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Approximating the coefficients of the Bessel functions

T0 review · 2 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Scaled Bessel coefficients are equivalent to Dunkl bilinear-form limits

desk verdict The type D and BC equivalences are new and structurally coherent, but the converse directions rest on remainder estimates that are sketched rather than proved; worth refereeing, not yet a finished paper. read the letter →

arxiv 2510.10370 v6 pith:RBSEEWWU submitted 2025-10-11 math.CA math-phmath.COmath.MPmath.OAmath.PR

Andrew Yao This is my paper · ORCID
classification math.CAmath-phmath.COmath.MPmath.OAmath.PR MSC 33C5205A1841A6046L54
keywords BesselfunctionsDunkloperatorsbilinearformnoncrossingpartitionsrootsystemsA_NBC_ND_Nasymptoticexpansionsfreeconvolutionpower-sumsymmetric
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 sets out to prove that, in the large-N limit, the asymptotic coefficients of Bessel functions attached to the type A, BC, and D root systems are governed by a single equivalence: scaling the log-coefficients of an exponential formal power series by powers of θN or θ₀N yields the same information as scaling evaluations of the Dunkl bilinear form under power-sum differential operators. The explicit limiting formulas are weighted noncrossing-partition products, with type BC adding a (1+c)^{o(π)} factor and type D carrying an extra product ∏(1+2(j−1)θ) in its odd-degree sector. A sympathetic reader would care because these equivalences convert the difficult task of estimating Bessel coefficients into checkable moment asymptotics, and they yield free-convolution and rectangular-free-convolution limits for the associated measures. The paper also proves analogues in the θN→c regime, generalizing earlier statements and adding new type D results.

What carries the argument

The central object is the Dunkl bilinear form [f,g]=[1]D(f)g, evaluated on power-sum symmetric functions p_λ. Its leading-order behavior is encoded in triangular infinite matrices whose entries are sums over noncrossing partitions, weighted by block sizes, powers of 2, and (for BC) factors (1+c)^{o(π)}. The proof controls all subleading terms with remainder-degree bounds, so that division by the appropriate powers of θN and N leaves only the leading triangular system, whose invertibility drives the equivalence.

What would settle it

Choose N=2, θ_N=N, and evaluate [p_(2),p_(1,1)]_{A_{N-1}(θ)} directly from the definition of the Dunkl operators. Compare the exact value with the main term in Theorem 6.1: if the difference, divided by (θ_N N)^2 N^2, does not tend to zero as N→∞, the theorem's remainder bound is false. More generally, any explicit pair (λ,ν) and sequence θ_N for which the claimed degree bounds on the remainder fail would falsify the equivalence.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: for exponential formal power series F_N = exp(Σ c_λ(N)p_λ), the scaled log-coefficients c_λ(N)/(θN)^{ℓ(λ)} converge if and only if the scaled Dunkl bilinear-form evaluations [1]∏ᵢ(Σ_jD_j)^{νᵢ}F_N / ((θN)^{|ν|}N^{ℓ(ν)}) converge. The common limit is the product over blocks of noncrossing partitions of |B|c(|B|), with type BC contributing (1+c)^{o(π)} and type D splitting into even and odd sectors; the odd sector is normalized by ∏_{j=1}^N(1+2(j−1)θ). Equivalent statements are proved for the θN→c regime, with non-degeneracy conditions such as c≠−1 for BC and c not a negative integer for the converse directions.

Load-bearing premise

The load-bearing premise is the leading-order expansion with remainder bounds: after removing the main term, every remainder monomial is lower order in x and y by at least one power, so it vanishes after division by (θN)^{|ν|}N^{ℓ(ν)}. Every converse direction is an induction that erases those remainders, so an off-by-one power of N or θ would collapse the equivalence.

Editorial extensions

If this is right

  • If the theorem is right, computing the asymptotic coefficients of J^A, J^{BC}, and J^D reduces to computing noncrossing-partition sums: for each ν the limit is ∏ᵢ Σ_{π∈NC(νᵢ)} ∏_{B∈π} |B|c(|B|).
  • Assuming the conjectural integral representation for products of Bessel functions, the paper's corollaries give weak convergence of the associated measures to the free convolution (type A), the rectangular free convolution (type BC), and a symmetrized free convolution (type D).
  • In the θN→c regime the same machinery works under explicit non-degeneracy conditions, so the paper covers both |θN|→∞ and θN finite in one framework.
  • For fixed-degree coefficients, the paper obtains asymptotic inverses of the Dunkl-bilinear-form matrices and, under moment bounds, uniform convergence of Bessel functions on compact sets.
  • The type D odd-degree sector is genuinely non-polynomial: its leading terms contain ∏_{j=1}^{N-k}(1+2(j−1)θ), so coefficient asymptotics there involve gamma-function-type factors.

Reading between the lines

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

  • The paper's leading-order matrices are triangular; an implication it does not spell out is that the same triangular structure should organize finite-N corrections, so one could test the remainder bounds by computing exact small-example bilinear forms and checking the stated x- and y-degree decay.
  • At c=−1 for type BC, the converse direction is deliberately excluded, suggesting a genuine loss of information at that parameter value; a natural extension would classify what extra data would restore equivalence there.
  • The odd-degree type D prefactor ∏(1+2(j−1)θ) hints at an interpretation of that sector as moments of a signed measure with a gamma-type density; checking the sign pattern for small N would be a direct test.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the asymptotics of the Dunkl bilinear form [pλ, pν] for type A, BC, and D root systems, with multiplicity parameters varying in N. Its main theorem (Theorem 1.1) asserts equivalences between convergence of scaled logarithmic coefficients cλ(N)/(θN)^{ℓ(λ)} and convergence of scaled Dunkl evaluations [1] D(pν) F_N / ((θN)^{|ν|} N^{ℓ(ν)}), with limits given by explicit products of noncrossing partition sums. Analogous equivalences are proved in the finite-θN regimes (Theorems 6.17, 7.8, 8.26), generalizing [BGCG22], [Xu25], and [Yao25]. The proofs pass through leading-order expansions (Theorems 6.1, 7.1, 8.16, 8.23), triangular matrices MA, MBC, MD, and explicit matrix inversion. Applications include free-convolution corollaries conditional on Conjecture 1.6, uniform convergence results, and coefficient asymptotics for the Bessel functions themselves.

Significance. If the results are fully established, this is a significant contribution: it provides a unified, parameter-free description of coefficient asymptotics for the Bessel functions J^R_a in several asymptotic regimes and for three root systems, with all limiting constants expressed as sums over noncrossing partitions rather than fitted quantities. It generalizes earlier work, answers a question raised in [BGCG22], and treats the type-D odd-degree sector, including the gamma-type denominator ∏_{j=1}^N (1+2(j−1)θ), as a genuinely new feature. The paper is also commendable for its explicit triangularity lemmas and concrete matrix inversions, which make the leading-order formulas independently checkable. The main caveat is the rigor of the remainder estimates, on which the converse directions of the equivalences depend.

major comments (2)
  1. [Section 6.1, Theorems 6.1, 7.1, 8.16, 8.23] The stated bounds on the remainder R are load-bearing for the central claim. Every converse direction in Theorems 6.10, 7.4, 8.21, and 8.26 proceeds by discarding R/((θN)^{|ν|} N^{ℓ(ν)}) (or the analogous scaled denominator), and an off-by-one in the claimed x- or y-degree would leave a surviving monomial and break the triangular-matrix inversion. As written, the proof is not complete: the first proof of Theorem 6.1 is explicitly said to "lack some details"; the second proof says "It is not challenging to determine that the remainder term R is a polynomial..." and defers to [Yao25, Section 6]; Theorems 7.1 and 8.16 make similar assertions without carrying out the case analysis. Since the leading-order coefficients and diagonal values spot-check correctly, I am not claiming the formula is false; rather, the equivalence theorems are not yet established without a complete proof of these rem
  2. [Section 8.3, Theorems 8.16 and 8.23; Corollaries 8.22 and 8.26] The type-D odd-sector argument is more delicate than the type-A case because the leading term contains the N-dependent factor ∏_{i=1}^{N-k}(1+2(i−1)θ), and the applications divide by the full product ∏_{j=1}^{N}(1+2(j−1)θ). The proof of Lemma 8.20 also uses a signed cancellation between contributions corresponding to ∂1 and −2θ∂1, and Lemma 8.24 uses an analogous cancellation involving (1+2Nθ)^{-1}. These cancellations are stated in words rather than proved by an explicit bijection or generating-function identity. Since Lemma 8.20 and Lemma 8.24 are load-bearing for part (C) of Theorem 1.1, the paper should give a fully detailed proof of these cancellations and of the remainder bounds after division by the gamma-type product.
minor comments (5)
  1. [Section 6.1, Theorem 6.1] The notation [∏_{l=1}^{ℓ(ν)} x^{ν_l}] is used in the leading-order formula and in the definition of MA, but it is not defined. It appears to denote coefficient extraction; please define it explicitly in Section 2.
  2. [Theorems 6.1, 7.1, 8.16] The variables x and y in the remainder bounds R(N,θ) are not explicitly identified as N and θ. State this identification in each theorem statement to make the degree bounds unambiguous.
  3. [Lemma 6.9] The lemma calls MA upper-triangular, but this depends on the ordering of Γ[k] (e.g., by decreasing ℓ(λ)). Make the ordering explicit so that the triangularity claim is unambiguous.
  4. [Theorems 8.10 and 8.12] Statements such as "Theorem 6.1 with A_{N-1}(θ) replaced by D_N(θ) and θ replaced by 2θ is true" are not self-contained. Please restate the full theorem or state precisely which substitutions are used in the proof.
  5. [Corollary 6.12] The proof invokes [Yao25, Theorem 1.6] with a Bessel generating function that may not exist in the measure-theoretic setting. Since the argument is formal, please explain why the proof of that theorem applies in the formal power series setting without the existence assumption.

Circularity Check

0 steps flagged · score 2.0 of 10

No constructional circularity: the equivalence is proved by inverting explicitly computed triangular matrices; the main caveat is a sketched remainder bound and a supporting self-citation, not a definitional shortcut.

full rationale

The central equivalence in Theorem 1.1 is not circular. Conditions (a)/(e)/(f) concern scaled log-coefficient limits, while (b)/(g)/(h) concern scaled Dunkl bilinear-form evaluations; the proofs in Theorems 6.10, 7.4, and 8.21 expand [p_λ,p_ν] into an explicit leading term plus a remainder R, define fixed triangular matrices M_A, M_BC, M_D, and invert them degree-by-degree. The limiting right-hand side is not set equal to the left-hand side by construction; it is derived from the operator combinatorics in Lemmas 6.4-6.7, 7.2, and 8.18-8.20. No parameter is fitted to data and no known result is merely renamed. The self-citations to [Yao25] provide an alternate type-A proof and 'more details of a similar argument' for the remainder estimate; they are prior work, not an assumption of Theorem 1.1. The genuine weakness is that Theorem 6.1's remainder-degree bounds are asserted with 'It is not challenging to determine that the remainder term R is a polynomial...' and the first proof is explicitly said to 'lack some details'; since every converse direction deletes R after scaling, this is a real rigor gap. But a deferred, sketched estimate is a completeness issue, not a circular reduction. The paper also recovers and generalizes [BGCG22], [Xu25], [BR25], so the central content is externally anchored. Score 2 reflects the nontrivial self-citation and the incompleteness of the remainder proof, not constructional circularity.

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

The derivations introduce no fitted numbers; the limiting quantities (c_λ, c, c_0, c_1) are bookkeeping limits under stated regime hypotheses. The real costs are: standard Dunkl/Bessel existence theory, the explicit invertibility exclusions used in the converse directions, the remainder bounds in the leading-order expansions (argued by case analysis and partly by reference to the author's earlier [Yao25]), and — for the free-convolution applications only — the open Conjecture 1.6. No new particles, forces, or entities are invented; e(x) = x_1···x_N and the W-polynomials are mathematical constructions, not postulates.

assumptions (6)
  • standard math Existence, uniqueness, and holomorphy of the nonsymmetric and symmetric Dunkl eigenfunctions E and J for θ ∈ Θ(R) (Opdam), and Rösler's integral representation for nonnegative multiplicities.
    Invoked throughout: the functions F_N in Theorem 1.1 are instances of J^R_a; the measure applications (Corollaries 1.7/1.8, Section 10) use the Rösler representation (Theorem 1.4) and require θ, θ0, θ1 ≥ 0.
  • standard math Invertibility (θ ∈ Θ(R)) of the Dunkl operator system, equivalently invertibility of the graded matrices M^{i;L}.
    Definition 2.4 and Theorems 2.5/2.8 require it; Section 9 inverts the [pλ,pν] matrices, and the asymptotic inversion lemmas (9.1/9.3/9.5/9.7/9.9–9.14) impose it. In the θN→c regime this becomes explicit exclusions ('c not a negative integer', etc.).
  • ad hoc to paper The remainder bounds in the leading-order expansions, e.g., Theorem 6.1: R has x-degree ≤ k+ℓ(λ)−ℓ(ν)−1 and y-degree ≤ k−ℓ(ν).
    Load-bearing premise: the (b)⇒(a) inductions in Theorems 6.10/6.17/7.4/8.21/8.26 delete summands using these bounds. Proven by case analysis in §6.1 ('It is not challenging to determine that the remainder term R...'), with type-A details partly deferred to [Yao25, Section 6].
  • domain assumption Conjecture 1.6: existence of a nonnegative product Bessel measure μ^{R(θ)}_{a1,a2}.
    Assumed explicitly in Corollaries 1.7, 1.8, 10.2 and the Dyson-Brownian-motion example; without it the free-convolution statements are unproven. Transparent in the paper, but load-bearing for the probabilistic headline.
  • domain assumption Asymptotic invertibility exclusions for the converse directions: c ≠ −1 (BC, Theorem 7.4/1.1(B)); c not a negative integer (A, Theorem 6.17; W^A([l])(c) = ∏_{j=1}^{l−1}(c+j)); 2c not a negative integer (D, Theorem 8.26); and ∏_{j=1}^N(1+2(j−1)θ) ≠ 0 for the D odd-degree part.
    These are stated hypotheses, not hidden assumptions, but they genuinely restrict the regimes where the equivalences are claimed; they match the diagonal formulas in Lemmas 6.16, 7.3, 8.25.
  • domain assumption θ, θ0, θ1 ≥ 0 whenever measure-theoretic or uniform-convergence conclusions are drawn (Theorem 10.8, Corollaries 1.7/1.8).
    Needed for Rösler's integral representation and the L² bound in Lemma 10.5; the paper notes that generalization to complex θ with D invertible is future work (§1.5).

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Pith. "Pith review of Approximating the coefficients of the Bessel functions." pith.science (2026). https://pith.science/paper/RBSEEWWU

@misc{pith2026251010370,
  author       = {Pith},
  title        = {Pith review of: Approximating the coefficients of the Bessel functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RBSEEWWU}},
  note         = {Machine review of arXiv:2510.10370}
}
abstract

We determine equivalent conditions between the asymptotic coefficients of the Bessel generating functions of a sequence of probability measures and the asymptotic expected values of power sums when their inputs are sampled from these measures. We establish these conditions over the $|\theta N| \rightarrow\infty$ regime for the type A and D root systems and over the $|\theta_0 N|\rightarrow \infty, \frac{\theta_1}{\theta_0 N}\rightarrow c\in\mathbb{C}$ regime for the type BC root system. We also establish equivalent conditions over the $\theta N \rightarrow c\in\mathbb{C}$ regime for the type A and D root systems and over the $\theta_0 N\rightarrow c_0\in\mathbb{C}, \frac{\theta_1}{\theta_0 N}\rightarrow c_1\in\mathbb{C}$ regime for the type BC root system that generalize existing results. Furthermore, we determine the asymptotics of the coefficients of the Bessel functions over the regimes that we have mentioned.

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