On some new moments of Gamma type

Min Wang; Tetyana Kadankova; Thomas Simon

REVIEW 3 major objections 4 minor 20 references

On some new moments of Gamma type

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Gamma-type moments exist exactly above a single non-increasing boundary curve, the paper proves, and two classical Bessel formulas produce new van Dantzig pairs and signed spectral measures.

desk verdict A clean Bessel-function note with genuinely new van Dantzig pairs and a simpler proof of a known Gamma-moment criterion; the advertised complete boundary in Proposition 3 is the soft spot because it leans on an unpublished external theorem and an omitted edge case. read the letter →

arxiv 1908.03428 v2 pith:IMLATHTF submitted 2019-08-09 math.PR math.CA

Tetyana Kadankova , Thomas Simon , Min Wang This is my paper · ORCID

classification math.PRmath.CA MSC 33C1033C2060E0760E10

keywords BesselfunctionsMomentsofGammatypevanDantzigproblemgeneralizedhypergeometricquasi-infinitedivisibilityWeber-SchafheitlinintegralMeijerG-functionMellintransform

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

The paper's goal is to decide, for a natural four-parameter family of moment problems, exactly when a positive random variable exists with Mellin transform of Gamma type. It proves that for the family $D\big[\begin{smallmatrix}a&b\\ (c,d)&-\end{smallmatrix}\big]$ — the law with $\mathbb{E}[X^s]=\frac{(a)_s(b)_{-s}}{(c)_s(d)_{-s}}$ on $s\in(-\min(a),\min(b))$ — existence is governed by a single non-increasing boundary curve $f_{a,b}$: the pair $(c,d)$ must lie symmetrically above it. In the linear regime the curve is the explicit line $f_{a,b}(u)=3a+b+1/2-u$. The same analysis gives the paper's second theme: the classical von Lommel formula packages the power semicircle distributions and first hitting times of Bessel processes into new explicit van Dantzig pairs. The consequence of the proof is that a subtle positivity question about special functions is reduced to a one-parameter curve, and two classical Bessel identities are connected to probability.

What carries the argument

The work is carried by two nineteenth-century Bessel identities. The von Lommel formula, $\frac{1}{\sqrt\pi\,\Gamma(\alpha+1/2)}\left(\frac z2\right)^\alpha\int_{-1}^1 e^{itz}(1-t^2)^{\alpha-1/2}\,dt=J_\alpha(z)$, supplies the Fourier representation of the power semicircle law and, with the product expansion of $J_\alpha$ over its positive zeros, turns reciprocals at imaginary arguments into products over Bessel zeros. The Weber–Schafheitlin formula, $\int_0^\infty z^{-2s}J_\alpha^2(z)\,dz=\frac{\Gamma(s)\Gamma(\alpha+1/2-s)}{2\sqrt\pi\,\Gamma(1/2+s)\Gamma(\alpha+1/2+s)}$, gives the extremal Gamma-type moments directly: the density of $D\big[\begin{smallmatrix}a&b\\ (2a+b,a+1/2)&-\end{smallmatrix}\big]$ is proportional to $J_{a+b-1/2}^2(x^{-1/2})$. Around these, the paper assembles the Mellin-transform formalism of Gamma-type moments, the Meijer $G$-function identification that converts existence into the ${}_1F_2$ non-negativity condition (8), and the concatenation rules for the Gamma-type moment laws that extend the extremal case to the full boundary curve $f_{a,b}$.

What would settle it

Take $a=b=1$ and $c,d$ on the boundary line $c+d=7.5$ with $\min(c,d)<1.5$, the case whose details are omitted. Compute ${}_1F_2(2;\,c+1,\,d+1;\,-x)$ numerically for large $x$ using the asymptotic formula 16.11.8 of [15]. If it ever stays non-negative for all $x\ge0$ for such a pair, the claimed boundary (and Proposition 3's asymptotic branch) is wrong; if it turns negative, the omitted case is confirmed.

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Extended reading notes

Core claim

On its own terms, the paper establishes that for every $a,b>0$ there is a continuous non-increasing function $f_{a,b}$ on $[(3a+b)/2+1/4,\infty)$ such that the Gamma-type distribution $D\big[\begin{smallmatrix}a&b\\ (c,d)&-\end{smallmatrix}\big]$ exists if and only if $f_{a,b}(d)\le c\le d$ or $f_{a,b}(c)\le d\le c$. For moderate parameters the curve is exactly $f_{a,b}(u)=3a+b+1/2-u$; for large parameters it lies in the wedge $\big]a,\,a+\frac{a+b}{2}(u-a)\big]$ and tends to $a$ as $u\to\infty$. Necessarily the distribution fails when $c+d<3a+b+1/2$ or $\min(c,d)\le a$. The paper also proves that existence is equivalent to non-negativity of the generalized hypergeometric function ${}_1F_2\big(a+b;\,c+b,\,d+b;\,-x\big)$ for all $x\ge0$, and that the extremal case has an explicit density proportional to a squared Bessel function $J_{a+b-1/2}^2(x^{-1/2})$. In the same note, the von Lommel formula yields new van Dantzig pairs: $(\hat h_\alpha(t),1/\hat h_\alpha(it))$ is such a pair for every $\alpha>-1/2$, where $h_\alpha$ is the power semicircle density.

Load-bearing premise

The sharp description of the boundary for large parameters, Proposition 3(b), is taken from an external result (Theorem 4.2 in [6]) that is not proved in this preprint; if that result is wrong or does not apply, the exact boundary claim fails, although the self-contained Proposition 2 still stands.

Editorial extensions

If this is right

The existence question for the four-parameter family is fully answered by one curve: no additional inequalities or case checks are needed once $f_{a,b}$ is known.
The moment problem and non-negativity of ${}_1F_2$ are the same problem, so any positivity test or diagram for these hypergeometric functions transfers to moment existence, and vice versa.
The extremal moments are non-trivial: $\log X_{a,b}$ is quasi-infinitely divisible with a signed spectral density, but is not infinitely divisible, so the implication "if $X$ exists then $\log X$ is infinitely divisible" is false.
New explicit van Dantzig pairs exist for every $\alpha>-1/2$, including the semicircle case $\alpha=1$ and the uniform case $\alpha=1/2$.
The convexity of the existence region $D_{a,b}$ is reduced to monotonicity of the boundary function $f_{a,b}$, giving a concrete route to settle the paper's conjecture.

Reading between the lines

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

The exact large-parameter branch of $f_{a,b}$ rests on Theorem 4.2 of the cited paper [6], which is not proved here; if that result does not hold, Proposition 3(b) could fail even though the self-contained Bessel analysis of Proposition 2 would stand.
The paper's equivalence suggests a numerical recipe for mapping the whole existence diagram: compute signs of ${}_1F_2$ on a grid for each $(a,b)$ and trace the zero set; discrepancies with the linear boundary near the omitted region would be a quick check of the unproved asymptotic.
One could push the same Weber–Schafheitlin mechanism to other pairs of Bessel indices (different powers of $J_\alpha J_\beta$) and ask whether the resulting moment families still admit a one-curve boundary.
The reformulation of the old positivity question for integrals of Bessel functions as membership $(a,1)\in D_{b,b}$ makes it a moment-existence problem, so progress on the convexity conjecture could feed back into Bessel-function inequalities.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a well-written probabilistic note built on Bessel-function integrals. Proposition 1 is the genuinely new piece — van Dantzig pairs for all power semicircle laws with index alpha > -1/2, obtained by reading von Lommel as the Laplace transform of the first hitting time of a Bessel process. The subordination argument is clean and goes beyond the range covered by Lukacs. Proposition 2 is not new as a statement (they explicitly acknowledge equivalence with Cho–Yun), but the proof is considerably shorter: identify the Mellin transform with a squared Bessel integral via Weber–Schafheitlin, then use hypergeometric asymptotics for the non-existence half. That is a real expository contribution. The appendix proof of Weber–Schafheitlin from the Fresnel and Selberg integrals is elegant and self-contained, and the quasi-infinite divisibility remark is a nice aside.

Soft spots: the sharp statement of the paper, Proposition 3, is not self-contained, and the stress-test concern is valid. The proof asserts that the cross-section D_{t,a,b} is a closed segment with left endpoint x_t in (a,m] by citing Proposition 2, but Proposition 2 only settles sums below threshold, min(c,d) ≤ a, and min(c,d) ≥ m. For t above threshold it does not rule out existence with a < min(c,d) < m. That gap is exactly where the omitted boundary analysis of Remark (b) and Theorem 4.2 of the 'to appear' paper [6] are needed. So (9) as an iff statement is conditional on an external proof the reader cannot check. If that theorem holds, Proposition 3 is fine; if not, Proposition 2 still stands on its own.

The reference to [6] is handled honestly, and the citation pattern looks normal. The paper does not overclaim: it flags the equivalence with [5] and the dependence on [6]. It is aimed at specialists in special functions and infinite divisibility. I would send it to a serious referee; the referee should ask for a proof of the boundary case or a clear statement that Proposition 3 is a conditional corollary of [6]. With that, it is publishable as a short note.

Referee Report

3 major / 4 minor

Summary. The paper studies two probabilistic consequences of classical Bessel-function integrals. Section 2 uses the von Lommel integral representation to construct explicit van Dantzig pairs: for each α > -1/2, the reciprocal of the characteristic function of the power semicircle law is shown to be the characteristic function of a Brownian-subordinated first hitting time of a Bessel process. Section 3 studies positive random variables with 'moments of Gamma type', that is, Mellin transforms that are ratios of Pochhammer symbols with two denominator parameters. Proposition 2 gives a sufficient condition and separate necessary conditions for existence of such distributions, built on the Weber-Schafheitlin integral, and proves an equivalence with non-negativity of a certain 1F2 hypergeometric function. Proposition 3 states that the existence region is bounded by a continuous non-increasing curve f_{a,b}, with a linear portion and a tail asymptotic to a. The appendix gives self-contained proofs of the von Lommel and Weber-Schafheitlin formulas, deriving the latter from the Selberg integral.

Significance. The van Dantzig construction in Proposition 1 is explicit and elegant, and the appendix proof of the Weber-Schafheitlin formula via the Selberg integral is a nice self-contained contribution. Proposition 2 appears to be a genuine new family of Gamma-type moments with a signed spectral measure, and the connection with 1F2 positivity is valuable; the final reformulation of the Askey-Szego problem is suggestive. The exact boundary result in Proposition 3 would be the sharpest claim of the paper, but its proof is currently conditional on an external to-appear theorem and on an omitted boundary analysis. If these gaps are filled, the paper would make a solid contribution to the existence problem for Gamma-type distributions.

major comments (3)

[§3.3, Proposition 3 and Remark (b)] The proof of Proposition 3 asserts that Proposition 2 shows D_{t,a,b} = D_{a,b} ∩ {c+d=t} is a closed segment with lower endpoint x_t ∈ (a, m], where m = min(2a+b, a+1/2). Proposition 2(a) establishes existence only under min(c,d) ≥ m, and Proposition 2(b) excludes existence only when c+d < 3a+b+1/2 or min(c,d) ≤ a. The intermediate case c+d = 3a+b+1/2 and a < min(c,d) < m is exactly the boundary case deferred in Remark (b) with 'We omit details'. Since this case is not resolved in the manuscript, the segment structure of D_{t,a,b}, and therefore the very definition of f_{a,b} on the linear part, is not established. The non-emptiness statement for t ≥ 3a+b+1/2 is fine; the gap is the exact left endpoint of the segment.
[§3.3, proof of Proposition 3(b)] Part (b) of Proposition 3, including the asymptotic f_{a,b}(u) → a as u → ∞, is justified by the sentence 'it suffices to combine (8) and Theorem 4.2 in [6]', where [6] is cited as 'to appear'. The theorem is not stated in the manuscript and its hypotheses are not verified there. This external result is doing the load-bearing work of excluding existence above the boundary and determining the tail. Proposition 3 as written is therefore a conditional statement. The authors should either state and prove the needed form of Theorem 4.2, or clearly mark Proposition 3(b) as conditional on the appearance of [6].
[§3.3, proof of Proposition 3, first paragraph] The proof states without argument that the set D_{a,b} is closed. This is not immediate from the definition of existence of a distribution with a prescribed Mellin transform on a strip, and it is used to obtain the continuity of x_t and y_t and hence of f_{a,b}. A justification can be supplied from (8) and the continuity of the 1F2 function in its parameters, but the manuscript should include the argument.

minor comments (4)

[§3.3, notation] The notation D[a b/(c,d)-] is used in Proposition 2 without being defined; the proof shows that the denominator is (c)_s(d)_s, which conflicts with the general convention in (1) where the second denominator set appears as (d)_{-s}. Please define the new notation explicitly.
[§3.3, proof of Proposition 2(b)] The asymptotic display writes cos(sqrt(x) + νπ/2) for the leading term of the 1F2 function; the standard DLMF 16.11.8 asymptotics for 1F2(A;B;C;-x) contain 2 sqrt(x) (unless the variable has been rescaled). Please check the phase and the definition of ν.
[§3.3, Proposition 3(b)] In the formula for f_{a,b}(u) there is a stray bracket in 'u > max(2a+b,a+1/2)]', and the interval notation ']a, a+(a+b)/2(u-a)]' mixes French and English conventions; use a uniform notation.
[References] Reference [6] is listed as 'To appear in Constructive Approximation'; please update the reference if it has appeared and include the precise statement of the theorem used.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the existence and non-existence arguments are built from independent Bessel-integral and hypergeometric-asymptotic inputs; Proposition 3's tail boundary is delegated to an external paper, which is a support gap, not a circular reduction.

full rationale

The paper does not exhibit circular reasoning. Proposition 2(a) constructs the Gamma-type moments explicitly: at the extremal parameters (2a+b, a+1/2), the density is written as a nonnegative multiple of x^{a-3/2} J_{a+b-1/2}^2(x^{-1/2}), and Lemma B (Weber-Schafheitlin, proved self-contained in the appendix from von Lommel, Fresnel and Selberg) gives the required Mellin transform. Extension to other parameter pairs uses independent convolution/concatenation rules for Gamma-type moments. Proposition 2(b) uses the Meijer G-function identification and the asymptotic formula 1F2(a+b; c+b, d+b; -x) = x^{nu/2}/sqrt(pi) cos(sqrt(x)+nu pi/2)+o(x^{nu/2}), which is an independent condition for negativity rather than an assumption of the target distributions. Equation (8) is a Mellin-transform equivalence, not a definitional coincidence. Proposition 3 is a parametrization of the existence boundary: f_{a,b} is defined from the geometry of D_{a,b}, and the linear regime (a) follows from Proposition 2; the tail bound (b) is explicitly delegated to the external paper [6, Theorem 4.2], while Remark (b) says the boundary-case analysis is omitted. This is a correctness and self-containedness concern, not circularity, particularly since [6] is by Cho, Chung and Yun rather than by the present authors, and no load-bearing step in the paper reduces to a self-citation. There is no fitted parameter renamed as a prediction and no equation whose conclusion is its input by construction.

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

The central claims rest on classical special function identities and external probabilistic results, all cited. There are no free parameters fitted to data and no new postulated entities. The main non-self-contained input is the to-appear result [6] used for Proposition 3.

assumptions (6)

standard math Hadamard factorization of the Bessel function J_alpha into an infinite product over its positive zeros (from [1], used in proof of Proposition 1).
Invoked in Section 2 to express 1/hat h_alpha(it) as a product over j_{alpha,n}; standard special-function background.
domain assumption Kent's formula (3.8) in [11] for the Laplace transform of the first hitting time T_alpha of a Bessel process of dimension 2alpha+1.
Used in Proposition 1 to identify the Laplace transform of T_alpha with the product over Bessel zeros; this is a probabilistic result from the literature.
standard math Selberg integral and Fresnel integral identities from [1] used in the proof of Lemma B (Weber-Schafheitlin).
These classical identities are assumed as background in the appendix; they are standard and cited.
domain assumption Sufficiency of conditions (3) for p=n=2 distributions from [4] and [7], used in Proposition 2(a).
The paper relies on the known existence of D[(2a+b,a+1/2)-; (c,d)-] for the intermediate cases; this is cited from the literature.
domain assumption Theorem 4.2 in [6] (cited as to appear) determining the exact boundary f_{a,b} in Proposition 3(b).
Proposition 3(b) depends on this external result, which is not proved in the preprint; also Remark (b) omits details for a boundary case.
standard math Meijer G-function identities and 1F2 asymptotics from DLMF [15] (Formulas 16.18.1, 16.19.2, 16.11.8), used for the equivalence (8) and the non-existence proof.
The negativity of the leading oscillatory term in the 1F2 asymptotics is used to show the hypergeometric function takes negative values; these are standard DLMF formulas.

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Pith. "Pith review of On some new moments of Gamma type." pith.science (2026). https://pith.science/paper/IMLATHTF

@misc{pith2026190803428,
  author       = {Pith},
  title        = {Pith review of: On some new moments of Gamma type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IMLATHTF}},
  note         = {Machine review of arXiv:1908.03428}
}

read the original abstract

We investigate certain positive random variables having moments of Gamma type. Some necessary and some sufficient conditions are given for their existence. In particular, we observe that the Weber-Schafheitlin formula for the Bessel function makes it possible to construct non-trivial moments of Gamma type having a signed spectral measure.

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[6]

Cho, S.-Y

Y.-K. Cho, S.-Y. Chung and H. Yun. Rational extension of Newton diagram for the positivity of 1F2 hypergeometric functions and Askey-Szeg¨ o problem. To appear in Constructive Approximation

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[5]

Cho and H

Y.-K. Cho and H. Yun. Newton diagram of positivity for 1F2 generalized hypergeometric functions. Int. Transf. Spec. Funct. 29 (7), 527-542, 2018

work page 2018
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G. E. Andrews, R. Askey and R. Roy. Special functions. Cambridge University Press, Cambridge, 1999

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Arizmendi and V

O. Arizmendi and V. P´ erez-Abreu. On the non-classical inﬁnite divisibility of power semicircle distributions. Comm. Stoch. Anal. 4 (2), 161-178, 2010

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W. N. Bailey. Some inﬁnite integrals involving Bessel functions. Proc. London Math. Soc. 40 (2), 37-48, 1936

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Chamayou and G

J.-F. Chamayou and G. Letac. Additive properties of the Dufres ne laws and their multivariate extension. J. Theoret. Probab. 12 (4), 1045-1066, 1999

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D. Dufresne. G distributions and the beta-gamma algebra. Elec. J. Probab. 15, No. 71, 2163-2199, 2010

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S. Janson. Moments of Gamma type and the Brownian supremum p rocess area. Probab. Surveys 7, 1-52, 2010

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NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov
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Wolfram Research, Meijer G. http://functions.wolfram.com/HypergeometricFunctions/MeijerG/ V akgroep Wiskunde, Vrije Universiteit Brussel, Gebauw G, P leinlaan 2, 1050 Elsene, Belgium. Email: tetyana.kadankova@vub.be Laboratoire Paul Painlev ´ e, Universit ´ e de Lille, Cit ´ e...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [6]

Cho, S.-Y

Y.-K. Cho, S.-Y. Chung and H. Yun. Rational extension of Newton diagram for the positivity of 1F2 hypergeometric functions and Askey-Szeg¨ o problem. To appear in Constructive Approximation

work page

[2] [5]

Cho and H

Y.-K. Cho and H. Yun. Newton diagram of positivity for 1F2 generalized hypergeometric functions. Int. Transf. Spec. Funct. 29 (7), 527-542, 2018

work page 2018

[3] [1]

G. E. Andrews, R. Askey and R. Roy. Special functions. Cambridge University Press, Cambridge, 1999

work page 1999

[4] [2]

Arizmendi and V

O. Arizmendi and V. P´ erez-Abreu. On the non-classical inﬁnite divisibility of power semicircle distributions. Comm. Stoch. Anal. 4 (2), 161-178, 2010

work page 2010

[5] [3]

W. N. Bailey. Some inﬁnite integrals involving Bessel functions. Proc. London Math. Soc. 40 (2), 37-48, 1936

work page 1936

[6] [4]

Chamayou and G

J.-F. Chamayou and G. Letac. Additive properties of the Dufres ne laws and their multivariate extension. J. Theoret. Probab. 12 (4), 1045-1066, 1999

work page 1999

[7] [7]

Dufresne

D. Dufresne. G distributions and the beta-gamma algebra. Elec. J. Probab. 15, No. 71, 2163-2199, 2010

work page 2010

[8] [8]

S. Janson. Moments of Gamma type and the Brownian supremum p rocess area. Probab. Surveys 7, 1-52, 2010

work page 2010