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On series expansions of zeros of the deformed exponential function

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Pith's one-line read Every coefficient in the q-expansion of a normalized zero of the deformed exponential function is a rational function of the zero index, with one explicit integer denominator and an integer-polynomial numerator.

desk verdict The paper is correct and publishable; the stress-test counterexample comes from reading A0 as A1, and the real fixes are a sign typo in Lemma 3, the sketch-level (18), and the unstated precision of the n<=300 positivity check. read the letter →

arxiv 2412.02462 v1 pith:WWNGYSER submitted 2024-12-03 math.CA math.COmath.CV

Alexey Kuznetsov This is my paper · ORCID
classification math.CAmath.COmath.CV MSC 30C1530E1534K06
keywords deformedexponentialfunctionpowerseriessumofdivisorssymboliccomputationzerosentirefunctionsrationalcoefficientsJacobitripleproductpositivityconjectures
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 deformed exponential function $f(x)=\sum_{n\ge1} x^n q^{n(n-1)/2}/n!$ has infinitely many negative zeros for $0

What carries the argument

The argument is carried by the fixed-point equation $w_k(q)=1+qF_k(w_k(q);q)$ for the normalized zero variable $w_k(q)=-x_k(q)/(kq^{1-k})$, together with a class $\mathcal{A}$ of formal power series whose $q^n$ coefficients become integer polynomials after multiplication by $Q_n(k)$. The class $\mathcal{A}$ is closed under multiplication and division, so the fixed-point iteration (and the identity $w_k(q)\times(1/w_k(q))=1$) preserves the structure; this yields Theorem 3(i) without any asymptotic input. The leading-coefficient part rests on the identity $A_0(q)=\sum_{i\ge1}(-1)^{i+1}[i(i+1)(2i+1)/6+(2i+1)A_0(q)]q^{i(i+1)/2}$, obtained from the logarithmic derivative of Jacobi's triple product, which gives the uniform $k^{-3}$ control in Lemma 3.

What would settle it

Expand both sides of the identity $A_0(q)=\sum_{i\ge1}(-1)^{i+1}[i(i+1)(2i+1)/6+(2i+1)A_0(q)]q^{i(i+1)/2}$ through $q^{50}$ and compare coefficients; any mismatch would break Lemma 3 and, with it, the leading-coefficient formulas of Theorem 3(ii).

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Extended reading notes

Core claim

The central result is Theorem 3. For $n\ge1$ set $\gamma_{n,l}=\lfloor 2n/(l(l+1))\rfloor$ and $Q_n(k)=k^n\prod_{l\ge1}(k+l)^{\gamma_{n,l}}$, and write the normalized zero expansions as $-x_k(q)/k = 1+\sum_{n\ge1} a_{k,n}q^n$ and $k/x_k(q)=1+\sum_{n\ge1}\hat a_{k,n}q^n$. The paper proves that $a_{k,n}=P_n(k)/Q_n(k)$ and $\hat a_{k,n}=\hat P_n(k)/Q_n(k)$ for all $k,n\ge1$, where $P_n$ and $\hat P_n$ are integer polynomials that can be computed recursively. It further proves the leading coefficients: both $P_n$ and $\hat P_n$ have leading term $\sigma(n)k^{M_n-2}$ followed by $\sigma(n)(\mu_1(n)-n)k^{M_n-3}$, where $M_n=\deg Q_n$, $\sigma$ is the sum-of-divisors function, and $\mu_1(n)=\sum_l l\gamma_{n,l}$; the second coefficient is positive for all $n\ge3$. Computations for $n\le300$ confirm that $P_n(k)$ and $\hat P_n(k)$ are non-negative for all $k\in\mathbb{N}$, offering evidence for the positivity conjectures.

Load-bearing premise

The proof of the leading-coefficient formulas rests on a single external identity about the divisor-sum series; if that identity is wrong, part (ii) collapses, while the structure theorem of part (i) still stands.

Editorial extensions

If this is right

  • Because both expansions share the same denominator $Q_n(k)$, the arithmetic of every coefficient in either series is controlled by a single explicit integer polynomial.
  • The positivity conjectures become finite polynomial checks: for a fixed $n$ it suffices to show $P_n(k)\ge0$ and $\hat P_n(k)\ge0$ for all positive integers $k$, and the recursive formulas plus root bounds make each check algorithmic.
  • The leading-coefficient formula gives a quantitative description of the first corrections in the large-$k$ expansion of each zero, matching the known asymptotic expansion and extending it to all $n$.
  • The verified positivity for $n\le300$ provides concrete evidence that both series in (7) converge for $|q|<1$, since positivity of the $\hat a$-coefficients is the strongest of the three conjectures and implies the other two.
  • The explicit polynomials and their roots, with roots clustering near integers, give a detailed picture of where the numerators can fail to be positive for real $k>1$, even though they are non-negative on the integers.

Reading between the lines

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

  • A natural next step is to search for a combinatorial interpretation of the coefficients of $P_n(k+1)$: the zebra-stripe sign pattern in Figure 3 suggests hidden structure that, if understood, could turn the verified inequalities into a proof.
  • The recursive formulas (44)-(45) may allow the polynomial data to be generated in exact integer arithmetic, extending verification well beyond $n=300$ without the three-week high-precision floating-point computation.
  • The identity for $A_0(q)$ is a self-contained functional equation for the divisor-sum generating function; the same logarithmic-derivative technique might produce analogous identities for $A_j(q)$ and thereby make the third and higher leading coefficients explicit.
  • The tendency of real roots of $P_n$ and $\hat P_n$ to cluster near integers hints that the numerators are related to products of factors $(k+m)$, connecting them to the explicit denominator $Q_n$ and possibly to a finite-factorization formula.
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Editorial analysis

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

Referee Report

2 major / 3 minor

Summary. The paper studies the q-series expansions of the zeros x_k(q) of the deformed exponential function f(x)=∑_{n≥1} x^n q^{n(n−1)/2}/n!, written as −x_k(q)/k = 1 + ∑_{n≥1} a_{k,n} q^n and k/(−x_k(q)) = 1 + ∑_{n≥1} \hat a_{k,n} q^n. Theorem 3(i) claims that a_{k,n} and \hat a_{k,n} are rational functions P_n(k)/Q_n(k) and \hat P_n(k)/Q_n(k) with an explicit integer denominator Q_n and integer polynomials P_n, \hat P_n, and Theorem 3(ii) identifies the top two coefficients of these polynomials as σ(n)k^{M_n−2} + σ(n)(μ_1(n)−n)k^{M_n−3}. The proof of part (i) is based on a fixed-point equation and a closure argument for a class A of q-series with denominators Q_n. The proof of part (ii) relies on a uniform k^{-3} asymptotic expansion of w_k(q) (Lemma 3), whose proof depends on identity (37) for A_0(q), allegedly derived from Jacobi's triple product. The paper also reports extensive numerical computations for n ≤ 300, with data made available online, supporting Sokal's non-negativity conjectures for the coefficients a_{k,n} and \hat a_{k,n}.

Significance. If correct, Theorem 3(i) is a striking structural result: the q-coefficients of the zeros are exactly governed by one explicit denominator sequence, with integer polynomial numerators that can be computed recursively. The proof via the class-A closure is self-contained and appears sound; the downloadable data for n ≤ 300 is a valuable resource. Theorem 3(ii) would strengthen the Wang–Zhang asymptotic expansion and convert Sokal's positivity conjectures into concrete polynomial non-negativity statements. The numerical verification of P_n(k), \hat P_n(k) ≥ 0 for n ≤ 300 is substantial empirical evidence, although it does not constitute a proof. The main obstacle is the proof of Theorem 3(ii), which depends critically on the unproved and, as stated, false identity (37).

major comments (2)
  1. [Lemma 3, Eq. (37)] The identity (37) is false as stated. With A_0(q) = ∑_{n≥1} n σ(n) q^n = q + 6q^2 + 12q^3 + …, the right-hand side of (37) begins (1+3A_0(q))q − (5+5A_0(q))q^3 + …, so its q^2 coefficient is 3, while the left-hand side has q^2 coefficient 2σ(2)=6. No later term with exponent i(i+1)/2 can affect the q^2 coefficient, so the mismatch is unconditional. This identity is described in the proof as 'the crucial step' that makes the k^{-3} control in Lemma 3 possible; without it, the cancellation that leads to the uniform bound k^3|w_k^{(1)}(q)−w_k^{(0)}(q)| ≤ C does not occur.
  2. [Proof of Theorem 3(ii), Eq. (42)] Because Lemma 3 is the sole source of the uniform boundedness of k^3(w_k(q)−1−A_0(q)k^{-2}) on |q| ≤ δ, the failure of (37) invalidates the Vitali–Porter passage from the pointwise limit (41) to the coefficientwise limits in (42). Consequently the two leading coefficients in (17), the positivity claim for the second coefficient, and the sketch leading to (18) are unsupported. Part (i) of Theorem 3 is unaffected because it does not use Lemma 3. The central claim of the paper therefore needs a corrected proof of Lemma 3 or an alternative argument for Theorem 3(ii) before the results can be accepted.
minor comments (3)
  1. [Section 3, Figure 2] The legend states 'we plot a white pixel if b_{n,i} > 0 and a black pixel if b_{n,i} > 0'; the second condition should presumably be b_{n,i} < 0.
  2. [Page 14, proof of Lemma 2(b)] The phrase 'for for m ≥ 1 |q| ≤δ' contains a duplicated 'for' and missing punctuation; it should read 'for m ≥ 1 and |q| ≤ δ'.
  3. [Equation (7)] The notation a_{k,n} and \hat a_{k,n} is used before the formal definitions in (7) are stated; the definitions are clear from context but could be stated more explicitly when introduced.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is proved from the fixed-point equation via a self-contained class-A induction, and the leading coefficients use independent external asymptotics rather than fitted inputs.

full rationale

The central result, Theorem 3(i), is derived directly from the defining fixed-point equation w_k(q) = 1 + q F_k(w_k(q); q) (Lemma 2) through the class A closure argument: the paper proves that certain rational-function coefficient classes are closed under the operations needed to carry out the fixed-point iteration, and then identifies the limiting coefficients by matching Taylor series orders. This is a genuine induction, not a fit, and it does not presuppose the form of P_n or Q_n beyond the explicit denominator definition. Theorem 3(ii) is obtained by combining the independent Wang–Zhang asymptotic expansion (9) with Lemma 3, whose crucial identity (37) is imported from Zhang [19] via Jacobi's triple product. That identity is external and independent of the paper's own conclusions; even if it were erroneous, that would be a correctness defect, not a circularity. The numerical computations and non-negativity verifications for n ≤ 300 are presented as evidence supporting Sokal's conjectures, not as inputs used to derive the theorem. The citations to Sokal and Zhang are not self-citations of the present author, and none of them smuggles in the target statement as an assumption. No step reduces by construction to a fitted parameter, a renamed known result, or a self-citation chain. The derivation is therefore self-contained in the relevant sense, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to target data: the constants t1,t2,t3,t* in the reproduced proof of Theorem 1 are explicit numerical certificates for inequalities G<2, and the finite check ranges in Section 3 are chosen for proof bounds, not to tune the claimed expansions. The proof imports the axioms above from prior literature; none is authored by the present author or contains the target theorem as an assumption.

assumptions (4)
  • domain assumption The zeros x_k(q) are simple and negative for q in (0,1), giving the Hadamard product representation (2).
    Invoked at the start to justify the indexing, simplicity, and product structure of the zeros; established in prior literature [5,8,10] and not reproved in this paper.
  • domain assumption The coefficients c_{k,1-k}=-k and the analyticity of x_k(q) near q=0, so that x_k(q)=-k q^{1-k} w_k(q) with w_k analytic.
    Used to define w_k and the coefficients a_{k,n} and \widehat a_{k,n}; the first coefficient is cited to Sokal [14,15,17].
  • standard math Wang-Zhang asymptotic expansion (9) with B_1(q)=A_0(q) and B_2(q)=-A_1(q).
    A theorem from [18], used in the proof of Theorem 3(ii) to fix the leading coefficients of P_n and \widehat P_n.
  • standard math Identity (37), obtained from the logarithmic derivative of Jacobi's triple product identity.
    Called 'the crucial step in the proof' of Lemma 3; cited from Zhang [19] and not proved in this paper.

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Pith. "Pith review of On series expansions of zeros of the deformed exponential function." pith.science (2026). https://pith.science/paper/WWNGYSER

@misc{pith2026241202462,
  author       = {Pith},
  title        = {Pith review of: On series expansions of zeros of the deformed exponential function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WWNGYSER}},
  note         = {Machine review of arXiv:2412.02462}
}
abstract

For $q \in (0, 1)$, the deformed exponential function $f(x) = \sum_{n \geq 1} x^n q^{n(n-1)/2}/n!$ is known to have infinitely many simple and negative zeros $\{x_k(q)\}_{k \geq 1}$. In this paper, we analyze the series expansions of $-x_k(q)/k$ and $k/x_k(q)$ in powers of $q$. We prove that the coefficients of these expansions are rational functions of the form $P_n(k)/Q_n(k)$ and $\widehat{P}_n(k)/Q_n(k)$, where $Q_n(k) \in {\mathbb Z}[k]$ is explicitly defined and the polynomials $P_n(k), \widehat{P}_n(k)\in {\mathbb Z}[k]$ can be computed recursively. We provide explicit formulas for the leading coefficients of $P_n(k)$ and $\widehat{P}_n(k)$ and compute the coefficients of these polynomials for $n \leq 300$. Numerical verification shows that $P_n(k)$ and $\widehat{P}_n(k)$ take non-negative values for all $k \in \mathbb{N}$ and $n\le 300$, offering further evidence in support of conjectures by Alan Sokal.

Figures

Figures reproduced from arXiv: 2412.02462 by the authors.

Figure 1
Figure 1. The number of decimal digits of the coefficients of [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. The sign of coefficients bn,i (left) and Bn,i (right). Here n = 1, 2, . . . , 300 is on the x-axis and i = 0, 1, . . . , Mn − 2 = deg(Pn) is on the y-axis. The corresponding graphs for the polynomials Pbn(k) and Pbn(k + 1) are shown on [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. The sign of coefficients ˆbn,i (left) and Bbn,i (right). Here n = 1, 2, . . . , 300 is on the x-axis and i = 0, 1, . . . , Mn − 2 = deg(Pn) is on the y-axis. Proposition 1. Let P(x) = Pn j=0 ajx j be a real polynomial of degree n such that aj ≥ 0 for m ≤ j ≤ n. For r > 0 denote νr(P) := max{r, 2c(r)}, where c(r) := max 1≤i≤m i s |am−i | b(r) and b(r) := Xn j=m ajr j−m. Then P(x) > 0 for all x > νr(P). Proof. Assume … view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The left plot shows the values of ν(Pn) (blue), ν10(Pn) (black) and ν20(Pn) (red) for n = 1, 2, . . . , 300. The right plot shows the corresponding values of ν(Pbn), ν50(Pbn) and ν100(Pbn). From this last inequality it follows that c(r)z −1 ≥ 1/2, so that z ≤ 2c(r). Th…
Figure 5
Figure 5. Figure 5: The left plot shows the number of real roots of [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

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