{"id":"49507ab1-5e6d-45e3-b0ea-4d15b96f6cee","arxiv_id":"1908.03428","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"New existence and non-existence criteria are given for distributions with Gamma-type moments, via Bessel and hypergeometric function analysis.","lead":"This paper constructs new examples of random variables whose moments follow a Gamma-type formula, using classical Bessel function identities that the authors re-prove. It also builds new pairs of characteristic functions satisfying the van Dantzig property.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3's exact boundary is unsupported: it relies on the external to-appear Theorem 4.2 of [6], and the proof does not rule out existence with a<min(c,d)<m.","rationale":"The reader's CONDITIONAL verdict is appropriate. The paper's main self-contained contribution, Proposition 2, is supported by a correct-looking Bessel-function computation: Lemma B is proved from Lemma A, the Fresnel integral and the Selberg integral, and the Mellin transform calculation matches the claimed Gamma-type moments. The non-existence direction via the 1F2 asymptotic is also coherent, and the connection with [5] is explicitly disclosed. The load-bearing weakness is Proposition 3, the exact boundary characterization. Its proof depends on the external paper [6] for the asymptotic shape of f_{a,b}, and the equality-boundary case needed to establish even the linear part of the boundary is explicitly deferred in Remark (b). Moreover, the claim that each diagonal cross-section is a closed segment with endpoint in (a,m] is not derivable from Proposition 2 alone, because Proposition 2 only covers min(c,d)≥m on the existence side and min(c,d)≤a on the non-existence side. Thus the remaining interval (a,m) is exactly the unproved regime. This makes Proposition 3 conditional rather than established. It does not undermine Proposition 2, and it does not suggest the main construction is wrong; it means the sharpest claim should not be accepted as fully proved in this preprint. The concrete numerical sign check proposed would settle the specific omitted boundary case and provide a direct test of the boundary curve.","tokens_in":10860,"tokens_out":31730,"duration_ms":313128,"concrete_test":"Numerically settle the disputed boundary: for a=1, b=1, c=1.2, d=3.3, which satisfies c+d=4.5=3a+b+1/2 and min(c,d)=1.2<1.5=m, evaluate 1F2(a+b; c+b, d+b; -x) at high precision on a fine grid up to large x, e.g. with mpmath. The omitted boundary case claims this function is negative for some x, while Proposition 2 is silent in this regime. If the function is nonnegative everywhere, Proposition 3's boundary is wrong; if it is negative, repeat the same sign check for a large-u point on the (b) branch, such as a=1,b=1,u=10, to test the claimed asymptotic bound. This directly tests the load-bearing gap that Proposition 2 alone cannot close.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 3 is the sharpest claim of the paper, but its proof is not self-contained and contains a structural gap that Proposition 2 does not fill. In the proof of Proposition 3, the authors assert that Proposition 2 shows each cross-section D_{t,a,b}=D_{a,b}∩{c+d=t} is a closed segment with left endpoint x_t∈(a,m], where m=min(2a+b,a+1/2). Proposition 2(a) gives existence only when min(c,d)≥m, and Proposition 2(b) gives non-existence only when c+d<3a+b+1/2 or min(c,d)≤a. For t>3a+b+1/2, Proposition 2 therefore does not exclude existence with a<min(c,d)<m; that exclusion is exactly the omitted equality-boundary analysis of Remark (b) and the external Theorem 4.2 of [6]. The final step of the proof of Proposition 3(b) is literally 'it suffices to combine (8) and Theorem 4.2 in [6]', with [6] cited as 'to appear'. If Theorem 4.2 is inapplicable, or if the omitted boundary case has the opposite sign, then the if-and-only-if characterization (9) and the linear boundary asserted in Proposition 3(a) are unsupported. Proposition 2 itself would remain valid because its proof uses only the self-contained Bessel analysis and the asymptotic formula; the risk is specifically to Proposition 3 and to the claimed complete boundary.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11140,"tokens_out":28098,"duration_ms":269753,"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":[{"comment":"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.","section":"§3.3, Proposition 3 and Remark (b)"},{"comment":"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].","section":"§3.3, proof of Proposition 3(b)"},{"comment":"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.","section":"§3.3, proof of Proposition 3, first paragraph"}],"minor_comments":[{"comment":"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.","section":"§3.3, notation"},{"comment":"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.","section":"§3.3, proof of Proposition 2(b)"},{"comment":"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.","section":"§3.3, Proposition 3(b)"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The core self-contained result (Proposition 2) appears correct, and the appendix is a strength. The main risk is the dependence of Proposition 3 on the external to-appear Theorem 4.2 of [6] and on the omitted equality-boundary case in Remark (b). I also suggest asking the authors to clarify the notation D[a b/(c,d)-], which is easy to misread as having denominator (c)_s(d)_{-s} when the proof uses (c)_s(d)_s."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nSoft 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.\n\nThe 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.","headline":"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.","tokens_in":11685,"tokens_out":3201,"would_cite":true,"duration_ms":30040,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["33C10","33C20","60E07","60E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Bessel functions","Moments of Gamma type","van Dantzig problem","generalized hypergeometric functions","quasi-infinite divisibility","Weber-Schafheitlin integral","Meijer G-function","Mellin transform"],"falsifier":"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.","tokens_in":23,"feed_emoji":"📐","tokens_out":14186,"duration_ms":191824,"temperature":0.7,"pith_summary":"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.","feed_headline":"One curve decides which Gamma-type moments exist","feed_subtitle":"Weber-Schafheitlin turns the moment problem into hypergeometric non-negativity, with an explicit linear boundary.","key_machinery":"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 $f_{a,b}$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the $D[a b; c d]$ notation and the concatenation and simplification rules used to build the full existence region.","marker":"[4]"},{"why":"Supplies the Meijer $G$-function formula identifying the density, the transformation to the ${}_1F_2$ condition, and the asymptotic formula 16.11.8 used for non-existence.","marker":"[15]"},{"why":"States the theorem the paper proves equivalent to its Proposition 2, connecting moments to the positivity diagram of ${}_1F_2$.","marker":"[5]"},{"why":"Contains Theorem 4.2, which supplies the large-parameter branch of the boundary in Proposition 3(b); the result is used without proof here.","marker":"[6]"},{"why":"Provides the product expansion of $J_\\alpha$ over its zeros and the gamma-function identities applied throughout the proofs.","marker":"[1]"},{"why":"Gives the Laplace transform of the first hitting time of a Bessel process, used for the van Dantzig pairs.","marker":"[11]"},{"why":"Supplies the Brownian subordination argument converting the hitting-time transform into a reciprocal characteristic function.","marker":"[17]"},{"why":"Sets the van Dantzig framework and the criterion used to locate the self-reciprocal pairs.","marker":"[14]"}],"fun_headline_variants":["A single curve tells when Gamma-type moments exist","Bessel squared density marks the extremal moment","Gamma moments: existence reduces to one non-negative curve","The boundary for Gamma moments is an explicit line","Weber-Schafheitlin draws the moment-existence curve"],"cache_read_input_tokens":13824,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["A single curve tells when Gamma-type moments exist","Bessel squared density marks the extremal moment","Gamma moments: existence reduces to one non-negative curve","The boundary for Gamma moments is an explicit line","Weber-Schafheitlin draws the moment-existence curve"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000248,"raw_usage":{"total_tokens":1526,"prompt_tokens":907,"completion_tokens":619,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":544}},"tokens_in":523,"tokens_out":619,"duration_ms":6597,"temperature":1.0,"reasoning_tokens":544,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:13:41.198729+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Chamayou and G","cited_arxiv_id":null,"evidence_quote":"Introduces the $D[a b; c d]$ notation and the concatenation and simplification rules used to build the full existence region."},{"cited_title":"http://dlmf.nist.gov","cited_arxiv_id":null,"evidence_quote":"Supplies the Meijer $G$-function formula identifying the density, the transformation to the ${}_1F_2$ condition, and the asymptotic formula 16.11.8 used for non-existence."},{"cited_title":"Cho and H","cited_arxiv_id":null,"evidence_quote":"States the theorem the paper proves equivalent to its Proposition 2, connecting moments to the positivity diagram of ${}_1F_2$."},{"cited_title":"Cho, S.-Y","cited_arxiv_id":null,"evidence_quote":"Contains Theorem 4.2, which supplies the large-parameter branch of the boundary in Proposition 3(b); the result is used without proof here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the product expansion of $J_\\alpha$ over its zeros and the gamma-function identities applied throughout the proofs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Laplace transform of the first hitting time of a Bessel process, used for the van Dantzig pairs."},{"cited_title":"Roynette and M","cited_arxiv_id":null,"evidence_quote":"Supplies the Brownian subordination argument converting the hitting-time transform into a reciprocal characteristic function."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Sets the van Dantzig framework and the criterion used to locate the self-reciprocal pairs."}],"review_version":1}