper's central claim is an if-and-only-if characterization of complete monotonicity. The sufficiency direction is an explicit construction: for 0<α<κ and μ≥0, the kernel K in (6) is shown to be strictly positive by the external Ferreira–Simon theorem, and the equal-step cases are derived as a powered-beta density and an atom from Euler's beta integral and coefficient cancellation. The necessity directions do not assume the conclusion. The excluded interior case μ<0 uses the Mellin identity (13), proved in Appendix A from Wright's negative-ray theorem and an integration-by-parts argument, to show that the continuous kernel has both signs; a finite signed Laplace uniqueness lemma then rules out any positive Bernstein measure. The boundary cases α=κ, β<γ and α>κ are excluded by moment-root/support arguments: the moments of any hypothetical Bernstein measure are forced by the series coefficients and Stirling asymptotics to grow or vanish in ways incompatible with a finite positive measure. No parameter is fitted to data, no input is renamed as a prediction, and no load-bearing result is justified by a citation to the present authors' own prior work. The cited external results (Wright [30], Ferreira–Simon [2], Wang [28], Schilling–Song–Vondraček [23]) are quoted with theorem numbers and are not equivalent to the target characterization. The skeptical concern about the precise scope of Wright's negative-ray theorem is a correctness risk about an external ingredient, not a circularity in the paper's derivation chain. The paper itself states its increment as synthesis and connection, which is consistent with the non-circular use of prior Wright-law results.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim introduces no free parameters; α, β, γ, κ are free variables. It rests on standard transform theorems and on prior probabilistic results for Wright and beta laws. No new entities are postulated.

assumptions (6)
  • standard math Bernstein's theorem identifies complete monotonicity with positive Laplace spectra.
    Invoked throughout: sufficiency from positive kernels and existence of a representing measure in necessity arguments.
  • standard math Wright's negative-ray theorem gives the stretched-exponential bound (14) for W_{-a,ν}(-t) for 0<a<1 and all real ν.
    Used in Appendix A and Proposition 4.2 to prove K∈L^1, absolute moments, and the Mellin identity; cited to [30] and [17].
  • standard math The second-kind Wright Mellin identity (13) with fundamental strip and removable gamma-pole cases.
    Load-bearing for moment formulas and for the μ<0 exclusion at z0=−μ/a in Proposition 5.1.
  • domain assumption Ferreira-Simon positivity and admissibility of the second-kind Wright density for μ≥0, and Wang's power-biased law.
    The sufficiency direction for the interior region rests on these prior probabilistic results; the paper explicitly labels them as prior in Remark 4.3.
  • standard math Finite signed Laplace uniqueness and the moment-root support lemma.
    Appendix C lemmas are used to exclude alternative positive measures and to force support endpoints in the α=κ and α>κ cases.
  • standard math Stirling's formula for gamma ratios.
    Used for the moment asymptotics in Propositions 5.2 and 5.3.

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Pith. "Pith review of Complete Monotonicity of the Srivastava-Tomovski Function and Its Laplace-Wright Realization." pith.science (2026). https://pith.science/paper/35YZO2MK

@misc{pith2026260808198,
  author       = {Pith},
  title        = {Pith review of: Complete Monotonicity of the Srivastava-Tomovski Function and Its Laplace-Wright Realization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/35YZO2MK}},
  note         = {Machine review of arXiv:2608.08198}
}
abstract

For positive parameters, we study complete monotonicity of the negative-axis Srivastava-Tomovski function while keeping its generalized-Wright series separate from a positive-axis Laplace-Wright realization. The series is entire only when $\Delta=1+\alpha-\kappa>0$; in that regime it is completely monotone precisely when $\alpha\leq\kappa$ and $\kappa\beta\geq\alpha\gamma$. For $0<\alpha<\kappa$, the Laplace-Wright realization extends through the finite-radius and zero-radius phases and obeys the same characterization. Its normalized interior measure is a prior law: with $a=\alpha/\kappa$, $\mu=\beta-a\gamma$, it is the pushforward under $t\mapsto t^\kappa$ of Wang's power-biased Wright law, based on the Ferreira-Simon Wright density. Their results supply the interior positivity, admissible parameter region, moments, and beta/atomic endpoints. We connect these laws to the analytic phases of the four-parameter realization and use finite signed Laplace uniqueness and moment-support arguments for the excluded regions. The specialization $\kappa=1$ consolidates known Prabhakar sufficiency and negative-value results with the prior support-collapse mechanism. Every admissible measure has mass $1/\Gamma(\beta)$.

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Works this paper leans on

32 extracted references · 5 linked inside Pith

  1. [1]

    Beghin, L

    L. Beghin, L. Cristofaro, and J. L. da Silva. Fox-h densities and completely monotone generalized Wright functions.Journal of Theoretical Probability, 38(1):18, 2025. ISSN 0894-9840. doi: 10.1007/s10959-024-01391-9. URL https://doi.org/10.1007/ s10959-024-01391-9

    doi:10.1007/s10959-024-01391-9 2025

  2. [2]

    R. A. C. Ferreira and T. Simon. On the log-concavity of the Wright function. Constructive Approximation, 60(2):309–338, 2024. ISSN 0176-4276. doi: 10.1007/ s00365-023-09666-w. URLhttps://doi.org/10.1007/s00365-023-09666-w

    doi:10.1007/s00365-023-09666-w 2024

  3. [3]

    Garra and R

    R. Garra and R. Garrappa. The Prabhakar or three parameter Mittag–Leffler func- tion: Theory and application.Communications in Nonlinear Science and Numerical Simulation, 56:314–329, Mar. 2018. ISSN 1007-5704. doi: 10.1016/j.cnsns.2017.08.018. URLhttps://doi.org/10.1016/j.cnsns.2017.08.018

    doi:10.1016/j.cnsns.2017.08.018 2018

  4. [4]

    NumericalevaluationoftwoandthreeparameterMittag–Lefflerfunctions

    R.Garrappa. NumericalevaluationoftwoandthreeparameterMittag–Lefflerfunctions. SIAM Journal on Numerical Analysis, 53(3):1350–1369, 2015. ISSN 0036-1429. doi: 10.1137/140971191. URLhttps://doi.org/10.1137/140971191

    doi:10.1137/140971191 2015

  5. [5]

    F. Giraldi. A class of positive FoxH-functions.Fractional Calculus and Applied Analysis, 29(2):1074–1095, 2026. ISSN 1311-0454. doi: 10.1007/s13540-026-00482-0. URLhttps://doi.org/10.1007/s13540-026-00482-0

    doi:10.1007/s13540-026-00482-0 2026

  6. [6]

    Giusti, I

    A. Giusti, I. Colombaro, R. Garra, R. Garrappa, F. Polito, M. Popolizio, and F. Mainardi. A practical guide to Prabhakar fractional calculus.Fractional Calculus and Applied Analysis, 23(1):9–54, 2020. ISSN 1311-0454. doi: 10.1515/fca-2020-0002. URLhttps://doi.org/10.1515/fca-2020-0002

    doi:10.1515/fca-2020-0002 2020

  7. [7]

    Gorenflo, A

    R. Gorenflo, A. A. Kilbas, F. Mainardi, and S. V. Rogosin.Mittag–Leffler Functions, Related Topics and Applications. Springer Monographs in Mathematics. Springer, Berlin and Heidelberg, 2 edition, 2020. ISBN 978-3-662-61549-2, 978-3-662-61550-8. doi: 10.1007/978-3-662-61550-8. URLhttps://doi.org/10.1007/978-3-662-61550-8

    doi:10.1007/978-3-662-61550-8 2020

  8. [8]

    Górska, A

    K. Górska, A. Horzela, A. Lattanzi, and T. K. Pogány. On complete monotonicity of three parameter Mittag–Leffler function.Applicable Analysis and Discrete Mathematics, 15(1):118–128, 2021. ISSN 1452-8630. doi: 10.2298/AADM190226025G. URL https://doi.org/10.2298/AADM190226025G

    doi:10.2298/aadm190226025g 2021

Show all 32 references
  1. [9]

    D. B. Karp and E. G. Prilepkina. Completely monotonic gamma ratio and infinitely divisible H-function of Fox.Computational Methods and Function Theory, 16(1): 135–153, 2016. ISSN 1617-9447. doi: 10.1007/s40315-015-0128-9. URL https: //doi.org/10.1007/s40315-015-0128-9

    2016 doi

  2. [10]

    D. B. Karp and E. G. Prilepkina. Some new facts concerning the delta neutral case of Fox’sH function.Computational Methods and Function Theory, 17(2):343–367, 2017. ISSN 1617-9447. doi: 10.1007/s40315-016-0183-x. URL https://doi.org/10.1007/ s40315-016-0183-x

    2017 doi

  3. [11]

    A. A. Kilbas and M. Saigo.H-Transforms: Theory and Applications, volume 9 of Analytical Methods and Special Functions. Chapman and Hall/CRC, Boca Raton, FL,

  4. [12]

    A. A. Kilbas, M. Saigo, and R. K. Saxena. Generalized Mittag–Leffler function and generalized fractional calculus operators.Integral Transforms and Special Functions, 15(1):31–49, Feb. 2004. ISSN 1065-2469. doi: 10.1080/10652460310001600717. URL https://doi.org/10.1080/1065246...

    2004 doi

  5. [13]

    Mainardi.Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models

    F. Mainardi.Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press and World Scientific, London and Singapore, 2010. ISBN 978-1-84816-329-4. doi: 10.1142/p614. URL https://doi. org/10.1142/p614. 14 REY R. CUENCA,...

    2010 doi

  6. [14]

    Mainardi and R

    F. Mainardi and R. Garrappa. On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics.Journal of Computational Physics, 293: 70–80, July 2015. ISSN 0021-9991. doi: 10.1016/j.jcp.2014.08.006. URL https: //doi.org/10.1016/j.jcp.2014.08.006

    2015 doi

  7. [15]

    K. Mehrez. Monotonicity patterns and functional inequalities for classical and generalized Wright functions.Mathematical Inequalities & Applications, 22(3): 901–916, 2019. ISSN 1331-4343. doi: 10.7153/mia-2019-22-61. URL https : //doi.org/10.7153/mia-2019-22-61

    2019 doi

  8. [16]

    K. Mehrez. Positivity of certain classes of functions related to the FoxH-functions with applications.Analysis and Mathematical Physics, 11(3):114, 2021. ISSN 1664-

    2021

  9. [17]

    R. B. Paris. The asymptotics of the generalised Bessel function.Mathematica Aeterna, 7(4):381–406, 2017. URLhttps://arxiv.org/abs/1711.03006

    2017 arXiv

  10. [18]

    R. B. Paris and D. Kaminski.Asymptotics and Mellin–Barnes Integrals, vol- ume 85 ofEncyclopedia of Mathematics and its Applications. Cambridge Univer- sity Press, Cambridge, 2001. ISBN 978-0-521-79001-7, 978-0-511-54666-2. doi: 10.1017/CBO9780511546662. URLhttps://doi.org/10.1...

    2001 doi

  11. [19]

    R. B. Paris, A. Consiglio, and F. Mainardi. On the asymptotics of Wright functions of the second kind.Fractional Calculus and Applied Analysis, 24(1):54–72, 2021. ISSN 1311-0454. doi: 10.1515/fca-2021-0003. URL https://doi.org/10.1515/ fca-2021-0003

    2021 doi

  12. [20]

    H. Pollard. The completely monotonic character of the Mittag–Leffler functionEa(−x). Bulletin of the American Mathematical Society, 54(12):1115–1116, Dec. 1948. ISSN 0002-9904. doi: 10.1090/S0002-9904-1948-09132-7. URL https://doi.org/10.1090/ S0002-9904-1948-09132-7

    1948 doi

  13. [21]

    T. R. Prabhakar. A singular integral equation with a generalized Mittag–Leffler function in the kernel.Yokohama Mathematical Journal, 19(1):7–15, 1971. ISSN 0044-0523. URL https://ynu.repo.nii.ac.jp/record/6514/files/YMJ_19_N1_ 1971_007-015.pdf

    1971

  14. [22]

    D. S. P. Salazar. Bernstein functions at work: Coalescents, copulas, and subordination. arXiv preprint, version 1, July 2026. URLhttps://arxiv.org/abs/2607.04467

    2026 arXiv

  15. [23]

    R. L. Schilling, R. Song, and Z. Vondraček.Bernstein Functions: Theory and Applications, volume 37 ofDe Gruyter Studies in Mathematics. De Gruyter, Berlin and Boston, 2 edition, 2012. ISBN 978-3-11-025229-3, 978-3-11-026933-8. doi: 10. 1515/9783110269338. URLhttps://doi.org/10...

    2012 doi

  16. [24]

    W. R. Schneider. Completely monotone generalized Mittag–Leffler functions.Exposi- tiones Mathematicae, 14(1):3–16, 1996. ISSN 0723-0869. URLhttps://www.tib.eu/ en/search/id/BLSE:RN005755256

    1996

  17. [25]

    N. K. Sibisi. A probabilistic perspective on Feller, Pollard and the complete mono- tonicity of the Mittag–Leffler function. arXiv preprint, version 2, Jan. 2023. URL https://arxiv.org/abs/2301.01466

    2023 arXiv

  18. [26]

    H. M. Srivastava and Ž. Tomovski. Fractional calculus with an integral operator containing a generalized Mittag–Leffler function in the kernel.Applied Mathematics and Computation, 211(1):198–210, Apr. 2009. ISSN 0096-3003. doi: 10.1016/j.amc. 2009.01.055. URLhttps://doi.org/10...

    2009 doi

  19. [27]

    Tomovski, T

    Ž. Tomovski, T. K. Pogány, and H. M. Srivastava. Laplace type integral expressions for a certain three-parameter family of generalized Mittag–Leffler functions with applications involving complete monotonicity.Journal of the Franklin Institute, 351 (12):5437–5454, Dec. 2014. I...

    2014 doi

  20. [28]

    M. Wang. Moments of gamma type and three-parametric Mittag–Leffler function. arXiv preprint, Oct. 2024. URLhttps://arxiv.org/abs/2410.19330

    2024 arXiv

  21. [29]

    M. Wang. Infinite divisibility ofα-Cauchy distributions. arXiv preprint, Dec. 2025. URLhttps://arxiv.org/abs/2512.23164. Version 3, revised 15 April 2026

    2025 arXiv

  22. [30]

    E. M. Wright. The generalized Bessel function of order greater than one.The Quarterly Journal of Mathematics, os-11(1):36–48, 1940. ISSN 0033-5606. doi: 10.1093/qmath/ os-11.1.36. URLhttps://doi.org/10.1093/qmath/os-11.1.36

    1940 doi

  23. [2004]

    doi: 10.1201/9780203487372

    ISBN 978-0-415-29916-9, 978-0-203-48737-2. doi: 10.1201/9780203487372. URL https://doi.org/10.1201/9780203487372

    doi

  24. [2368]

    URL https : / / doi

    doi: 10.1007/s13324-021-00553-w. URL https : / / doi . org / 10 . 1007 / s13324-021-00553-w

    doi

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