Appell Polynomials in Shifted Asymptotic Expansions: the Mills ratio, Hermite polynomials, and Stieltjes bounds

Lenka Mihokovi\'c; Neven Elezovi\'c; Tomislav Buri\'c

Appell Polynomials in Shifted Asymptotic Expansions: the Mills ratio, Hermite polynomials, and Stieltjes bounds

T0 review · 0 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2607.26636 v1 pith:JDUANJJ4 submitted 2026-07-29 math.CA

Tomislav Burić , Neven Elezović , Lenka Mihoković This is my paper · ORCID

classification math.CA MSC 41A6033C4530E0541A2160E05

keywords AppellpolynomialsasymptoticexpansionMillsratioHermiteBorel–LaplacetransformStieltjesmomentsequencePadéapproximantsnon-centralGaussiantail

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 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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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 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.

The 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.

Soft 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.

The 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.

Who 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.

Referee Report

0 major / 4 minor

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.

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.

minor comments (4)

[Section 5, Theorem 5.1] 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 4.2] 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 3, Theorem 3.2] 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 5.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: core derivations are self-contained or rest on classical external theorems.

full rationale

The paper's central chain is not circular. Theorem 2.1 (from [4]) is stated and given a complete proof in the text by Taylor expanding (x+t)^{-k-1}; it is not imported as an unverified premise. Proposition 2.3 is a direct Watson-lemma argument: the Appell addition formula and the Laplace integral of s^n give the coefficients, and the stated hypotheses are substantive (growth, Taylor behaviour, convergence) rather than equivalent to the conclusion. Theorem 3.2 is fully self-contained: Lemma 3.1 provides the integral representation M(x+t)=∫ e^{-xs} e^{-ts-s^2/2} ds, the Hermite generating function supplies the Taylor coefficients, and the explicit derivative identity g^{(N)}(s)=(-1)^N He_N(s+t)e^{-ts-s^2/2} yields the remainder bound |ρ_N| ≤ M_N(t)/x^{N+1}. The finite-difference identities (Prop. 2.4) are algebraic consequences of Appell recurrences. The Stieltjes/Padé bracketing (Theorem 5.1) depends on the classical monotone Padé theory for Stieltjes series [2,1], which is an external theorem accurately cited, and the paper explicitly flags the one-sided shift condition; this is a dependency, not a circular definition. Self-citations [3],[4],[5] are either proved here ([4]) or used only as illustrative examples (De Moivre's formula), so they are not load-bearing. No parameter is fitted and then renamed a prediction; equations are not assumed equivalent to their targets. Hence no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

Central derivation uses no fitted constants: Hermite coefficients, moments, and Padé coefficients are fixed by the generating function or measure. The only imported machinery is standard (Watson's lemma, Appell/Hermite identities, Padé and moment theory), and the paper's contribution is a repackaging of these into shifted asymptotics and Stieltjes bounds.

assumptions (5)

standard math Watson's lemma: the Laplace transform of a function with an asymptotic Taylor series at s=0 yields an asymptotic expansion in inverse powers of x.
Used in the proof of Proposition 2.3, §2.2, to convert the local expansion of A(-s)e^{-ts} into the x^{-n-1} expansion.
standard math Appell addition formula: A(z)e^{tz}=Σ R_n(t) z^n/n! and the coefficient convolution that gives R_n(t) from (a_n).
Central to the shift theorem in §2 and to the Gaussian and Stieltjes examples in §§3–5.
standard math Hermite polynomial identities: generating function e^{ys-s²/2}=Σ He_n(y)s^n/n!, derivative d^N/dz^N e^{-z²/2}=(-1)^N He_N(z)e^{-z²/2}, and parity He_n(-z)=(-1)^n He_n(z).
Core to Theorem 3.2 and Theorem 4.1; these identities drive the Mills–Hermite expansion and the non-central tail formula.
standard math Padé bracketing and monotonicity for Stieltjes series, cited from [2,1].
Load-bearing for Theorem 5.1; the inequalities (5.3) are classical and are cited rather than proved in the paper.
standard math Determinacy criteria for Stieltjes moment problems, including Carleman's condition.
Used in §5 to discuss when the Padé staircase converges to f and when the measure is uniquely determined.

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Pith. "Pith review of Appell Polynomials in Shifted Asymptotic Expansions: the Mills ratio, Hermite polynomials, and Stieltjes bounds." pith.science (2026). https://pith.science/paper/JDUANJJ4

@misc{pith2026260726636,
  author       = {Pith},
  title        = {Pith review of: Appell Polynomials in Shifted Asymptotic Expansions: the Mills ratio, Hermite polynomials, and Stieltjes bounds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JDUANJJ4}},
  note         = {Machine review of arXiv:2607.26636}
}

abstract

A theorem of Buri\'c, Elezovi\'c and Vuk\v si\'c states that translating the argument in an asymptotic expansion, $f(x)\sim\sum(-1)^na_nx^{-n-1}$, to $f(x+t)$, replaces the constant coefficients $a_n$ by the Appell polynomials $R_n(t)$ generated by $(a_n)$. Their motivating examples came from the gamma and polygamma functions, where Bernoulli polynomials occur. We extend this construction beyond that setting and identify the Borel--Laplace representation $L_A(x)=\int_0^\infty e^{-xs}A(-s)\,\dd s$, whenever the integral exists. For the Gaussian kernel $A(-s)=e^{-s^2/2}$, this representation gives the \emph{Mills--Hermite} expansion $M(x+t)\sim\sum(-1)^n\He_n(t)x^{-n-1}$, extending the classical scalar expansion. We establish an explicit remainder estimate and derive finite-difference formulas that remain in the Hermite polynomial algebra. As an application, we obtain a large-threshold expansion of the non-central Gaussian tail with explicit polynomial dependence on the mean. If $A(-s)$ is itself the Laplace transform of a positive measure, then the coefficients form a Stieltjes moment sequence. The associated Pad\'e convergents give a systematic hierarchy of two-sided bounds in which successive lower and upper approximants incorporate the moments one at a time.

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Works this paper leans on

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [4]

Burić, N

T. Burić, N. Elezović, L. Vukšić,Appell’s polynomials and asymptotic expansions, Mediterr. J. Math.13(2016) 899–912

2016

[2] [1]

N. I. Akhiezer,The Classical Moment Problem and Some Related Questions in Analysis, Oliver & Boyd, 1965

1965

[3] [2]

G. A. Baker, P. Graves-Morris,Padé Approximants, 2nd ed., Cambridge University Press, 1996

1996

[4] [3]

Burić, N

T. Burić, N. Elezović,Bernoulli polynomials and asymptotic expansions of the quotient of gamma functions, J. Comput. Appl. Math.235(2011) 3315–3331

2011

[5] [5]

Elezović,Local binomial expansions with an Appell shift, and the mean absolute deviation of the binomial distribution, arXiv:2607.18494, 2026

N. Elezović,Local binomial expansions with an Appell shift, and the mean absolute deviation of the binomial distribution, arXiv:2607.18494, 2026

arXiv 2026

[6] [6]

Gasull, F

A. Gasull, F. Utzet,Approximating Mills ratio, J. Math. Anal. Appl.420(2014) 1832–1853

2014

[7] [7]

J. J. Heckman,Sample selection bias as a specification error, Econometrica47(1979) 153–161

1979

[8] [8]

N. L. Johnson, S. Kotz, N. Balakrishnan,Continuous Univariate Distributions, Vol. 1, 2nd ed., Wiley, 1994

1994