{"id":"ab7841dc-df34-47ef-ae9a-b394aa31826e","arxiv_id":"2607.26636","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Shifting an asymptotic expansion replaces its coefficients by Appell polynomials, yielding a Mills–Hermite expansion with explicit remainder and Stieltjes Padé bounds.","lead":"This paper extends a known theorem: shifting the argument of an asymptotic expansion replaces its constant coefficients by Appell polynomials, and makes the mechanism precise through a Borel–Laplace integral. The payoff is a shifted expansion for the Gaussian Mills ratio with an explicit error bound, plus a hierarchy of two-sided rational bounds for Stieltjes-type transforms.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the core shift theorem, the Mills-Hermite remainder bound, and the Stieltjes-Pade bracketing all survive scrutiny; the only external dependency is classical and correctly applied.","rationale":"I read the full text and focused on the central mathematical claims. The shifted asymptotic theorem is elementary and correctly proved; Corollary 2.2 is justified by the uniform error on compact shift sets. Proposition 2.3 is a standard Watson-lemma application. The main novel result, Theorem 3.2, checks out in detail: the Hermite derivative identity, the finite supremum, and the integral remainder all work, giving the stated explicit remainder bound with local uniformity. The non-central tail expansion follows correctly by the parity of Hermite polynomials. For the Stieltjes section, the only point of external reliance is the classical Pade bracketing theorem; the paper cites it, states the required positivity condition, and the numerical examples satisfy the claimed moment matching and inequalities. I found no internal inconsistency or hidden assumption that would undermine the central claim. The reader's weakest-assumption flag identifies the same external dependency, but I do not regard it as a load-bearing correctness risk. Therefore the ACCEPT verdict stands unchanged.","tokens_in":12491,"tokens_out":32067,"duration_ms":291301,"concrete_test":"Independently verify the Stieltjes bracket outside the paper's examples: take F(z)=int_0^infty e^{-sigma}(1+sigma z)^{-1} dsigma, compute the exact [m-1/m] and [m/m] Pade approximants from the moments for m=1,2,3, and check [m-1/m] <= F <= [m/m] and the monotone ordering at z=0.1,1,10 with high precision. In parallel, recompute several Theorem 3.2 remainders (e.g., x=2,5, t=-2,0,3, N=1..5) and compare |rho_N| with M_N(t)/x^{N+1} to confirm the bound is not violated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The central components check out. Theorem 3.2 is self-contained: differentiating g(s)=exp(-ts-s^2/2) gives g^(N)(s)=(-1)^N He_N(s+t)exp(-ts-s^2/2), the sup M_N(t) is finite, and the integral remainder bound leads exactly to |rho_N| <= M_N(t)/x^{N+1}. Proposition 2.3 is a standard Watson-lemma argument, and Theorem 2.1 is elementary and correct. The only genuine dependency is in Theorem 5.1, where the Pade bracketing and monotone ordering for Stieltjes series are cited from [2,1] rather than proved. This is a real external assumption, but it is classical, accurately stated, and the paper explicitly flags the one-sided shift condition t+supp mu subset [0,infty). I checked the first-level Jensen/Pade bounds and the example approximants: they match the moments to the claimed orders. The Stieltjes determinacy remark is ancillary. Thus the reader's identified weak assumption is a dependency, not a correctness risk.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general framework for shifting the argument in alternating asymptotic expansions. Starting from the known result that f(x+t) has coefficients given by the Appell polynomials generated by the coefficients of f, it identifies a Borel--Laplace representation whenever the corresponding kernel exists. The main applications are: (i) the Mills ratio of the Gaussian distribution, where the shifted expansion is expressed in terms of Hermite polynomials and is accompanied by an explicit remainder bound (Theorem 3.2); (ii) finite-difference formulas for the shift; (iii) a large-threshold expansion of the non-central Gaussian tail; and (iv) for Stieltjes kernels, two-sided Padé bounds built from the shifted moment sequence (Theorem 5.1). The paper also treats moving shifts in Corollary 2.2 and gives worked examples for exponential, Poisson, and Marchenko--Pastur laws.","tokens_in":12746,"tokens_out":14968,"duration_ms":135290,"significance":"If the results stand, the paper gives a clean algebraic treatment of how a translation in the asymptotic variable transforms the coefficients, with concrete benefits for the Mills ratio and for Stieltjes functions. The main mathematical content is sound: Theorem 3.2 is self-contained and correctly proved via the Hermite derivative identity, and Proposition 2.3 is a standard but correctly applied Watson-lemma argument. A particular strength is that the Mills--Hermite expansion is obtained with an explicit, finite remainder bound, rather than just as a formal series. The Stieltjes section organizes the classical Padé bracketing in terms of the Appell-generating moment sequence, which is a useful perspective. No parameters are fitted and no circular reasoning is apparent. The only external dependency is the classical Padé bracketing theorem cited in Theorem 5.1; the authors state the needed result and the citation is appropriate, so this is a dependency rather than a correctness risk.","major_comments":[],"minor_comments":[{"comment":"The proof of the bracketing inequalities (5.3) is delegated entirely to [2,1]. Since this theorem is the central tool of Section 5, it would be helpful to state the precise classical form used (e.g., the Stieltjes-series Padé alternation theorem) and to verify that the hypotheses hold for F(z). The current one-sentence citation is mathematically acceptable, but a more explicit statement would improve readability and make the dependency transparent.","section":"Section 5, Theorem 5.1"},{"comment":"The sentence 'Since M is convex, r is convex in µ' is used to identify the sign of the second difference. A one-line justification, such as M''(z)>0 via the identity M''(z)=(1+z^2)M(z)-z, would make the argument self-contained and avoid relying on an unstated fact.","section":"Section 4.2"},{"comment":"The definition of M_N(t) in (3.2) is clear, but the proof of finiteness is compressed. It may be worth explicitly displaying the equivalent form M_N(t)=e^{t^2/2} sup_{s≥0}|He_N(s+t)|e^{-(s+t)^2/2}, which makes the decay immediate and also clarifies the compact-uniformity claim.","section":"Section 3, Theorem 3.2"},{"comment":"The examples are well chosen, but the numerical values in Table 1 and the bracketing examples are reported without indicating how many digits are trustworthy. Adding a short note on the precision (e.g., mpmath settings) would be useful for reproducibility, especially because the Hermite series is only asymptotic.","section":"Section 5.1"}],"recommendation":"accept","confidential_remarks":"I concur with the reader's assessment. The central arguments are correct, the exposition is careful, and the caveats concerning the one-sided shift and the classical Padé result are stated honestly. The paper is within scope for math.CA and represents a solid incremental contribution. The minor comments above do not require another full round of review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is solid and worth your time. The main new result is Theorem 3.2: the shifted Mills ratio has an asymptotic expansion in Hermite polynomials with an explicit remainder bound, M(x+t) = sum (-1)^n He_n(t) x^{-n-1} + rho_N, |rho_N| <= M_N(t)/x^{N+1}. The proof is direct and correct — the Hermite derivative identity and the sup bound on the remainder work. This is genuinely useful: it turns the shift into an algebraic operation on the coefficients and gives uniform control on compact shift sets.\n\nThe paper does several other things well. Proposition 2.3 places the shift theorem in a clean Borel–Laplace framework, so the Appell polynomials appear naturally from multiplying the kernel by e^{-ts}. The finite-difference formulas in Section 2 are correct and stay inside the Hermite algebra. The non-central Gaussian tail expansion (4.2) is a nice application, and the formal derivative check with the Hermite recurrence is a good consistency test. The Stieltjes section correctly applies classical Padé bracketing to shifted moment sequences, and the three examples (exponential, Bell, Catalan) with shared low moments make the bounding staircase transparent. The numerical table matches the expansion.\n\nSoft spots are few and mostly minor. The Stieltjes bracketing in Theorem 5.1 rests on the classical Padé monotonicity theorem cited from Baker–Graves-Morris and Akhiezer, not proved here. That is a real external dependency, but it is standard and the paper flags it. The shift is one-sided for Stieltjes sequences (t >= 0), which the paper also states clearly; that limitation is intrinsic, not a flaw. The non-central tail expansion is essentially a corollary of the main theorem with t = -mu, but the paper says as much. The moving-shift corollary and the De Moivre example are interesting but somewhat separate; they do not affect the core results.\n\nThe citation pattern is honest: the prior shift theorem [4] is properly attributed, and the self-citations are illustrative. No fitted parameters, no circularity, no invented entities. The proofs I checked are sound.\n\nWho is this for? Anyone working on asymptotics of special functions, Mills-ratio approximations, or moment-problem bounds. It deserves a serious referee rather than a desk reject. I would send it to peer review and expect a minor-revision outcome. I would likely cite it for the shifted Mills–Hermite expansion in my own work on Gaussian tail approximations.","headline":"A clean, correct paper that gives the shifted Mills–Hermite expansion with an explicit remainder and organizes Stieltjes Padé bounds under the same Appell framework; worth refereeing and worth citing for the Mills-ratio expansion.","tokens_in":13248,"tokens_out":1092,"would_cite":true,"duration_ms":13005,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["41A60","33C45","30E05","41A21","60E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Shifting the argument in an asymptotic expansion converts the constant coefficients into Appell polynomials, and this one algebraic rule drives both the Hermite/Mills-ratio expansions and the two-sided Stieltjes bounds.","keywords":["Appell polynomials","asymptotic expansion","Mills ratio","Hermite polynomials","Borel–Laplace transform","Stieltjes moment sequence","Padé approximants","non-central Gaussian tail"],"falsifier":"Numerically evaluate the Mills ratio M(x+t) at, say, x=10, t=3, and check whether |M(x+t)-sum_{n=0}^{N-1} (-1)^n He_n(t) x^{-n-1}| exceeds M_N(t)/x^{N+1} for any N; a violation would falsify Theorem 3.2. Similarly, compute the Stieltjes transform of a compactly supported measure and test whether the level-m Padé approximants indeed bracket it for all x>0; because the bracketing theorem is cited rather than proved, any counterexample would refute Theorem 5.1.","tokens_in":12388,"feed_emoji":"🧮","tokens_out":5172,"duration_ms":50731,"temperature":0.7,"pith_summary":"The paper establishes that shifting the argument in an asymptotic expansion f(x) ~ sum (-1)^n a_n x^{-n-1} is a purely algebraic operation: the constants a_n are replaced by the Appell polynomials R_n(t) generated by the sequence (a_n). It embeds this in a Borel–Laplace representation, showing that the shifted coefficients follow from a kernel A(-s) whose Taylor coefficients at 0 are (-1)^n a_n. For the Gaussian kernel e^{-s^2/2}, this yields a shifted Mills-ratio expansion in Hermite polynomials with an explicit remainder bound, and leads to finite-difference formulas and a large-threshold expansion of the non-central Gaussian tail. For kernels that are themselves Laplace transforms of positive measures, the coefficients form a Stieltjes moment sequence, and Padé approximants produce a monotone hierarchy of two-sided bounds. The paper thus unifies two previously separate classes—gamma-related Bernoulli expansions and the Mills-ratio/Hermite setting—under one Appell-generating-function mechanism.","feed_headline":"Shift rule turns asymptotic coefficients into Appell polynomials","feed_subtitle":"The same algebraic shift yields Hermite expansions for the Mills ratio and two-sided Padé bounds for Stieltjes transforms.","key_machinery":"The central objects are the Appell polynomials R_n(t) (generated by A(z)e^{tz} = sum R_n(t) z^n/n!, so R_n(0)=a_n and R'_n=nR_{n-1}) together with the Borel–Laplace representation f(x)=int_0^infty e^{-xs} A(-s) ds. The Appell property turns a shift of the argument into multiplication of the generating function by e^{tz}, moving the Taylor coefficients from numbers to polynomials; the Laplace representation then makes the remainder amenable to Watson's lemma. For Stieltjes-admissible sequences, the same coefficients are moments, so Padé approximants built from them give rigorous two-sided approximations.","core_discovery":"The central claim is that the Appell polynomials R_n(t), defined by A(z)e^{tz} = sum R_n(t) z^n/n! with A(z)=sum a_n z^n/n!, are exactly the coefficients that appear after shifting the argument in an asymptotic expansion: f(x+t) ~ sum (-1)^n R_n(t) x^{-n-1}. The paper proves this in a Borel–Laplace framework (Proposition 2.3) and, in the Gaussian case A(-s)=e^{-s^2/2}, obtains the Mills–Hermite expansion with the explicit remainder bound |rho_N| <= M_N(t)/x^{N+1}, where M_N(t)=sup_{s>=0} |He_N(s+t)| e^{-ts-s^2/2}. In the Stieltjes case, where A(-s) is the Laplace transform of a positive measure, the shifted coefficients are moments of a shifted measure, and Padé convergents bracket the trans","pith_inferences":["The one-sidedness of the Stieltjes shift (only t>=0 is treated) suggests that asymmetric versions of the staircase might hold for t<0 if the shifted moments remain a Stieltjes sequence under a different representing measure; the paper leaves this open.","The formal identity d_x F = d_t F and the Hermite recurrence suggest the Appell shift construction may extend to multi-parameter shifts or to differential operators in the shift variable, yielding higher-order finite-difference schemes beyond the symmetric ones derived here.","The explicit remainder bound M_N(t) might be used to design optimal truncation rules in the shifted Hermite approximation, analogous to the least-term truncation for the alternating scalar Mills series.","Because the same Appell sequence governs both the Gaussian (Hermite) and Stieltjes cases, the method may transfer to other kernels (e.g., Euler polynomials or generalized Hermite polynomials) to produce new bracketing identities for special functions."],"forward_implications":["Shifting the argument in any asymptotic expansion with coefficients a_n is a purely algebraic Appell operation; the result holds uniformly for bounded shifts that may depend on x (moving shifts), covering cases like De Moivre's mean-absolute-deviation expansion.","The Mills ratio has a shifted asymptotic expansion M(x+t) ~ sum (-1)^n He_n(t) x^{-n-1} with a computable remainder bound, giving uniform asymptotic control on compact t-intervals.","The non-central Gaussian tail P(X>a) with X~N(mu,1) has a large-threshold expansion in which each coefficient is a Hermite polynomial in mu, and the corresponding formal derivative satisfies d_pi/d_mu = phi(a-mu) exactly.","For any positive measure mu, the coefficients a_n = integral sigma^n d_mu(sigma) produce a monotone Padé staircase [m-1/m] <= f <= [m/m] whose successive brackets improve the estimate by one moment at a time; the error is O(x^{-2m-1}).","The same first moments give identical initial brackets for different distributions; for example, the exponential, Poisson(1), and Marchenko–Pastur (parameter one) measures all share the same level-one bracket."],"fun_headline_variants":["Shift trick yields Appell polynomials for asymptotic expansions","New error bound for Mills ratio via Hermite polynomials","Padé hierarchy yields two-sided bounds for Stieltjes transforms","Gaussian shift yields Mills-Hermite expansion with tight error bound","Appell polynomials via shifting: from Bernoulli to Hermite"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The two-sided Stieltjes bounds rest on an unproved classical theorem—that Padé approximants of Stieltjes series form a monotone bracketing staircase—and on the shift staying nonnegative so the shifted coefficients remain a Stieltjes moment sequence; if either fails, the inequalities collapse.","fun_headline_variants_meta":{"raw":{"variants":["Shift trick yields Appell polynomials for asymptotic expansions","New error bound for Mills ratio via Hermite polynomials","Padé hierarchy yields two-sided bounds for Stieltjes transforms","Gaussian shift yields Mills-Hermite expansion with tight error bound","Appell polynomials via shifting: from Bernoulli to Hermite"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000777,"raw_usage":{"total_tokens":3340,"prompt_tokens":877,"completion_tokens":2463,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":621,"completion_tokens_details":{"reasoning_tokens":2382}},"tokens_in":621,"tokens_out":2463,"duration_ms":17476,"temperature":1.0,"reasoning_tokens":2382,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T11:49:03.638417+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically evaluate the Mills ratio M(x+t) at, say, x=10, t=3, and check whether |M(x+t)-sum_{n=0}^{N-1} (-1)^n He_n(t) x^{-n-1}| exceeds M_N(t)/x^{N+1} for any N; a violation would falsify Theorem 3.2. Similarly, compute the Stieltjes transform of a compactly supported measure and test whether the level-m Padé approximants indeed bracket it for all x>0; because the bracketing theorem is cited rather than proved, any counterexample would refute Theorem 5.1.","supporting_citations":[],"review_version":1}