{"id":"abe029c7-f8f8-4b11-9ea1-3060feac8bba","arxiv_id":"2501.01364","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Two Stieltjes integral characterizations for Sheffer-Dunkl sequences are presented, extending Thorne's and Sheffer's classical theorems.","lead":"This paper proposes two integral tests that identify Sheffer-Dunkl polynomials, a Dunkl-operator version of classical Sheffer polynomials. The intended use is to find the right measure for a given polynomial family, with examples for Bernoulli, Euler, and truncated Dunkl polynomials.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Reader's counterexample is invalid: Boas/Widder does give a BV measure for g(t)=1-t. The real gap is that the proof treats ∫Eν(xt)dα as an ordinary Stieltjes integral without hypotheses ensuring convergence or a stated formal interpretation.","rationale":"The reader's central objection—that the Boas/Widder moment theorem is false for g(t)=1-t—is itself false. A simple Fourier construction gives a bounded-variation signed measure with the required moments, so the alleged counterexample does not land. However, the reader is pointing in a region where a real issue exists: the proof uses the equality g(t)=∫Eν(xt)dα(x), and this equality is not implied by the moment conditions alone for an ordinary Stieltjes integral. The manuscript never states whether the integral is taken formally, coefficientwise in the formal power series, or as a convergent ordinary integral. The examples in §4.1 and §4.3, which use derivatives of the Dirac delta, reinforce this ambiguity because such distributions are outside the stated class of bounded-variation functions.\n\nMy recommendation is CONDITIONAL rather than REJECT: the central characterization is likely correct if the generating-function integrals are understood formally, since all conclusions involving (2.1) and (3.1) are finite polynomial integrals and can be derived algebraically from the moment functional. But as written, the proof has an unjustified step and the examples do not satisfy the theorem's hypotheses. The paper should be revised to state the formal interpretation explicitly, or to impose additional exponential-integrability conditions on α and β, and to replace the distributional examples with genuine BV representatives. Once that is done, the theorems appear sound and the novelty of extending Thorne and Sheffer characterizations to the Dunkl setting is preserved.","tokens_in":10500,"tokens_out":40514,"duration_ms":438806,"concrete_test":"Construct the BV witness for g(t)=1-t: fix ψ ∈ C_c^∞(R) with ψ ≡ 1 near 0, set h(x)=∫ iξψ(ξ)e^{ixξ}dξ, and take ν = δ_0 + γ_1 h(x)dx. Verify ∫ x^n dν = 1, -γ_1, 0, 0, ... and that ν has finite total variation. Then check whether the ordinary Stieltjes integral ∫ Eν(xt) dν(x) converges for t in a neighborhood of 0. If it does not converge, then equation (2.2) in the proof of Theorem 1 cannot be an ordinary integral identity, and the theorem is only justified if the generating-function integrals are interpreted formally or if α is additionally required to have exponential integrability.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The reader's weakest assumption is not the actual weak point. The cited Boas/Widder theorem, in the form needed here, is true: every real sequence (μ_n) is the moment sequence of some function of bounded variation on R. For g(t)=1-t, an explicit BV witness exists: take ν = δ_0 + γ_1 h(x)dx, where h is the inverse Fourier transform of iξψ(ξ) with ψ ∈ C_c^∞(R) identically 1 near 0. Then ∫ x^n h(x)dx = -δ_{n,1}, so ν has moments 1, -γ_1, 0, 0, ... and finite total variation. The functional p ↦ p(0) - γ_1 p'(0) is represented by ν on polynomials, even though it is unbounded in the sup norm on C[-1,1]; the tails of h compensate. Hence g(t)=1-t is not a counterexample to the moment-representation step.\n\nThe load-bearing difficulty lies in the step (2.2)/(3.3): the proof asserts g(t) = ∫ Eν(xt) dα(x) for a BV function α known only through its moments. The hypotheses in Theorem 1 guarantee only that the moment integrals ∫ x^n dα exist; they do not imply that the infinite-series kernel Eν(xt) is integrable with respect to α, or that termwise integration is legitimate. For the Boas construction, α is typically only Schwartz, not exponentially decaying, so the ordinary Stieltjes integral of the exponentially growing Dunkl exponential need not converge. Thus the equality (2.2) is at best a formal power-series identity, but the manuscript does not say this. Moreover, the paper's own examples in §4.1 and §4.3 use δ'_0 and δ''_0, which are distributions rather than functions of bounded variation, confirming that the stated BV hypotheses are not actually used in those computations.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10892,"tokens_out":17193,"duration_ms":149247,"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":[{"comment":"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.","section":"§2, eq. (2.2)"},{"comment":"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.","section":"§2, proof of Theorem 1, converse"},{"comment":"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.","section":"§4.1, §4.3"}],"minor_comments":[{"comment":"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.","section":"§4.4, §4.6"},{"comment":"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.","section":"§3, eq. (3.2)"},{"comment":"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.","section":"§2, eq. (2.1)"},{"comment":"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.","section":"§4.1"},{"comment":"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.","section":"§3, proof of Theorem 3"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper has a sound algebraic core and the examples are instructive, but the proof deficiencies are substantial. I do not share the reader's view that the Boas/Widder moment representation fails for g(t)=1-t; the real issue is the unproved interchange of integral and series. I recommend major revision rather than outright rejection, because the results may be repairable by reformulating the identities as formal power series and by restricting Theorem 1 to the Appell-Dunkl case or by adding hypotheses that make L_f explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nWhere 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.\n\nThis 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.\n\nCitation pattern is fine. Reliance on [11] is appropriate, and the Widder citation is correctly used for the moment existence. No sign of circularity.\n\nBottom 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.","headline":"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.","tokens_in":11456,"tokens_out":6468,"would_cite":false,"duration_ms":60999,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11B83","44A60","11B68"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two Stieltjes integrals characterize exactly when a polynomial sequence is Sheffer-Dunkl","keywords":["Sheffer-Dunkl sequences","Dunkl operator","Stieltjes integrals","moment problems","Appell-Dunkl polynomials","Bernoulli-Dunkl polynomials","Euler-Dunkl polynomials"],"falsifier":"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.","tokens_in":10273,"feed_emoji":"📐","tokens_out":10630,"duration_ms":85536,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two Stieltjes integrals pin down Sheffer-Dunkl sequences","feed_subtitle":"Two moment tests identify Sheffer-Dunkl families and recover their generating functions, for Bernoulli and Euler too.","key_machinery":"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 $E_\\nu(x\\bar f(t))\\int E_\\nu(u\\bar f(t))\\,d\\beta_\\nu(u)$.","core_discovery":"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 $L_f$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the theorem used in both proofs to pass from a formal series $g(t)=\\sum \\mu_n t^n/\\gamma_n$ to a bounded-variation function with those moments.","marker":"[25]"},{"why":"Establishes the Sheffer-Dunkl umbral-calculus framework, including $L_f$, the Dunkl translation, and the binomial identity for associated Dunkl polynomials.","marker":"[11]"},{"why":"Gives the classical Sheffer integral characterization that Theorem 3 extends to the Dunkl context.","marker":"[21]"},{"why":"Gives the classical Appell characterization by derivative-moment orthogonality that Theorem 1 extends.","marker":"[23]"},{"why":"Supplies the Fourier technique used in Section 4 to construct the functions $\\alpha_\\nu$ from their moment sequences.","marker":"[5]"},{"why":"Provides the positive measures $\\beta_\\nu$ for the truncated and other Appell-Dunkl moment problems used in Examples 4.1 and 4.2.","marker":"[6]"},{"why":"Introduces the Dunkl operator whose action replaces the derivative throughout the paper.","marker":"[4]"}],"fun_headline_variants":["Two Stieltjes integrals characterize Sheffer-Dunkl sequences","Sheffer-Dunkl sequences pinned by two moment integrals","Twin integral tests define Sheffer-Dunkl sequences","Moment-based Stieltjes integrals specify Sheffer-Dunkl","Two Stieltjes integral characterizations of Sheffer-Dunkl"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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 $g$.","fun_headline_variants_meta":{"raw":{"variants":["Two Stieltjes integrals characterize Sheffer-Dunkl sequences","Sheffer-Dunkl sequences pinned by two moment integrals","Twin integral tests define Sheffer-Dunkl sequences","Moment-based Stieltjes integrals specify Sheffer-Dunkl","Two Stieltjes integral characterizations of Sheffer-Dunkl"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00054,"raw_usage":{"total_tokens":2578,"prompt_tokens":923,"completion_tokens":1655,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":1569}},"tokens_in":539,"tokens_out":1655,"duration_ms":12229,"temperature":1.0,"reasoning_tokens":1569,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:32:27.902757+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the theorem used in both proofs to pass from a formal series $g(t)=\\sum \\mu_n t^n/\\gamma_n$ to a bounded-variation function with those moments."},{"cited_title":"Gil Asensi, J","cited_arxiv_id":null,"evidence_quote":"Establishes the Sheffer-Dunkl umbral-calculus framework, including $L_f$, the Dunkl translation, and the binomial identity for associated Dunkl polynomials."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the classical Sheffer integral characterization that Theorem 3 extends to the Dunkl context."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the classical Appell characterization by derivative-moment orthogonality that Theorem 1 extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Fourier technique used in Section 4 to construct the functions $\\alpha_\\nu$ from their moment sequences."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the positive measures $\\beta_\\nu$ for the truncated and other Appell-Dunkl moment problems used in Examples 4.1 and 4.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Dunkl operator whose action replaces the derivative throughout the paper."}],"review_version":1}