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REVIEW 3 major objections 4 minor 1 cited by

Popa's "Recurrent Sequences" and Reciprocity

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Switching to reciprocals computes a dozen asymptotic constants to 25 digits

desk verdict Plausible high-precision constants for a dozen recurrences, several genuinely new, but the Mavecha–Laohakosol algorithm is applied beyond the paper's own stated hypotheses, and the extension is asserted without proof or independent verification. read the letter →

arxiv 2412.11806 v3 pith:WZCPWOSJ submitted 2024-12-16 math.CA cs.DMmath.COmath.NT

Steven Finch This is my paper · ORCID
classification math.CAcs.DMmath.COmath.NT MSC 41A6039A10
keywords nonlinearrecurrencesasymptoticconstantsreciprocalsubstitutionexpansionsofiteratesMavecha-Laohakosolalgorithmhigh-precisioncomputationrecurrentsequences
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

This paper tackles a dozen nonlinear recurrences whose iterates grow like $x_k \sim \alpha k + \beta \ln(k) + C$, where the constant $C$ was previously known to exist but not numerically evaluated. The key move is to study the reciprocal sequence $y_k = 1/x_k$, which obeys its own recurrence and whose asymptotic series can be produced by a standard algorithm for iterates of analytic functions. The paper reports 25-digit values of $C$ for all twelve cases, including several boundary cases where the original asymptotic expansion for $x_k$ does not apply. A sympathetic reader cares because the constants are now concrete numbers, and the reciprocal viewpoint is shown to be the essential computational step rather than a cosmetic shortcut.

What carries the argument

The central object is the reciprocal substitution $y_k=1/x_k$ paired with the Mavecha–Laohakosol algorithm, a method for expanding iterates of an analytic map $g$ whose Taylor series at the origin begins $y+a_1 y^{\tau+1}+a_2 y^{2\tau+1}+\cdots$ (with $a_1<0$ and integer $\tau\ge1$) into an asymptotic series in powers of $k^{-1/\tau}$ and $\ln k$. The substitution converts $x_{k+1}=f(x_k)$ into $y_{k+1}=g(y_k)$ with $g(y)=1/f(1/y)$; the algorithm then delivers a series whose first coefficients are polynomials in $C$, so comparing the series to an exactly iterated value $y_K$ determines $C$. The paper's reported constants all come from this recipe.

What would settle it

For the reciprocal $p$-sequence with $p=1/2$ (where $g(y)=y/\sqrt{1+y}$), compute $y_K$ exactly for $K=10^6$ and for $K=2\cdot10^6$, using the six-term series in Section 5 to solve for $C$ both times; if the two estimates disagree beyond the 25th decimal, the paper's precision claim fails. Alternatively, compare the reported $C=0.8615711875687117305317813\ldots$ for the $q=2$ sequence against an independent high-precision extrapolation of $\sqrt{2}\,x_k - 2\sqrt{k} - \frac{1}{4}\ln(k)/\sqrt{k}$ from direct iteration; any discrepancy in the last digits would reveal underestimated error.

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Extended reading notes

Core claim

The paper's central claim is that, for any recurrence $x_{k+1}=f(x_k)$ with $x_k \sim \alpha k + \beta \ln(k) + C$, the constant $C$ can be computed to 25 digits by passing to $y_k = 1/x_k$. The reciprocal sequence satisfies $y_{k+1}=g(y_k)$ with $g(y)=1/f(1/y)$, and the Mavecha–Laohakosol algorithm expands $y_k$ in an asymptotic series in powers of $k^{-1/\tau}$ and $\ln k$; setting that series equal to an exactly computed $y_K$ and solving for $C$ yields the constants. Twelve cases are worked out in detail—$p$-sequences, radicals, exponentials, logarithms, and $q$-sequences—and in two boundary cases ($f(x)=x\exp(\sqrt{1/x})$ and $f(x)=x(1+\ln(1+1/x^2))$) the algorithm succeeds even where the original expansion for $x_k$ fails. The paper also corrects two coefficient typos in the source theorems and notes a duality between the $q=1/2$ and $p=1/2$ recurrences.

Load-bearing premise

The numerical scheme assumes that, at the chosen index $K$, the truncated asymptotic series for $y_K$ differs from the true value by less than the claimed $10^{-25}$ error, and no rigorous bound on the remainder is provided.

Editorial extensions

If this is right

  • The twelve constants are now available to 25 digits, providing numerical anchors for further asymptotic study of these recurrences and for checking future symbolic derivations.
  • The reciprocal method extends to boundary cases where Popa's original expansions do not apply, enlarging the class of recurrences whose constants can be computed.
  • The same procedure can be applied to any recurrence whose reciprocal map $g$ satisfies the algorithm's hypotheses, giving a general numerical tool for constants in similar asymptotic forms.
  • Correcting the two coefficient typos in the source theorems changes the numerical values derived from the affected corollary, so any prior computations based on those formulas need revision.
  • The observed duality between the $q=1/2$ and $p=1/2$ recurrences, and between $q=2$ and $p=2$, shows that seemingly different recurrences can share constants up to simple factors, suggesting underlying transformation identities.

Reading between the lines

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

  • Not pursued in the paper: the claimed 25-digit accuracy is asserted without a rigorous remainder bound, so a numerical-analysis follow-up could certify or slightly revise the final digits of each constant.
  • The method's success suggests that the open problem of finding a nonlinear recurrence whose constant $C$ has a closed-form expression (independent of the sequence) might be approached by looking for reciprocal maps $g$ whose asymptotic series coefficients simplify.
  • The same reciprocal-plus-algorithm recipe could be applied to recurrences with asymptotic forms like $x_k \sim \alpha k^p + \beta \ln k + C$ for other values of $p$, provided the corresponding $g$ is analytic at the origin with the required leading behavior.
  • The connection between $g(y)=y e^{-y}$ and the Lambert $W$ function hints that some of the computed constants may be expressible in terms of special functions; that link is not developed in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops a computational method for evaluating the constant term C in asymptotic expansions of positive sequences satisfying nonlinear recurrences. For x_{k+1}=f(x_k) with x_k ~ α k + β ln k + C (or analogous growth), the substitution y_k=1/x_k transforms the recurrence to y_{k+1}=g(y_k), and the author applies the Mavecha–Laohakosol algorithm to obtain a truncated asymptotic series for y_k containing C. The constant is then found by matching the series to an exactly computed value y_K. The paper presents expansions and 25-digit constants for a collection of recurrences (p-sequences, radicals, exponential/logarithmic cases, and q-sequences), and corrects two typos in Popa's theorems. The p=1/2 case is worked in full detail in the appendix.

Significance. If the expansions and constants are correct, the paper provides a useful collection of high-precision constants for a family of recurrences that have appeared in various contexts. The reciprocal transformation is a clever and potentially reusable device, and the p=1/2 appendix gives a genuinely worked example with explicit coefficient arrays and the resulting series to order k^{-6}. The procedure is a legitimate shooting/matching method rather than a data fit, since C enters as an integration constant of the asymptotic expansion. The claimed constants, however, are not accompanied by rigorous error bounds, and several cases use the Mavecha-Laohakosol formalism outside the hypotheses stated in the appendix; these gaps must be closed before the numerical claims can be regarded as established.

major comments (3)
  1. [§5 vs. §3.7, §4.3, §4.5] Section 5 states the Mavecha-Laohakosol algorithm applies to an analytic g(y) whose Taylor series at the origin starts as y + a_1 y^{τ+1} + a_2 y^{2τ+1} + ... with a_1<0 and τ≥1 an integer. However, §3.7 uses g(y)=y exp(-√y), which has expansion y - y^{3/2} + y^2/2 - y^{5/2}/6 + ... and is not analytic at 0, and §4.3 and §4.5 use g(y)=y/(1+y^q) with q=3/2 and q=4/3, corresponding to τ=3/2 and τ=4/3, which are not integers. The text asserts without proof that the algorithm 'does, however, apply' in these cases. Since the constants C=0.8791712792948618603132189, C=0.8010888849039666437110775, and C=0.8248745112329031526004762 rest on these expansions, a proof of the extended validity or an independent verification is required.
  2. [§5] The procedure 'We then set the value y_K equal to our series and numerically solve for C' is used to obtain the 25-digit constants, but the paper gives no remainder bound for the truncated series and no analysis of the numerical solve. The assertion that terms to order 1/k^6 with K≈10^6 might suffice is plausible but unproved. Without such an error estimate, the claimed number of correct digits is not justified. The author should supply a rigorous bound or provide an independent high-precision check, e.g., a second method or comparison of results at different K with rigorous intervals.
  3. [§1.2–§4.5] Most of the expansions in Sections 1–4 are presented as 'We find' with no derivations or input data (coefficients a_m, a_{0j}, b_j, or their analogues for the nonstandard cases). The reader cannot reproduce or audit the computations except for the p=1/2 example in the appendix. Since the paper's contribution is precisely these expansions and constants, the author should include enough algorithmic detail or a supplement, such as a Mathematica notebook, to make all results checkable.
minor comments (4)
  1. [References] Reference [7] has the obvious typo 'arXiv:24409.15175'; it should presumably be 'arXiv:2409.15175'.
  2. [§1.1] The displayed series for y_k contains the expression '− (− 1/24 + C + 10C^2 + 16C^3) 1/k^4' with a doubled minus sign; this should be simplified for readability.
  3. [General] The paper does not number its equations, which makes it awkward to refer to specific displayed results; adding equation numbers would help readers and referees.
  4. [§3.7, §3.8] The introductory reciprocal asymptotic formula is stated only for x_k ~ α k + β ln k + C, but in §3.7 and §3.8 the growth of x_k is quadratic or square-root; a short statement of the generalized reciprocal asymptotic relation used in those sections would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the constants C are independently defined limits, and the matching procedure determines C from exact iteration rather than fitting it.

full rationale

The paper's derivation chain is: for each recurrence x_{k+1}=f(x_k), define the reciprocal sequence y_k=1/x_k, which satisfies y_{k+1}=g(y_k); apply the Mavecha–Laohakosol algorithm (external reference [5], via the author's [9]) to obtain an asymptotic expansion for y_k containing C as an undetermined integration constant; then compute y_K exactly by iteration and solve the series equation for C. The constant C is defined independently as the limit of x_k - α k - β ln k (or the analogous limit in the non-logarithmic cases), not as the parameter that makes the series fit the data. Thus the 'prediction' is not equivalent to an input by construction. The paper's reliance on Finch's own prior work [9] for the explicit M–L formula is a self-citation, but the underlying algorithm is due to Mavecha and Laohakosol [5], and the 25-digit constants are new outputs, not restatements of the cited papers. The acknowledged extension of the algorithm to non-analytic g(y)=y exp(-√y) and non-integer τ in §3.7 and §4.3/§4.5 is an unproved validity assertion and a numerical error-bound concern, not a circular reduction: the series is still evaluated at an exactly iterated point and solved for a limit-defined constant. There is no fitted parameter renamed as a prediction and no self-citation chain that forces the result.

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

No free parameters are introduced: C is the target constant determined by the initial condition and computed, not fitted. The paper rests on external theorems (Popa's expansions, the Mavecha-Laohakosol algorithm) and on an unproved numerical remainder bound.

assumptions (3)
  • standard math Popa's asymptotic expansion theorems (Theorems 3/4 and 6/5 of [1]/[2]) give the form x_k ~ α k + β ln(k) + C + ... for the recurrences treated.
    The paper relies on these published theorems to write every expansion without proof; they are external standard results.
  • standard math The Mavecha-Laohakosol algorithm (and the formula from Finch [9]) gives the asymptotic expansion of y_k = 1/x_k for the reciprocal maps g(y)=1/f(1/y).
    The algorithm is cited [5] and its formula is 'assumed' from [9] in Section 5; the paper does not re-derive it.
  • ad hoc to paper The truncated series remainder after 1/k^6 terms is negligible at the index K used, so solving y_K = series(C) gives C to 25 digits.
    No error bound is given; the paper states only that K ≈ 10^6 suffices for 25 digits with the 1/k^6 series, which is a plausible but unproved numerical assumption.

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Cite this review

Pith. "Pith review of Popa's "Recurrent Sequences" and Reciprocity." pith.science (2026). https://pith.science/paper/WZCPWOSJ

@misc{pith2026241211806,
  author       = {Pith},
  title        = {Pith review of: Popa's "Recurrent Sequences" and Reciprocity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZCPWOSJ}},
  note         = {Machine review of arXiv:2412.11806}
}
read the original abstract

Dumitru Popa found asymptotic expansions for certain nonlinear recurrences, but left open the numerical evaluation of associated constants. We address this issue. A change of variables involving reciprocals and the algorithm of Mavecha & Laohakosol play a key role in our computations.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Exercises in Iterational Asymptotics

    math.NT 2024-11 conditional novelty 5.0 of 10

    For three families of nonlinear recurrences, the paper gives asymptotic expansions and high-precision numerical constants, including a new reciprocity relation for the q=3/2 case.

Reference graph

Works this paper leans on

20 extracted references · 19 canonical work pages · cited by 1 Pith paper

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