Sheffer sequences with zeros on a line

G.-S. Cheon; K. Tran; T. Forg\'acs

REVIEW 3 major objections 4 minor 13 references

Sheffer sequences with zeros on a line

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Zeros of a broad Sheffer polynomial family fall on one vertical line

desk verdict A genuine generalization of Bump et al. with a real proof gap: the final zero-counting step is outsourced to an unpublished companion, so the main theorem is not yet established as written. read the letter →

arxiv 2508.18229 v1 pith:GPXGQCEM submitted 2025-08-25 math.NT math.CV

G.-S. Cheon , T. Forgács , K. Tran This is my paper

classification math.NTmath.CV MSC 05A1505A4030C1530E15

keywords SheffersequenceszerosonalineMellintransformRiemannzetafunctionLaguerrepolynomialsRiordanmatricessaddle-pointmethodinterlacing

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 central claim is that a wide class of Sheffer sequences—polynomial families generated by (1−α_0z)^(p+s)(1+α_0z)^(p*−s)∏_{i=1}^N(1−α_i²z²)^(p_i)—have all but finitely many zeros on the vertical line Re(s)=c=(p*−p)/2. This extends a result of Bump and collaborators, which had been proved only for the single-factor generating function behind the generalized Laguerre polynomials. The extension matters because it shows the 'zeros on a line' phenomenon survives when the generating product is multiplied by further even factors, and because a particular member of the family is linked, through a scaled Mellin transform, to the Riemann zeta function: the transform of the sum ψ*_j(x)=∑_{n≥1}Φ_j^*(n√x) equals q_j(s/2)π^(−s/2)Γ(s/2)ζ(s). A sympathetic reader would care because this makes the zero location of these polynomials a concrete cousin of the Riemann hypothesis.

What carries the argument

The central object is the Sheffer sequence {h_n(s)} for the pair (g(z), f(z)) with g(z)=∏_{i=0}^N G_i(z) and f(z)=log((1−α_0z)/(1+α_0z)). The load-bearing identity is the exponential Riordan matrix factorization ⟨g(z),z⟩ = ⟨∏_{i=1}^N G_i(z),z⟩ ⟨G_0(z),z⟩, which writes h_n(s)=∑_{k=0}^n n!b_{n−k}q_k(s)/k! and isolates the two-factor core q_k(s) whose zeros are tractable. That core is studied through the contour integral p_n(t)=∫_{Γ_2}ψ(z)e^{nϕ(z,t)}dz, where ϕ(z,t)=it Log(1+z)−it Log(1−z)−Log z; the saddle-point curve z(y) with −y²=ϕ(z(y),t)−ϕ(ζ,t) and the sign of Im z′(y) are what make the central piece dominate the tails. The zeta connection rides on the Mellin-transform chain (2.13), (2.19)

What would settle it

For a concrete case with p+p*>0 (e.g. p=0, p*=1, N=0, α_0=1), compute the zeros of q_n(s) for n up to, say, 40 and compare the number of off-line zeros with 2⌈c+p⌉; additionally, recompute the constrained extrema in Lemma 18 with high precision—if the minimum t is less than 0.924256... or the maximum exceeds 0.707107..., the sign of Im z′(y) can be wrong and the tail estimate fails.

Watch

Extended reading notes

Core claim

Theorem 1 states that for real parameters α_i, p_i, p, p* with |α_0|<|α_1|<...<|α_N|, the polynomials h_n(s) defined by relation (2.4) have all their zeros on Re(s)=c=(p*−p)/2 when p*+p≤0, and all but a specified finite set of 2⌈c+p⌉ zeros there when p*+p>0. The proof splits h_n through an exponential Riordan factorization into a finite linear combination of intermediate polynomials q_k(s) generated by the two-factor core (1−z)^(p+s)(1+z)^(p*−s); the zero-location theorem is proved for q_k by an integral representation on a Hankel contour, a saddle-point deformation through ζ(t)=−it+√(1−t²), and saddle-point asymptotics, with the off-line exception count obtained by comparing the change of a

Load-bearing premise

The final zero count and the asymptotic tail estimates rely on arguments in the authors' companion manuscript [5], which is not yet available, and on numerical Lagrange-multiplier extrema (0.924256... and 0.707107...) in Lemma 18; if the analogy to [5] fails for the generalized family, the exceptional-zero count 2⌈c+p⌉ is not proved.

Editorial extensions

If this is right

For p*+p≤0 every q_n(s) has all of its zeros on Re(s)=c; for p*+p>0, the only off-line zeros are the 2⌈c+p⌉ exceptions counted in Theorem 1.
The q_n(s) interlace after the affine change s=c+it: q_n⪯q_{n+1} for n≥1, so they behave like real-rooted polynomials in the vertical coordinate.
The scaled Mellin identity ∫_0^∞ ψ*_j(x)x^(s/2−1)dx = q_j(s/2)π^(−s/2)Γ(s/2)ζ(s) means information about zeros of these transforms is information about zeros of ζ(s).
The limiting density of zeros of h_n(c−int) on t∈(0,1) is D(x)=1/(2π) ln((1+√(1−x²))/(1−√(1−x²))).
Bump's local Riemann hypothesis is recovered as the special case N=0, α_0=1, p=−1−α/2, p*=−α/2.

Reading between the lines

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

Inference: the exceptional count 2⌈c+p⌉ suggests a phase transition controlled by p*+p; a cheap test is to compute the zeros of q_n(s) for p=0, p*=1, N=0 for moderate n and count off-line zeros.
Inference: the Mellin identity points to a family of Dirichlet-series-like objects whose non-vanishing on Re(s)=1/2 could be equivalent to a Riemann hypothesis for that family; constructing an explicit Euler product for ∫ψ*_j would make that precise.
Inference: the proof's tail estimates hinge on the numerically verified extrema in Lemma 18; replacing those Lagrange-multiplier checks with an analytic inequality would remove a computational dependency from the theorem.
Inference: the same Riordan splitting should in principle work for other products of even factors, so a natural conjecture is that the zero-line statement persists for more general palindromic generating functions.

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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

Colleague,

The headline: this is a real generalization of Bump et al., but the proof as it stands is not self-contained at the decisive step, so I wouldn't take Theorem 1 as proven yet.

What's new: Theorem 1 extends the known zero-line result from the two-factor generating function to a product with N additional factors and separate exponents on (1 - α_0 z) and (1 + α_0 z). That's a genuinely broader family, and the interpolating polynomials q_n(s) with their three-term recurrence and interlacing are a nice addition. The limiting zero density in Section 3.4 is a clean byproduct. The zeta connection (Theorem 11) is a true Mellin factorization, but it is formal; it rewrites the zeta function in terms of the q_n(s) and does not open a new route to zeta zeros. The paper does not oversell this, for once.

The soft spots: the proof of Theorem 1 depends on the unpublished companion [5] at three specific points. Lemma 14 uses Lemma 10 of [5] for the Hankel asymptotics; Lemma 17 uses Lemmas 10 and 12 of [5] for the Gamma argument change; and Section 3.3 ends by saying the final zero count follows by 'arguments completely analogous to those in [5]' after comparing (3.17) to (3.21) of [5]. That comparison is not shown. So the exceptional-zero count 2⌈c+p⌉ is asserted, not established, in this manuscript. This is a completeness gap, not a demonstrated contradiction; the theorem may well be true. A secondary gap is Lemma 18, where the proof that Im z'(y) > 0 relies on two numerical Lagrange-multiplier extrema (0.924256... and 0.707107...) reported without a rigorous certificate. That's a smaller issue but still a gap in the saddle-point tail control. There are also some formula typos in the display equations (for instance, the product in (3.2) doesn't quite match the statement of Theorem 1).

The citation pattern: heavy self-citation is not itself a problem; the companion [5] is the right place for the earlier proof. But the dependency should have been made more explicit in the introduction, and ideally the companion should be posted on the arXiv so referees can check the analogy.

Bottom line: this deserves a serious referee, but the referee should be asked to verify whether the analogy to [5] actually carries over. If the authors can fold the needed arguments into the paper, or at least make the companion publicly available, I'd be happy to accept the result. As submitted, it is a promising but incomplete manuscript.

Referee Report

3 major / 4 minor

Summary. The paper claims a substantial generalization of Bump et al.'s result on zeros of Sheffer polynomials. For real parameters satisfying the hypotheses of Theorem 1, the sequence {h_n(s)} generated by (1 - α_0 z)^{p+s}(1 + α_0 z)^{p*-s} ∏_{i=1}^N (1 - α_i^2 z^2)^{p_i} is asserted to have all zeros on the vertical line Re(s) = (p* - p)/2, except possibly for 2⌈c+p⌉ zeros when p*+p > 0. The authors also derive recurrences for the auxiliary polynomials q_n(s), prove an interlacing result for p*+p ≤ 0, establish a Mellin-transform identity connecting q_n with the Riemann zeta function, and compute a limiting zero density. The proof of Theorem 1 follows the contour-deformation/saddle-point strategy of Bump et al., but several decisive asymptotic lemmas and the final zero-counting step are explicitly delegated to the unpublished companion manuscript [5].

Significance. If completely proved, the zero-location theorem would be a meaningful extension of the known local Riemann hypothesis results, and the Mellin-transform factorization in Theorem 11 is an explicit and verifiable identity. The paper has clear strengths: the recurrences in Theorems 4 and 5 are correctly derived, the Riordan-matrix viewpoint is informative, and the calculation of the limiting density is elegant. However, as submitted, the main theorem is not proven within the paper: key Lemmas 14 and 17 depend on [5], the final comparison with Eq. (3.21) of [5] is not shown, and the Appendix relies on unsupported numerical extrema. These are load-bearing gaps in the central claim, so the significance of the paper cannot be assessed as a proof.

major comments (3)

[§3.3 (after Eq. (3.17))] The proof of Theorem 1 is not contained in the paper. Lemmas 14 and 17 invoke Lemma 10 and Lemmas 10/12 of the unpublished manuscript [5], and the final jump from (3.17) to the exceptional-zero count 2⌈c+p⌉ is left to 'arguments completely analogous to those in [5]' without showing the comparison to Eq. (3.21) of [5]. Since [5] is not available to the reader, the claim that all but 2⌈c+p⌉ zeros lie on Re(s)=c is unproven as submitted. This is the central result, so the gap is load-bearing.
[Appendix, Lemma 18] The proof that Im z'(y) > 0 for y > 0 rests on two 'numerical' Lagrange-multiplier extrema (0.924256... and 0.707107...) computed from algebraic constraints. No rigorous certificate, interval bound, or exact rational enclosure is provided. Floating-point output is not a proof. This inequality is used in Lemma 15 to show the central saddle-point section dominates the tails, and hence Lemma 16 also depends on it. Without a rigorous proof of these extrema, the saddle-point argument is incomplete.
[§2.1, Theorem 7] The proof begins 'It can be readily shown that bq_n(s) ∈ R[s] and all zeros of bq_n(s) are real.' The real-rootedness is exactly the conclusion of Corollary 3/Theorem 1 for p*+p ≤ 0, which is not proved at that point and is the main theorem of the paper. As written, Theorem 7 is either circular or lacks the key input. The theorem should be stated after Theorem 1, or the real-rootedness should be proved independently.

minor comments (4)

[Throughout] There are numerous typos and corrupted display artifacts ('poynomials', 'fucntions', 'generealized', 'Appel', 'NY' in the abstract/intro). The paper needs careful proofreading.
[Lemma 13] The assertion tπ/2 > ln z* for all t ∈ (0,1) is justified only by 'perhaps with a CAS'. This is elementary but should be proved explicitly.
[Theorem 11 proof] In the last line of the proof, the index switches from j to n: 'qn(s/2)' should be 'qj(s/2)'. Also, the interchange of summation and integration in the derivation of (2.23) should be justified by a convergence or dominated-convergence argument.
[§3 notation] The symbol p_n(t) is used for the Hankel integral in (3.4), which conflicts with the earlier use of p_n for polynomial sequences. Consider denoting the integral by I_n(t) or similar.

Circularity Check

1 steps flagged · score 6.0 of 10

Theorem 1's decisive zero-counting step is outsourced to the same authors' unpublished companion [5], so the central claim rests on a load-bearing self-citation.

self citation load bearing [Section 3.3, final paragraph after equation (3.17); cf. Lemma 14 and Lemma 17 proofs]
"The completion of the proof of Theorem 1, i.e. the claim about the location of the zeros of the polynomials pn(t), is obtained using arguments completely analogous to those in [5], after comparing the right hand side of equation (3.17) to the absolute value of (3.21) in [5]."

This is the step that converts the asymptotic change-of-argument in (3.17) into the claimed zero count on Re(s)=c except for 2⌈c+p⌉ zeros. The conversion is not carried out in the paper; it is asserted by analogy to [5]. The same dependency appears earlier: Lemma 14's Hankel asymptotics invoke 'Lemma 10 in [5]', and Lemma 17 invokes 'Lemma 10 in [5] once more' and 'the computations in the proof of Lemma 12 in [5]' for the Gamma argument change. Reference [5] is the authors' own manuscript 'under review', not machine-checked or independently reproduced, so the central theorem's exceptional-zero count is justified by a self-citation chain rather than by a proof contained in the paper.

full rationale

Most of the paper is not circular. The interlacing proof (Theorem 7) uses the three-term recurrence and Lemma 6 directly. The zeta connection (Theorem 11) is an identity: substituting y=n√x into the definition ψ*_j(x)=∑Φ*_j(n√x) and using (2.21) literally produces q_j(s/2)π^{-s/2}Γ(s/2)ζ(s); this is a factorization, not a derivation of one side from the other in a circular sense. The contour deformations and saddle-point estimates in Lemmas 12, 13, 15 are self-contained. However, the completion of the proof of Theorem 1—the argument-principle count that yields the finite exceptional set 2⌈c+p⌉—is not proved in this paper. The final sentence delegates it to the same authors' unpublished companion [5], and Lemmas 14 and 17, which feed the change-of-argument estimate (3.17), borrow Lemmas 10 and 12 from [5]. Because [5] is under review with overlapping authorship and is not independently certified, the central claim's decisive step is load-bearing self-citation. This is a partial circularity/completeness dependence rather than an identity-of-inputs circularity. The numerical Lagrange-multiplier extrema in Lemma 18 are an additional rigor gap, but that is a correctness risk, not circularity. Score 6 reflects that the main theorem's exceptional-zero count rests on the self-citation chain while substantial independent content remains.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central theorem depends on the unpublished companion paper [5] for the final argument and two lemmas, and on numerical extrema calculations in the Appendix for a monotonicity claim. The zeta identity is derived from standard Mellin and Meixner identities.

assumptions (4)

domain assumption The lemmas and final argument of the unpublished companion paper [5] apply verbatim to the generalized family of Theorem 1.
Used in Lemmas 14 and 17 and the final zero count; [5] is not publicly available.
ad hoc to paper The numerical Lagrange multiplier computations in Lemma 18 (minimum 0.924256..., maximum 0.707107...) are accurate enough to prove the required inequalities for all t in (0,1).
Underpins Im z'(y)>0, essential for tail estimates; no analytic proof or code supplied.
domain assumption Bump et al. theorem [4]: zeros of P_n^{(α)}(s) generated by (1-z)^{s-1-α/2}(1+z)^{-s-α/2} lie on Re(s)=1/2.
The base case being extended; cited from [4].
standard math Mellin transform identities (2.13) and (2.15) as stated in [2] and [11].
Used to derive Theorems 9 and 11.

invented entities (1)

Polynomial families {h_n(s)} and {q_n(s)}
purpose: Central objects of the theorem; q_n connects to zeta via Mellin transform.
Defined by generating functions (2.4) and (2.7); existence and properties are the paper's own construction with no external falsifiable handle.

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Cite this review

Pith. "Pith review of Sheffer sequences with zeros on a line." pith.science (2026). https://pith.science/paper/GPXGQCEM

@misc{pith2026250818229,
  author       = {Pith},
  title        = {Pith review of: Sheffer sequences with zeros on a line},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GPXGQCEM}},
  note         = {Machine review of arXiv:2508.18229}
}

read the original abstract

We extend a result of Bump et al. to show that a large family of Sheffer sequences has their zeros - up to perhaps a finite number of exceptions - on a vertical line. We connect a particular such sequence to the Riemann zeta function via a product representation of a scaled Mellin transform, analogously to the product decomposition of a Mellin transform involving the generalized Laguerre polynomials into factors of the Gamma function and Meixner polynomials.

Figures

Figures reproduced from arXiv: 2508.18229 by the authors.

**Figure 2.1.** The zeros of q9(s) and q10(s), with p = −1 and p ∗ = 0 2.2. A connection to the Riemann zeta function ζ(s). This section establishes a connection between the Riemann zeta function, our sequence of polynomials {qn(s)} and a scaled Mellin transforms of a family of fucntions reltated to the generalized Laguerre polynomials. We first recall (see [2]) that the Mellin transform of a family of functions related to the gene… view at source ↗

**Figure 3.1.** Γ1 and Γ2 curves On the other hand, the substitution z → z yields ˆ Γ2 ψ(z)e nϕ(z,−t) dz = (−1)n+1 ˆ Γ2 ψ(z)e nϕ(z,t)dz. We deduce from (3.3) that πhn(c − int) is either the imaginary part, or −i times the real part of (3.4) pn(t) = ˆ Γ2 ψ(z)e nϕ(z,t) dz, depending of whether n is even or odd. We now commence the analysis of the asymptotic behavior of pn(t) as n → ∞ using the saddle point method. Observe that the tw… view at source ↗

Figures from the paper (5 more)

**Figure 3.2.** Figure 3.2: The new contour of integration: the z(y) curve along with a vertcial line replacing Γ2 Proof. By the definition of ζ, expanding ϕ(z, t) about z = ζ yields ϕ (z, t) − ϕ (ζ, t) = ϕz 2 (ζ, t) 2 (z − ζ) 2 (1 + b(z)), [PITH_FULL_IMAGE:figures/full_fig_p012_3_2.png]

**Figure 3.3.** Figure 3.3: The Hankel contours before and after the change of variable z → e z Suppose ϵ > 0 is such that nϵ = o(1). We break up the range of integration of the last integral as (3.11) ˆ ln5n/n±iϵ (0−) ψ(e z )e nϕ(e z ,t) e z dz + ˆ +∞+iϵ ln5 n/n+iϵ ψ(e z )e nϕ(e z ,t) e z dz +…

**Figure 3.4.** Figure 3.4: The limiting zero distribution density function Appendix The work in this Appendix is dedicated to showing that for any y ∈ (−∞, L), Im z ′ (y) > 0. This result is used in establishing asymptotic estimates of the tails of the integral representations of the pn(t). As…

**Figure 3.5.** Figure 3.5: The three essential curves in our proof: the ζ curve, the z(y) curve and the γ(y) curve γ(L˜) ∈ (0, 1), then on the one hand Im ϕz(γ(L˜), t) = 0, since ϕz is real along the curve γ(y). On the other hand, Im ϕz(γ(L˜), t) = Im it 1 + γ(L˜) + it 1 − γ(L˜) − 1 γ(L˜) …

**Figure 3.6.** Figure 3.6: The triangle with vertices 0, 1 − z, and 1 + z Using the identities sin(π/2 + x − y) = cos(x − y) = cos x cos y + sin x sin y we find that sin π 2 + Arg(1 + z) − Arg(1 − z) = cos (Arg(1 + z)) cos (Arg(1 − z)) + sin (Arg(1 + z)) sin (Arg(1 − z)) = 1 + Re z |1 + z| …

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Works this paper leans on

13 extracted references · 13 canonical work pages

[5]

Cheon, T

G.-S. Cheon, T. Forgács, K. Tran, On the zeros of certain Sheffer sequences and their cognate sequences,under review

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [5]

Cheon, T

G.-S. Cheon, T. Forgács, K. Tran, On the zeros of certain Sheffer sequences and their cognate sequences,under review

work page

[2] [1]

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

work page 1999

[3] [2]

M. K. Atakishiyeva and N. M. Atakishiyev, On the Mellin transform of hypergeometric polynomials, J. Phys. A: Math. Gen. 32 (1999), L33-L41

work page 1999

[4] [3]

Bump and Eugene K.-S

D. Bump and Eugene K.-S. Ng, On Riemann’s zeta funcion, Math. Z. 192 (1986), 195-204

work page 1986

[5] [4]

Bump, K.-K

D. Bump, K.-K. Choi, P. Kurlberg and J. Vaaler, A local Riemann hypothesis, I, Math. Z. 233 (2000), 1-19

work page 2000

[6] [6]

Cheon, T

G.-S. Cheon, T. Forgács, A. Mwfase, and K. Tran, On combinatorial properties and the zero distri- bution of certain exponential Sheffer sequences, J. Math. Anal. App , 2024, 128513, ISSN 0022-247X, doi.org/10.1016/j.jmaa.2024.128513

work page arXiv 2024

[7] [7]

Cheon, T

G.-S. Cheon, T. Forgács, H. Kim and K. Tran, On combinatorial properties and the zero distribution of certain Sheffer sequences, J. Math. Anal. Appl. 514 (1) (2022), DOI: 10.1016/j.jmaa.2022.126273

work page arXiv 2022

[8] [8]

L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007), 542-560. Sons, INC., New York, 1958

work page 2007