Two characterizations of Sheffer-Dunkl sequences

Alejandro Gil Asensi; Judit Minguez Ceniceros

REVIEW 3 major objections 6 minor 25 references

Two characterizations of Sheffer-Dunkl sequences

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Two Stieltjes integrals characterize exactly when a polynomial sequence is Sheffer-Dunkl

desk verdict A natural and probably true Dunkl-analog of Thorne and Sheffer, but the proof skips a convergence step and the examples use distributions rather than the stated BV functions. read the letter →

arxiv 2501.01364 v1 pith:P35QH2IU submitted 2025-01-02 math.CA

Alejandro Gil Asensi , Judit Minguez Ceniceros This is my paper · ORCID

classification math.CA MSC 11B8344A6011B68

keywords Sheffer-DunklsequencesDunkloperatorStieltjesintegralsmomentproblemsAppell-DunklpolynomialsBernoulli-DunklEuler-Dunkl

topics The Collatz Conjecture

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

Classical Sheffer polynomials sit between ordinary derivatives and generating functions; this paper carries that description over to the Dunkl setting, where the derivative is replaced by the Dunkl operator $\Lambda_\nu$ and factorials by the sequence $\gamma_{n,\nu}$. It proves two if-and-only-if characterizations: a sequence $\{s_{n,\nu}\}$ is Sheffer-Dunkl for a pair $(g,f)$ exactly when a bounded-variation function $\alpha_\nu$ makes the moments of $g$ reproduce $\int L_f^r s_{n,\nu}\,d\alpha_\nu = \gamma_{n,\nu}\delta_{n,r}$, and exactly when a bounded-variation $\beta_\nu$ writes each polynomial as a Dunkl-translation integral of the associated Dunkl polynomials. In both cases the generating function $g(t)$ is recovered from the moments, so the integral data and the pair $(g,f)$ determine each other. The paper applies the characterizations to truncated, discrete, Bernoulli, Euler, and Boole-Dunkl families, giving explicit moment measures for several of them and identifying the remaining Bernoulli/Euler $\beta_\nu$ moment problems as open.

What carries the argument

The machinery is the Dunkl analogue of exponential generating functions. The Dunkl operator $\Lambda_\nu f(x)=f'(x)+\frac{2\nu+1}{2}\frac{f(x)-f(-x)}{x}$ replaces $d/dx$; the gamma-type numbers $\gamma_{n,\nu}$ replace $n!$, with $\gamma_{n,-1/2}=n!$; and the Dunkl kernel $E_\nu(t)=\sum_{n\ge 0} t^n/\gamma_{n,\nu}$ replaces $e^t$. The translation $\tau_y f(x)=\sum \Lambda_\nu^n f(x)\,y^n/\gamma_{n,\nu}$ replaces ordinary translation and obeys the binomial identity $\tau_t((\cdot)^n)(x)=\sum_k \binom{n}{k}_\nu t^k x^{n-k}$, with $\binom{n}{k}_\nu=\gamma_{n,\nu}/(\gamma_{k,\nu}\gamma_{n-k,\nu})$. The integral characterizations work because $L_f$ and $\tau_t$ commute and because the associated Dunkl polynomials satisfy the same binomial convolution identity, so substituting the integral representation into the generating function factorizes into $E_\nu(x\bar f(t))\int E_\nu(u\bar f(t))\,d\beta_\nu(u)$.

What would settle it

Check the theorem against the paper's own truncated example, $g(t)=1-t$. Its would-be moment functional $p \mapsto p(0)-\gamma_1 p'(0)$ is not representable by any bounded-variation signed measure on $\mathbb{R}$: it is unbounded on $C[-1,1]$ (the polynomials $x(1-x^2)^m$ witness this), even though the moments $1,-\gamma_1,0,0,\ldots$ are those of $\delta_0+\gamma_1\delta'_0$, which is a distribution. This computation settles the scope of the theorem as stated.

Watch

Extended reading notes

Core claim

The paper's central result is a pair of equivalent integral characterizations. Theorem 1 says that for formal series $g(t)=\sum \mu_{n,\nu}t^n/\gamma_{n,\nu}$ and $f(t)=\sum a_n t^n/\gamma_{n,\nu}$ with nonzero leading coefficients, a sequence $\{s_{n,\nu}\}$ is the Sheffer-Dunkl sequence for $(g,f)$ if and only if some function $\alpha_\nu$ of bounded variation on $\mathbb{R}$ has moments $\mu_{n,\nu}=\int x^n\,d\alpha_\nu(x)$ and satisfies $\int L_f^r s_{n,\nu}(x)\,d\alpha_\nu(x)=\gamma_{n,\nu}\delta_{n,r}$. Theorem 3 says $\{s_{n,\nu}\}$ is Sheffer-Dunkl if and only if some bounded-variation $\beta_\nu$ with nonzero zeroth moment represents $s_{n,\nu}(x)=\int \tau_t(p_{n,\nu})(x)\,d\beta_\nu(t)$, where $p_{n,\nu}$ are the associated Dunkl polynomials for $f$. In the first form the moments of $\alpha_\nu$ are the coefficients of $g(t)$; in the second, the moments of $\beta_\nu$ are the coefficients of $1/g(t)$. The proofs run through the generating identity $1/(g(\bar f(t)))\,E_\nu(x\bar f(t))=\sum s_{n,\nu}(x)t^n/\gamma_{n,\nu}$ and the degree-lowering action of $L_f$.

Load-bearing premise

The load-bearing premise is that any allowed generating series can be written as the moment sequence of a signed measure of bounded total variation on the real line; if some allowed $g(t)$ has no such function, the theorem as stated does not cover that $g$.

Editorial extensions

If this is right

For any Sheffer-Dunkl sequence, the function $g(t)$ is recoverable from the $\alpha_\nu$ moments by $g(t)=\int E_\nu(xt)\,d\alpha_\nu(x)$, and $1/g(t)$ is recoverable from the $\beta_\nu$ moments by $1/g(t)=\int E_\nu(xt)\,d\beta_\nu(t)$; the integral data and the generating pair determine each other.
The two characterizations give a practical test: to show a polynomial family is Sheffer-Dunkl it is enough to produce a bounded-variation moment function satisfying the degree-lowering orthogonality, or to represent each polynomial as a Dunkl-translation average of the associated polynomials.
In the limit $\nu=-1/2$, both theorems recover the classical Thorne and Sheffer characterizations, with $\gamma_{n,\nu}=n!$, $E_\nu(t)=e^t$, and $\tau_t$ equal to ordinary translation.
For the truncated, discrete truncated, and second-kind families in Section 4, the same $\alpha_\nu$ works for both the continuous and discrete operators $L_f$ and $L_{G_\nu}$, so the characterization transfers across the discrete Dunkl calculus.
For Bernoulli-Dunkl and Euler-Dunkl polynomials, the $\alpha_\nu$ measures are constructed explicitly, while the corresponding $\beta_\nu$ measures require solving moment problems for $B_{n,\nu}(0)$ and $E_{n,\nu}(0)$, which the paper leaves open.

Reading between the lines

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

Editorial inference: The examples that use $\delta'_0$ and $\delta''_0$ suggest the theorems would extend naturally to moment functionals or distributions rather than bounded-variation functions; read that way, bounded variation becomes a sufficient but unnecessary hypothesis, and the two characterizations are really dual statements about polynomial moment functionals.
Editorial inference: The duality between the $\alpha_\nu$ moments (coefficients of $g$) and $\beta_\nu$ moments (coefficients of $1/g$) suggests a transfer principle: a construction for one moment sequence yields the other by inversion in the ring of formal series, so solving the $\beta_\nu$ problem for truncated Appell-Dunkl polynomials would automatically give a solution for a related family wit
Editorial inference: The open $\beta_\nu$ moment problems for Bernoulli-Dunkl and Euler-Dunkl sequences invite comparison with the classical Bernoulli and Euler moment representations; if a positive measure exists, it would give integral formulas for $B_{n,\nu}(0)$ and $E_{n,\nu}(0)$ and likely extend to the discrete Boole-Dunkl families by the same transfer.
Editorial inference: Since the Dunkl kernel has subexponential growth, the $\alpha_\nu$ constructed by the Fourier technique are often tempered distributions; one testable extension is to check whether the same integral identities hold with the Dunkl transform pairing, which would place the characterizations in a harmonic-analysis setting rather than a purely formal one.

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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

The two characterizations are a genuinely useful extension of the classical Thorne and Sheffer theorems to Sheffer-Dunkl sequences, and I suspect the main theorems are true. The umbral algebra is clean: the moment functional built from g(t) annihilates the Sheffer-Dunkl polynomials, so the integral conditions (2.1) and (3.1) are the right formal conditions. That part is worth publishing, once the analytic wrapper is fixed.

What is new: Theorems 1 and 3 are not in the cited literature, and the use of the Dunkl translation and the ν-binomial theorem from [11] is natural. The examples section does real work, giving explicit moment representations for truncated, Bernoulli, Euler, and Boole Dunkl families. Some of those, like the Bernoulli-Dunkl measure (4.4), are directly checkable.

Where it gets soft: the proof of the forward direction in Theorem 1 asserts g(t)=∫Eν(xt)dα(x) from equality of moments. Boas/Widder does give a bounded-variation α with the right moments, so the reader's proposed counterexample with g(t)=1-t does not hold up—there are BV measures on R with moments 1, -γ1, 0, ... . The real problem is that the proof gives no reason why ∫Eν(xt)dα converges or why the power series can be integrated term by term. That is a load-bearing gap. And the paper's own examples make it worse: for g(t)=1-t and g(t)=1-t², the computed α is a distribution (δ'0 or δ''0), not a function of bounded variation. So the stated hypotheses of Theorem 1 are not satisfied by the paper's own examples, and the key integral equality is never established under those hypotheses.

This is repairable. The algebraic content does not depend on the convergence step; you can restate the theorems for signed measures or distributions, or add a representation theorem for the Dunkl kernel with explicit convergence conditions. But as written, the proof and the examples are out of sync with the stated BV framework.

Citation pattern is fine. Reliance on [11] is appropriate, and the Widder citation is correctly used for the moment existence. No sign of circularity.

Bottom line: this deserves a serious referee, not a desk reject. I would send it out and ask for a revision that either proves the integral representation under the stated hypotheses or changes the hypotheses to match the distributional examples. The characterization itself looks solid, but the analytic framing needs work.

Referee Report

3 major / 6 minor

Summary. The paper proposes two Stieltjes-integral characterizations of Sheffer-Dunkl sequences, paralleling the classical theorems of Thorne and Sheffer. Theorem 1 asserts that a sequence {s_n,ν} is Sheffer-Dunkl for (g,f) if and only if there is a bounded-variation function α on R such that the moment integrals exist and ∫ L_f^r s_n dα = γ_{n,ν} δ_{n,r}. Theorem 3 asserts the analogous characterization s_n(x)=∫ τ_t(p_n)(x)dβ(t) for a bounded-variation β whose moments are the coefficients of 1/g. The proofs use the Boas/Widder moment representation and formal generating-function identities. Section 4 applies the results to truncated, discrete, Bernoulli, Euler, and Boole-Dunkl polynomials, constructing the corresponding 'measures' in several cases.

Significance. Should the theorems hold, they would extend to the Dunkl setting a classical bridge between umbral calculus and the moment problem, and they would give a practical way to identify Sheffer-Dunkl sequences from their moments. The algebraic computations in Section 4 are original and useful, and the paper is clearly written at the formal-expansion level. However, the proofs do not currently justify the key analytic step of integrating the Dunkl exponential against a bounded-variation measure, and the converse of Theorem 1 is established only for the Appell-Dunkl case. These gaps leave the main claims only partially supported. The explicit construction of distributions in examples 4.1 and 4.3 is a further discrepancy with the stated function-of-bounded-variation hypotheses.

major comments (3)

[§2, eq. (2.2)] The equality g(t)=∫ Eν(xt)dα is asserted immediately after defining α through its moments, but the stated hypotheses only ensure existence of the moment integrals ∫ x^n dα. They do not imply that the Dunkl exponential Eν(xt) is integrable with respect to α, nor that the integral equals its termwise moment series. Consequently the subsequent application of ∫ dα to the generating function in (1.9) is not a justified operation. The same issue recurs in §3 and in Corollaries 2 and 4, where ∫ Eν(xf(t))dβ is treated as an ordinary integral. The proofs would need either additional growth assumptions on α and β, or an explicit statement that all such identities are interpreted as formal power series, together with a justification of termwise integration.
[§2, proof of Theorem 1, converse] The linear system displayed after 'we obtain the following system of equations' does not contain the coefficients a_n of f(t) from (1.7). For r=n it gives c_n (γ_n/γ_0) μ_0 = γ_n, but the actual condition ∫ L_f^n s_n dα = γ_n includes a factor (a_1)^n from the leading term of L_f, and for r<n the mixed powers Λ^k in L_f produce additional contributions. The system is therefore the one appropriate to L_f=Λ (the Appell-Dunkl case f(t)=t), and the construction of s_n satisfying (2.1) is not valid for a general f. Thus the 'if' direction of Theorem 1 remains unproved for the full class of Sheffer-Dunkl sequences.
[§4.1, §4.3] The 'measures' α_ν constructed in these examples are δ0 + γ1 δ'_0 and δ0 − (γ2/2)δ''_0, which are distributions, not functions of bounded variation on R as required by Theorems 1 and 3. The moment integrals exist in a distributional sense, but the theorems as stated do not apply to such objects. The authors either need to extend the theorems to a distributional formulation or provide genuine bounded-variation functions with the same moment properties that also satisfy the necessary integral identities.

minor comments (6)

[§4.4, §4.6] The statements 'we do not know to solve it' are honest limitations, but they should be flagged as open cases in the text; this is not a technical error.
[§3, eq. (3.2)] In equation (3.2), the notation Q_n(t) conflicts with the use of t as the formal variable; using u for the integration variable would remove the ambiguity.
[§2, eq. (2.1)] In Theorem 1, equation (2.1) writes L_r^f while the rest of the paper uses L_f^r; the notation should be unified.
[§4.1] The phrase 'we start giving the function α_ν(x) corresponding to the Theorem 1' is misleading because the object described is not a function but a distribution.
[§3, proof of Theorem 3] The proof uses the commutativity of L_f and τ_t with a citation to [11]; it would help to state precisely which result in [11] is being used.
[Throughout] There are numerous typographical errors (e.g., 'Sheffer' in the abstract vs the title, 'Stieltjes' spelling, and inconsistent spacing in equations); a careful proofreading is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the theorems are substantive equivalences and the cited prior work is background, not an assumed conclusion.

full rationale

The paper proves two iff characterizations of Sheffer-Dunkl sequences. The forward directions construct the measure from the Taylor data of g via the Widder/Boas theorem and derive the integral identities by formal manipulation of the generating function; the backward directions solve triangular systems or use the translation action to recover the operator condition. No parameter is fitted from the target data and no equality is assumed that is identical to the conclusion. The reliance on [11] for the Dunkl translation commutation and binomial identities is background theory from the same authors' earlier work, but those facts are not the characterization being proved and do not by themselves force the theorem; they are cited building blocks. The examples in Section 4 use distributional objects such as delta_0' and delta_0'' where the theorems state functions of bounded variation, which is a rigor gap, and the proof of (2.2)/(3.3) does not justify convergence of the Stieltjes integral of the Dunkl exponential; these are correctness concerns rather than circular reductions. The paper also honestly notes unsolved moment problems in Sections 4.4 and 4.6. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The only numerical parameters are the coefficients of the formal power series g and f, which are part of the definition of a Sheffer-Dunkl sequence, not fitted. The load-bearing assumptions are the representability of arbitrary formal series by bounded-variation integrators (false as stated) and the Dunkl-calculus identities imported from [11]. No new particles, forces, or physical entities are introduced.

assumptions (3)

domain assumption Boas-Widder representation: any formal series g(t)=Σ μ_n t^n/γ_n admits a bounded-variation function α_ν on R with μ_n=∫ x^n dα_ν.
Invoked in Theorem 1 and Theorem 3 proofs. This is false for g(t)=1-t, whose moments 1, -γ_1, 0, 0, ... define an unbounded polynomial functional; no bounded-variation signed measure exists.
domain assumption Dunkl translation and operator commute: L_f τ_t = τ_t L_f, and the binomial identity τ_u(p_n)(x)=Σ_k (n k)_ν p_k(x)p_{n-k}(u) holds.
Used in the proof of Theorem 3; cited to [11]. These properties are stated by the authors' prior work and are not proved in this paper.
domain assumption Moment functions for the examples can be found with Durán's technique [5] and the results of [6].
The examples rely on references for constructing measures; some resulting 'functions' are distributions, such as δ_0+γ_1 δ'_0.

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Cite this review

Pith. "Pith review of Two characterizations of Sheffer-Dunkl sequences." pith.science (2026). https://pith.science/paper/P35QH2IU

@misc{pith2026250101364,
  author       = {Pith},
  title        = {Pith review of: Two characterizations of Sheffer-Dunkl sequences},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P35QH2IU}},
  note         = {Machine review of arXiv:2501.01364}
}

read the original abstract

Sheffer polynomials can be characterized using different Stieltjes integrals. These families of polynomials have been recently extended to the Dunkl context. In this way some classical operators as the derivative operator or the difference operator are replaced as analogous operators in the Dunkl universe. In this paper we establish two Stieltjes integrals that help us to characterize the Sheffer-Dunkl polynomials.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

25 extracted references · 25 canonical work pages

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [11]

Gil Asensi, J

A. Gil Asensi, J. M ´ ınguez Ceniceros, J. L. Varona, Sheﬀer-Du nkl Sequences Via Umbral Calculus in the Dunkl Context, Bull. Malays. Math. Sci. Soc. 47 (2024), no.6, Paper No. 172

work page 2024

[2] [1]

Carrillo and M

S.A. Carrillo and M. Hurtado, Appell and Sheﬀer sequences: on th eir characterizations through functionals and examples, C. R. Math. Acad. Sci. Paris 359 (2021), 205–217

work page 2021

[3] [2]

Ciaurri, A

´O. Ciaurri, A. J. Dur´ an, M. P´ erez and J. L. Varona, Bernoulli-Dun kl and Apostol-Euler- Dunkl polynomials with applications to series involving zeros of Bessel functions, J. Approx. Theory 235 (2018), 20–45

work page 2018

[4] [3]

Ciaurri, J

´O. Ciaurri, J. M ´ ınguez Ceniceros and J. L. Varona, Bernoulli-Dunkland Euler-Dunkl poly- nomials and their generalizations, Rev. R. Acad. Cienc. Exactas F ´ ıs. Nat. Ser. A Mat. RACSAM 113 (2019), 2853–2876

work page 2019

[5] [4]

C. F. Dunkl, Diﬀerential-diﬀerence operators associated to reﬂ ection groups, Trans. Amer. Math. Soc. 311 (1989), 167–183

work page 1989

[6] [5]

A. J. Dur´ an, Functions with given moments and weight functions for orthogonal polynomi- als,, Rocky Mountain J. Math. 23 (1993), no. 1, 87–104

work page 1993

[7] [6]

A. J. Dur´ an, A. Gil Asensi and J. L. Varona, Moment Problems A ssociated to the Dunkl factorial, Proc. Amer. Math. Soc. , to appear. https://doi.org/10.1090/proc/16852

work page doi:10.1090/proc/16852

[8] [7]

A. J. Dur´ an, M. P´ erez and J. L. Varona, Fourier-Dunkl syst em of the second kind and Euler-Dunkl polynomials, J. Approx. Theory 245 (2019), 23–39

work page 2019