Pith. sign in

REVIEW 2 major objections 5 minor 55 references

Two sliding least-squares estimators compute unknown constants inside proven asymptotic expansions with explicit convergence-rate guarantees.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-07-31 23:53 UTC pith:AJ3GEI4F

load-bearing objection A genuinely rigorous convergence analysis for sliding least squares, but the advertised learning of DLW-type constants works only when you already have the hard part (high-order expansions); sT-LLSQ only certifies the dominant parameter. the 2 major comments →

arxiv 2607.23287 v1 pith:AJ3GEI4F submitted 2026-07-25 stat.ML cs.LGcs.NAmath.COmath.NA

Learning Asymptotics with Convergence-Rate Guarantees using Linear Least Squares

Christos N. Efrem This is my paper
classification stat.ML cs.LGcs.NAmath.COmath.NA MSC 05-0805A1668W4076M4582B2682B27
keywords asymptotics learning theorysliding least squaresconvergence-rate guaranteesasymptotic expansionsanalytic combinatoricsTikhonov regularizationratio methodparameter estimation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper's central claim is that unknown constants inside a proven asymptotic expansion can be computed numerically with rigorous convergence-rate guarantees by fitting the expansion's exponential-product form to data through sliding least-squares problems. It proves that the constants are unique, that the sliding estimator y*_n = A†b converges to the transformed parameter γ whenever a per-component condition ϑ_j(n) = o(1) holds, and that the Tikhonov variant y*_n = Cb always converges for the dominant component at rate O(φ_{k−1}(n)/φ_k(n)) = o(1). The methods turn non-constructive existence proofs into computable values, demonstrated on Fibonacci, Catalan, and factorial asymptotics. The paper also documents a failure mode: sLLSQ can diverge completely even with zero residuals when an oscillating lower-order term is hidden in the expansion.

Core claim

The discovery is a reduction: after taking logarithms, the exponential-product asymptotic form becomes linear in the unknown parameters, so estimating them is exactly an ordinary least-squares problem with a design matrix built from the known basis functions φ_j. The main theorems give closed-form estimators and matching error bounds: for sLLSQ, ||y*_n − γ|| = O(η(n)), where η(n) is a maximum of two terms involving the pseudoinverse of the design matrix; for sT-LLSQ, ||y*_n − γ|| = O(1) for all components and O(φ_{k−1}(n)/φ_k(n)) = o(1) for the dominant one. Since the map u is bijective with nonzero derivative, these rates transfer to the original parameters α via the convergence-rate invari

What carries the argument

The central object is the exponential-product asymptotic form bf(n;x) = exp(Σ φ_j(n) u_j(x_j)), whose logarithm is linear in the transformed parameters γ_j = u_j(x_j). The sliding estimator solves min_y ||A y − b||², with A the windowed matrix of the φ_j's and b the log-data vector; its closed-form solution A†b (and Tikhonov variant Cb, with C = (AᵀA + μI)⁻¹Aᵀ) is what carries the argument. The key identities are the decomposition b = Aγ + log(1 + Σβ_l g_l) + z, which splits the estimate into the true parameter plus a controllable perturbation, and the asymptotic invariants that transfer Θ-equivalent error rates through the inverse transformation x = u^{-1}(y).

Load-bearing premise

The entire guarantee rests on the assumption that an asymptotic expansion of sufficiently high order, with the exact basis functions φ_j and error scale g_l, has already been proven for the sequence; without those higher-order terms, sLLSQ's bound can diverge.

What would settle it

For a known sequence such as a_n = 7 n^{1/3} (5/2)^n (1 + (−1)^n / √n), compute the sLLSQ estimate y*_n = A†b with m = k = 3 and compare ||y*_n − γ|| to the theorem's O(n^{3/2} log n) bound: if the error does not grow like Θ(n^{3/2} log n), or if it converges despite ϑ_j(n) not being o(1), the claimed sufficient conditions would be wrong.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any asymptotics of the form (2.1) with the exponential-product structure can be numerically inverted with certified rates, complementing the ratio method where only rough estimates were available.
  • The convergence-rate invariant means one proven expansion yields simultaneous estimates of all parameters in a single vector, rather than separate per-parameter sequences.
  • The sT-LLSQ bound O(φ_{k−1}/φ_k) = o(1) gives an unconditional, if slow, way to compute the dominant parameter even when the asymptotics order is low.
  • Higher-order asymptotic knowledge strictly improves or preserves the sLLSQ error bound (η(n;p) = O(η(n;p−1))), so investing in deeper expansions is provably rewarded.
  • The counterexample shows that sLLSQ's global-minimum value is not a reliable indicator of success; zero residuals can coexist with divergence, shifting practical attention to the ϑ_j conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A natural extension is to other loss functions or regularization penalties (e.g., ℓ1/LASSO) that preserve the sliding-window idea; the paper's universal dominant rate suggests the method is robust to the choice of penalty.
  • The approach may adapt to asymptotics with unknown basis functions, first estimating the φ_j themselves, though the current proof assumes the basis is known a priori.
  • Since the estimators are closed-form, the theory could combine with exact arithmetic to certify constants to arbitrary precision, turning the asymptotic bound into a finite-time guarantee once explicit constants are extracted.
  • The structure of the error bound suggests the same technique may work for multivariate or multi-scale expansions where the ratio method breaks down.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper introduces a framework called Asymptotics Learning Theory (ALT) for computing the finite parameter vector α in a proven asymptotic expansion f(n) = bf(n;α)(1 + Σ_{l=1}^p β_l g_l(n) + O(g_{p+1}(n))) from exact values of f. The theory specializes to exponential-product forms bf(n;x) = exp(Σ_j φ_j(n)u_j(x_j)), covering α_1 n^{α_2} α_3^n, stretched-exponential, and power-law asymptotics. Two sliding-window estimators are studied in closed form: sLLSQ, y*_n = (A^T A)^{-1}A^T b = A†b, and sT-LLSQ, y*_n = (A^T A + μI)^{-1}A^T b, where A is the windowed matrix of the φ_j. Theorem 3.14 establishes uniqueness, residual decay F*_n=O(||g_1||^2), and the bound ||y*_n − γ||=O(η(n)) with per-component rates ϑ_j(n); convergence is certified exactly when ϑ_j(n)=o(1), and Proposition 3.7 transfers rate bounds from the transformed parameters γ to the original α. Theorem 3.27 gives ||y*_n − γ||=O(χ(n))=O(1) and an unconditional dominant-component rate y*_{k,n} − γ_k = O(φ_{k−1}(n)/φ_k(n)) = o(1). Applications to analytic combinatorics (Fibonacci, Catalan, factorials, and an eleven-form family in Tables 4–6) are compared with the ratio method (Prop. 5.1) and tested numerically. Section 6.4 constructs a p=0 example with a hidden alternating term where sLLSQ diverges despite zero residuals, showing that the sufficiency conditions are genuinely necessary.

Significance. If the results hold up, the paper contributes a fairly general, closed-form family of estimators for the parameters of exponential-product asymptotics, with explicitly stated sufficient conditions, per-component rate bounds, and a Θ-transfer theorem (Prop. 3.7) that lets rates for y* convert to rates for the original parameters. The appendix proofs are coherent, and I independently verified several representative computations: det(A^T A)=6 for m=3 and 20 for m=4 in Remark 3.23, the Θ(1/n), Θ(1/n^2) vs Θ(1), Θ(1/n) components of A† log(1+β_1 g_1), and the p=0 AF-1 bound ||y*_n − γ||=O(n^2 log n) of Remark 4.1. The framework is not circular: γ enters only through the assumed expansion (2.1); the bounds are functions of A† and the decreasing scale g_l, with β_l as symbolic constants; there are no self-citations. The strongest clean result is the unconditional dominant-component rate y*_{k,n} − γ_k = O(φ_{k−1}/φ_k) (Lemma 3.26/Theorem 3.27), valid for any μ>0, any norm, and any ν≥1 for the general regularized problem. The §6.4 counterexample (zero residuals, diverging sLLSQ) is a valuable negative result that correctly identifies the sufficiency conditions as the load-bearing checkpoin

major comments (2)
  1. [Abstract; §1.1; Theorems 3.14/3.27; Remarks 4.1–4.3] The paper's motivating settings (DLW theorem, catalytic equations, chordal graphs; §1.1 and eq. (1.3)) provide only leading-order asymptotics, i.e., AF-2 with p=1. For that order, Table 4 gives ||y*_n − γ||=O(log n) for sLLSQ, so the condition ϑ_j(n)=o(1) in Theorem 3.14 fails and full-vector convergence is not certified. Theorem 3.27 certifies only the dominant component γ_k (Table 6, Remark 3.34); for j<k it gives only y*_{j,n} − γ_j = O(1), and the o(1) in the Table 6 footnote describes convergence to the regularized bias 0, not to γ_j. Thus, in the very settings that motivate the paper, certified recovery is limited to the leading exponential parameter (via sT-LLSQ) unless higher-order terms have been proven in advance — precisely the 'hard part' flagged in Remarks 4.1–4.3. The abstract's 'convergence (to the correct values of unknown parameters)' should be qualified. This is a scope
  2. [Section 4, Tables 4–6] Tables 4–6 carry the applications and operationalize the ϑ_j(n)=o(1) checkpoints, but the Puiseux estimates are asserted without derivations or scripts. Remark 3.23 shows the rates are fragile: with the same φ_j and g_l, m=3 gives η(n)=Θ(1/n) while m=4 gives η(n)=Θ(1). With the many symbolic rate entries (11 forms × orders p=0..3, plus det(A^T A) values and the §6.4 Θ-estimates), the reader cannot audit those entries. Please provide the CAS notebook/script (e.g., the MATLAB symbolic commands) or a supplementary derivation for each row family. Without such artifacts, the rate tables function as trusted assertions rather than verified results.
minor comments (5)
  1. [Title] Typo in the manuscript header: 'CONVERGENCE-RA TE GUARANTEES' should be 'CONVERGENCE-RATE GUARANTEES'.
  2. [Abstract] The phrase 'introduce a new research area' is stronger than this first paper supports; 'framework' or 'methodology' would be more defensible and would not undercut the technical contribution.
  3. [Appendix B.3, proof of Lemma 3.26] The step following (B.4) moves from entrywise O(1) and O(φ_{k−1}/φ_k) estimates, valid for arbitrary (eγ, w), to uniform bounds on the matrices C and D. Since the O-constants may in principle depend on eγ and w, the argument should restrict to unit basis vectors and use finiteness of the test set; the conclusion is correct, but this step is compressed.
  4. [Section 4, Tables 4–6] All rate tables use m=k. Given Remark 3.23 (increasing m from 3 to 4 flips η from Θ(1/n) to Θ(1)), practitioners need explicit guidance on whether and when m>k is safe; Proposition 3.22 covers only the regime where convergence already holds. A sentence in Section 4 would prevent misreading the tables as valid for all m.
  5. [Table 7] For the rows 'AF-2 with known α_2 / α_3', the sT-LLSQ rate columns follow from the transformations noted in parentheses, but the dimension change of the transformed problem is not shown; a pointer to the corresponding AF-6/AF-10 rows would make the comparison self-contained.

Circularity Check

0 steps flagged

No significant circularity: the estimators are derived from the assumed asymptotic expansion and their error bounds follow from that expansion; the target γ is never used to construct the estimators.

full rationale

The paper's derivation chain is self-contained in the sense required by the circularity criteria. The input is a proven asymptotic expansion (2.1) with unknown α (equivalently γ) plus a known asymptotic scale {g_l} and known basis functions {φ_j}. The estimators y*_n = A†b (sLLSQ, eq. (3.17)) and y*_n = Cb (sT-LLSQ, eq. (3.34)) depend only on the data vector b and the design matrix A built from φ_j(n); they do not use the target γ. The main error estimates are obtained by substituting the assumed expansion into b, yielding b = Aγ + log(1 + Σ β_l g_l) + z, and then bounding A† times the remainder terms (Theorem 3.14, eqs. (B.1), (3.20)) or C times the remainder plus the regularization bias -μDγ (Theorem 3.27, eqs. (B.5), (3.36), (3.40)). The β_l are treated as symbolic constants whose existence is assumed, not fitted values, so there is no fitted-input-called-prediction pattern. The sufficient convergence conditions ϑ_j(n)=o(1) and ψ_k(n)=o(1) are expressed in terms of the known matrix A† or C and the known error scale {g_l}, not in terms of the unknown γ. The paper explicitly documents where convergence fails (Remark 4.1, Section 6.4), which further confirms that the guarantees are not vacuously forced by construction. There are no self-citations: the reference list is entirely external, and no uniqueness theorem or ansatz is imported from the author's prior work. The only substantive limitation is that sLLSQ's guarantee requires sufficiently high-order proven asymptotics, and sT-LLSQ's unconditional guarantee covers only the dominant component; this is an applicability caveat, not circular reasoning. The numerical examples use known constants for verification, which is an external consistency check rather than a circular construction.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

No invented entities: ALT introduces no new mechanism, force, or conserved quantity — it re-labels a class of estimators. Free parameters are limited to hyperparameters µ and m plus the externally-sourced expansion inputs (p, φ_j, g_l); β_l are treated symbolically. Axioms are the four numbered assumptions plus the CAS-asserted correctness of Tables 4–6. The exact-data requirement is acknowledged by the author as an open problem (Section 7), so it is not a hidden flaw, but it is a genuine load-bearing premise for all rate guarantees.

free parameters (2)
  • Tikhonov regularization parameter µ = 1 in all numerical experiments; any fixed µ>0 allowed
    Chosen by hand. The sT-LLSQ rate is universal in µ but hidden O-constants depend on it; µ=0 degenerates to sLLSQ (Remark 3.30 treats the k=1 bias m/(m+µ)).
  • Sliding window length m = m=k in Tables 4-6 and most examples; m=2,3,4,5,10 in figures
    User-chosen; sLLSQ needs m≥k (Assumption 3.10); guarantees are m-dependent and can be lost when m increases (Remark 3.23 counterexample).
axioms (5)
  • domain assumption The data sequence {f(n)} is exact or of sufficiently high accuracy
    All convergence claims are asymptotic in n with exact arithmetic; finite-precision/noise analysis is deferred to Section 7 (open problem on numerical stability). Remark 2.1.
  • domain assumption Assumptions 3.1–3.3: f has the proven expansion (2.1) with bf of exponential-product form (3.1); φ_j form an increasing asymptotic scale with φ_1=Ω(1); u_j are bijective differentiable with nonzero derivative
    The entire ALT framework. Existence of α is 'guaranteed in advance' (§2.1); if the basis φ_j or order p is misspecified the method can diverge (§6.4).
  • domain assumption Assumption 3.10/3.11: the design matrix A(n) has full column rank, det(A^TA)=Ω(h(n))
    Needed for unique LLSQ solution and well-defined A†; holds for the AF forms in Table 3 (det scales listed in Table 4).
  • domain assumption Assumption 3.24: φ_1(n)→∞ when k=1; for k≥2, φ₁=Ω(1) suffices
    Required for the universal rate in Lemma 3.26; the k=1 constant-φ case is handled separately in Remark 3.30 with a bias m/(m+µ).
  • ad hoc to paper Tables 4–6 symbolic Puiseux estimates are correct as stated
    The Θ-estimates (e.g., ∥y*_n−γ∥=O(log n/n) for AF-1 p=3; ∥D(k,:)∥=Θ(1/log n) for AF-1) are asserted as outputs of a computer algebra system with no derivation or code; the applied claims (Remarks 4.1–4.3, Sections 6.1–6.3) rely on them. I verified a few (AF-1 det scale, Remark 3.23), but most remain uncheckable from the text.

pith-pipeline@v1.3.0-alltime-deepseek · 60345 in / 39657 out tokens · 335621 ms · 2026-07-31T23:53:59.319741+00:00 · methodology

0 comments
read the original abstract

We introduce a new research area that is called Asymptotics Learning Theory (ALT) and combines optimization with asymptotic analysis. In particular, ALT provides a unified approach for computing unknown constants/parameters in proven asymptotic expansions using optimization theory. In this paper, we focus on a general asymptotic form which includes a broad class of asymptotics. Furthermore, we study two powerful numerical methods, namely, sliding Linear Least Squares (sLLSQ) and sliding Tikhonov Linear Least Squares (sT-LLSQ). For these techniques we rigorously prove asymptotic estimates that lead to sufficient conditions for convergence (to the correct values of unknown parameters) and convergence-rate guarantees. Despite their strengths, both methods have also limitations, e.g., slow convergence---or even, counterintuitively, divergence---in some cases. Moreover, we present fundamental applications in analytic combinatorics, a beautiful field of mathematics that deals with asymptotic enumeration of discrete structures using complex analysis. The proposed techniques complement existing approaches, such as the ratio method and its variants. Numerical examples also verify the theoretical results. Finally, we discuss interesting research directions in ALT.

Figures

Figures reproduced from arXiv: 2607.23287 by Christos N. Efrem.

Figure 1
Figure 1. Figure 1: Application to the Fibonacci numbers (k = 2). 6.2. The Catalan Numbers. The n-th Catalan number is defined recursively by Cn = 2(2n − 1) n + 1 Cn−1, for all n ∈ N with initial condition C0 = 1, or equivalently by the explicit formula (6.1) Cn = 1 n + 1 2n n  = (2n)! (n + 1)! n! . By applying the de Moivre-Stirling approximation of the second order, n! = √ 2πn n e n  1 + 1 12n + 1 288n2 + O  1 n3  ,… view at source ↗
Figure 2
Figure 2. Figure 2: Application to the Catalan numbers (k = 3). ζn := nrn − (n − 1)rn−1 = 4 − 12 n2 + O  1 n3  =⇒ ζn − α3 = ζn − 4 = Θ  1 n2  = o(1), κn := n 2  1 − rn rn−1  = − 3 2 + 9 4n + O  1 n2  =⇒ κn − α2 = κn −  − 3 2  = Θ  1 n  = o(1). As a consequence, κn and ζn have similar convergence rates with x ∗ 2,n and x ∗ 3,n of sLLSQ, respectively, whereas the convergence of rn is much slower compared to ζn and x… view at source ↗
Figure 3
Figure 3. Figure 3: Application to the factorial function (k = 4). 6.4. A Counterintuitive Example of Complete Divergence of sLLSQ. Af￾terwards, we study a counterexample to the assertion that “sLLSQ can always compute the unknown parameters”, thus revealing its fundamental limitations as a numerical method. Interestingly, we will see that sLLSQ may not converge at all to the desired values of parameters. In other words, it m… view at source ↗
Figure 4
Figure 4. Figure 4: An example of complete divergence of sLLSQ (k = 3). where hnew(n) := h(2n) = √ 1 2n , and we use the sampled sequence {fnew(n)}n∈N. The convergence rates achieved by sLLSQ with m = k = 3 are given by21 ∥y ∗ n − γ∥ = [PITH_FULL_IMAGE:figures/full_fig_p042_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Turning the complete divergence of sLLSQ into conver￾gence by using the sampled sequence {f(2n)}n∈N. Absolute errors of sLLSQ, sT-LLSQ and RM (k = 3). =  Θ(1), Θ(1), Θ  log(n) n T , since log(α1) = log(7) ̸= 0 and α2 = 1 3 ̸= 0, where Cnew := DnewAT new and Dnew := (AT newAnew + µI) −1 . Finally, the modified RM, with rn := fnew(n+1) fnew(n) , ζn := nrn − (n − 1)rn−1 and κn := n 2  1 − rn rn−1  , ach… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

55 extracted references

  1. [1]

    Banderier and M

    C. Banderier and M. Drmota,Formulae and asymptotics for coefficients of algebraic functions, Combinatorics, Probability and Computing24(2015), no. 1, 1–53

    2015

  2. [2]

    E. A. Bender,Asymptotic methods in enumeration, SIAM Review16(1974), no. 4, 485–515

    1974

  3. [3]

    D. P. Bertsekas,Nonlinear programming, Athena Scientific, 1999

    1999

  4. [4]

    Boyd and L

    S. Boyd and L. Vandenberghe,Convex optimization, Cambridge University Press, 2004

    2004

  5. [5]

    Brassesco and M

    S. Brassesco and M. A. M´ endez,The asymptotic expansion forn!and the Lagrange inversion formula, The Ramanujan Journal24(2011), no. 2, 219–234

    2011

  6. [6]

    Castellv ´ ı, M

    J. Castellv ´ ı, M. Drmota, M. Noy, and C. Requil´ e,Chordal graphs with bounded tree-width, Advances in Applied Mathematics157(2024), 1–27

    2024

  7. [7]

    A. R. Conway and A. J. Guttmann,Counting occurrences of patterns in permutations, The Electronic Journal of Combinatorics32(2025), no. 1, P1.3

    2025

  8. [8]

    N. G. de Bruijn,Asymptotic methods in analysis, Bibliotheca Mathematica, Vol. 4, North- Holland Publishing Co., 1958 (third edition by Dover, 1981)

    1958

  9. [9]

    Defant, A

    C. Defant, A. Elvey Price, and A. J. Guttmann,Asymptotics of 3-stack-sortable permutations, The Electronic Journal of Combinatorics28(2021), no. 2, P2.49

    2021

  10. [10]

    Domb and M

    C. Domb and M. F. Sykes,On metastable approximations in co-operative assemblies, Pro- ceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences235 (1956), no. 1201, 247–259

    1956

  11. [11]

    1221, 214–228

    ,On the susceptibility of a ferromagnetic above the Curie point, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences240(1957), no. 1221, 214–228

    1957

  12. [12]

    1, 63–67

    ,Use of series expansions for the Ising model susceptibility and excluded volume prob- lem, Journal of Mathematical Physics2(1961), no. 1, 63–67

    1961

  13. [13]

    Drmota,Systems of functional equations, Random Structures & Algorithms10(1997), 103–124

    M. Drmota,Systems of functional equations, Random Structures & Algorithms10(1997), 103–124

    1997

  14. [14]

    ,Random trees: An interplay between combinatorics and probability, Springer, 2009

    2009

  15. [15]

    Drmota, B

    M. Drmota, B. Gittenberger, and J. F. Morgenbesser,Infinite systems of functional equations and Gaussian limiting distributions, Discrete Mathematics & Theoretical Computer Science (2012), 453–478

    2012

  16. [16]

    Drmota and E.-M

    M. Drmota and E.-M. Hainzl,Universal asymptotic properties of positive functional equations with one catalytic variable, La Matematica2(2023), no. 3, 692–742

    2023

  17. [17]

    Drmota, M

    M. Drmota, M. Noy, and G.-R. Yu,Universal singular exponents in catalytic variable equa- tions, Journal of Combinatorial Theory, Series A185(2022), 105522. LEARNING ASYMPTOTICS WITH CONVERGENCE-RATE GUARANTEES USING LLSQ 61

    2022

  18. [18]

    Van Dyke,Analysis and improvement of perturbation series, The Quarterly Journal of Mechanics and Applied Mathematics27(1974), no

    M. Van Dyke,Analysis and improvement of perturbation series, The Quarterly Journal of Mechanics and Applied Mathematics27(1974), no. 4, 423–450

    1974

  19. [19]

    Flajolet and A

    P. Flajolet and A. Odlyzko,Singularity analysis of generating functions, SIAM Journal on Discrete Mathematics3(1990), no. 2, 216–240

    1990

  20. [20]

    Flajolet and R

    P. Flajolet and R. Sedgewick,Analytic combinatorics, Cambridge University Press, 2009

    2009

  21. [21]

    D. S. Gaunt and A. J. Guttmann,Asymptotic analysis of coefficients, Phase transitions and critical phenomena, Vol. 3: Series expansions for lattice models (C. Domb and M. S. Green, eds.), Academic Press, 1974, pp. 181–243

    1974

  22. [22]

    G. H. Golub and C. F. Van Loan,Matrix computations, third ed., Johns Hopkins University Press, 1996

    1996

  23. [23]

    R. L. Graham, D. E. Knuth, and O. Patashnik,Concrete mathematics: A foundation for computer science, second ed., Addison-Wesley, 1989

    1989

  24. [24]

    D. H. Greene and D. E. Knuth,Mathematics for the analysis of algorithms, third ed., Birkh¨ auser, 1990

    1990

  25. [25]

    A. J. Guttmann,Asymptotic analysis of power-series expansions, Phase transitions and criti- cal phenomena, Vol. 13 (C. Domb and J. L. Lebowitz, eds.), Academic Press, 1989, pp. 1–234

    1989

  26. [26]

    ,Analysis of series expansions for non-algebraic singularities, Journal of Physics A: Mathematical and Theoretical48(2015), no. 4, 1–33

    2015

  27. [27]

    41, 1–27

    ,Series extension: Predicting approximate series coefficients from a finite number of exact coefficients, Journal of Physics A: Mathematical and Theoretical49(2016), no. 41, 1–27

    2016

  28. [28]

    A. J. Guttmann and I. Jensen,Series analysis, Polygons, polyominoes and polycubes, Springer, 2009, pp. 181–202

    2009

  29. [29]

    48, 485201

    ,Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, Journal of Physics A: Mathematical and Theoretical55(2022), no. 48, 485201

    2022

  30. [30]

    N. J. Higham,Accuracy and stability of numerical algorithms, second ed., Society for Indus- trial and Applied Mathematics (SIAM), 2002

    2002

  31. [31]

    E. J. Hinch,Perturbation methods, Cambridge Texts in Applied Mathematics, Vol. 6, Cam- bridge University Press, 1991

    1991

  32. [32]

    R. A. Horn and C. R. Johnson,Matrix analysis, Cambridge University Press, 1985

    1985

  33. [33]

    Hwang,On convergence rates in the central limit theorems for combinatorial structures, European Journal of Combinatorics19(1998), no

    H.-K. Hwang,On convergence rates in the central limit theorems for combinatorial structures, European Journal of Combinatorics19(1998), no. 3, 329–343

    1998

  34. [34]

    D. E. Knuth,The art of computer programming: Fundamental algorithms, third ed., vol. 1, Addison-Wesley, 1997

    1997

  35. [35]

    2, Addison-Wesley, 1998

    ,The art of computer programming: Seminumerical algorithms, third ed., vol. 2, Addison-Wesley, 1998

    1998

  36. [36]

    3, Addison- Wesley, 1998

    ,The art of computer programming: Sorting and searching, second ed., vol. 3, Addison- Wesley, 1998

    1998

  37. [37]

    S. P. Lalley,Finite range random walk on free groups and homogeneous trees, The Annals of Probability21(1993), no. 4, 2087–2130

    1993

  38. [38]

    ,Random walks on regular languages and algebraic systems of generating functions, Contemporary Mathematics287(2001), 201–230

    2001

  39. [39]

    ,Algebraic systems of generating functions and return probabilities for random walks, Dynamics and randomness II, Springer, 2004, pp. 81–122

    2004

  40. [40]

    Marsaglia and J

    G. Marsaglia and J. C. W. Marsaglia,A new derivation of Stirling’s approximation ton!, The American Mathematical Monthly97(1990), no. 9, 826–829

    1990

  41. [41]

    G. N. Mercer and A. J. Roberts,A centre manifold description of contaminant dispersion in channels with varying flow properties, SIAM Journal on Applied Mathematics50(1990), no. 6, 1547–1565

    1990

  42. [42]

    J. F. Morgenbesser,Square root singularities of infinite systems of functional equations, Dis- crete Mathematics & Theoretical Computer Science (2010), 513–526

    2010

  43. [43]

    Nemes,On the coefficients of the asymptotic expansion ofn!, Journal of Integer Sequences 13(2010), no

    G. Nemes,On the coefficients of the asymptotic expansion ofn!, Journal of Integer Sequences 13(2010), no. 6, 1–5

    2010

  44. [44]

    A. M. Odlyzko,Asymptotic enumeration methods, Handbook of Combinatorics, vol. 2, Else- vier, 1995, pp. 1063–1229

    1995

  45. [45]

    Pemantle, M

    R. Pemantle, M. C. Wilson, and S. Melczer,Analytic combinatorics in several variables, vol. 212, Cambridge University Press, 2024. 62 CHRISTOS N. EFREM

    2024

  46. [46]

    A. E. Price, W. Fang, and M. Wallner,Compacted binary trees admit a stretched exponential, Journal of Combinatorial Theory, Series A177(2021), 105306

    2021

  47. [47]

    Santosa and W

    F. Santosa and W. W. Symes,Linear inversion of band-limited reflection seismograms, SIAM Journal on Scientific and Statistical Computing7(1986), no. 4, 1307–1330

    1986

  48. [48]

    Szpankowski,Average case analysis of algorithms on sequences, John Wiley & Sons, 2001

    W. Szpankowski,Average case analysis of algorithms on sequences, John Wiley & Sons, 2001

    2001

  49. [49]

    N. M. Temme,Asymptotic methods for integrals, Series in Analysis, Vol. 6, World Scientific Publishing Co., 2015

    2015

  50. [50]

    Tibshirani,Regression shrinkage and selection via the lasso, Journal of the Royal Statistical Society, Series B, Statistical Methodology58(1996), no

    R. Tibshirani,Regression shrinkage and selection via the lasso, Journal of the Royal Statistical Society, Series B, Statistical Methodology58(1996), no. 1, 267–288

    1996

  51. [51]

    A. N. Tikhonov,Solution of incorrectly formulated problems and the regularization method, Soviet Mathematics4(1963), 1035–1038

    1963

  52. [52]

    Wallner,The decompressed tree size ofk-ary chains, Annals of Combinatorics (2026), 1–21

    M. Wallner,The decompressed tree size ofk-ary chains, Annals of Combinatorics (2026), 1–21

    2026

  53. [53]

    H. S. Wilf,Generatingfunctionology, Academic Press, 1990

    1990

  54. [54]

    Wong,Asymptotic approximations of integrals, Classics in Applied Mathematics, Vol

    R. Wong,Asymptotic approximations of integrals, Classics in Applied Mathematics, Vol. 34, Society for Industrial and Applied Mathematics (SIAM), 2001

    2001

  55. [55]

    A. R. Woods,Coloring rules for finite trees, and probabilities of monadic second order sen- tences, Random Structures & Algorithms10(1997), no. 4, 453–485. National Technical University of Athens, Greece Email address:{the first two letters of the author’s first name followed by the first three letters of the author’s last name (all in lowercase)}@mail.ntua.gr

    1997