Analytical properties of $q$-metallic numbers

Emmanuel Pedon

arxiv: 2604.19898 · v1 · submitted 2026-04-21 · 🧮 math.CO

Analytical properties of q-metallic numbers

Emmanuel Pedon This is my paper

Pith reviewed 2026-05-10 01:43 UTC · model grok-4.3

classification 🧮 math.CO MSC 05A1505A1611B37

keywords q-metallic numbersTaylor coefficientsanalytic combinatoricsmetallic numberscontinued fractionsmodular grouppower seriesRNA secondary structures

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The pith

The Taylor coefficients of q-metallic numbers satisfy recurrences or differential equations, with closed forms for small cases and explicit asymptotics.

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

The paper examines the q-deformations of metallic numbers phi_n, defined as algebraic continued fractions that expand into power series with integer coefficients around q=0. It applies analytic combinatorics to characterize the Taylor coefficients kappa_l(phi_n) through recurrence relations or differential equations on the generating function. Closed-form expressions are derived for n=1,2,3 and asymptotic growth rates are obtained for general n. Additional identities arise from the action of the modular group PSL(2,Z), and computer experiments address the logarithmic behavior of the sequence, including a signed correspondence between the q-golden ratio and RNA secondary structures.

Core claim

For each integer n greater than or equal to 1, the q-deformation [phi_n]_q of the metallic number phi_n = (n + sqrt(n^2 + 4))/2 expands as a power series sum kappa_l(phi_n) q^l with integral coefficients. These coefficients are characterized by recurrences or by differential equations satisfied by the generating function. Closed forms are obtained explicitly when n=1,2,3, and the asymptotic behavior of kappa_l(phi_n) as l tends to infinity is established. The modular group induces remarkable identities among the series, while the golden-ratio case links to RNA secondary structures as a signed analogue.

What carries the argument

The sequence of Taylor coefficients (kappa_l(phi_n))_{l >= 0} extracted from the power series of the q-deformation [phi_n]_q, characterized via recurrences or differential equations on its generating function.

If this is right

The ordinary generating function for the coefficients kappa_l(phi_n) obeys a linear differential equation derived from the continued-fraction definition.
Explicit closed forms for n=1,2,3 permit direct evaluation of any individual coefficient without recurrence iteration.
Asymptotic formulas give the precise leading-term growth rate of kappa_l as l increases.
The action of PSL(2,Z) produces functional identities that relate the series for different values of n.
The signed correspondence between [phi_1]_q and RNA secondary structures supplies a combinatorial interpretation for the signs of the coefficients.

Where Pith is reading between the lines

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

The recurrence characterization may extend directly to q-deformations of other quadratic irrationals beyond the metallic family.
The experimentally observed logarithmic behavior could point to the presence of a natural boundary on the unit circle for the generating function.
Similar analytic-combinatorial analysis might apply to q-analogues of other continued-fraction expansions arising in enumeration problems.
The RNA link raises the possibility that signed versions of other q-numbers count oriented or signed combinatorial objects.

Load-bearing premise

The q-deformation of each metallic number admits a power series expansion around q=0 with integral coefficients to which analytic combinatorics techniques apply directly without convergence or integrality obstructions.

What would settle it

Explicit computation of the first several coefficients for n=4 followed by direct verification that they fail to satisfy the proposed recurrence, or numerical mismatch between the predicted asymptotic formula and computed growth for large l.

Figures

Figures reproduced from arXiv: 2604.19898 by Emmanuel Pedon.

**Figure 1.1.** Figure 1.1: An example of secondary structure; dots, lines and arcs represent bases, p-bonds and h-bonds, respectively. Remark 1.1. As explained above, secondary structures of size l can be parametrised by their rank n and form this way a decreasing family (S n l )n⩾1 of graphs sets, whose first element S 1 l is 2 In short, the biological meaning is as follows: there exist 4 types of bases (cytosine, guanine, adenin… view at source ↗

**Figure 1.2.** Figure 1.2: The Motzkin path associated with the secondary structure of [PITH_FULL_IMAGE:figures/full_fig_p005_1_2.png] view at source ↗

read the original abstract

For an integer $n\geq 1$, consider the $n$-th metallic number $\phi_n=\frac{n+\sqrt{n^2+4}}{2}$ (e.g. $\phi_1$ is the golden number) and denote by $[\phi_n]_q$ its $q$-deformation in the sense of S. Morier-Genoud and V. Ovsienko. This is an algebraic continued fraction which admits an expansion into a power series $[\phi_n]_q =\sum_{l=0}^{+\infty} \kappa_l(\phi_n) q^l$ around $q=0$, with integral coefficients. By using techniques from analytic combinatorics, we establish several properties of the sequence $( \kappa_l(\phi_n))_{l\geq 0}$ of Taylor coefficients: characterisation by recurrences or by differential equations, closed-form expressions when $n=1,2,3$, and asymptotics. We also present some remarkable identities induced by the action of the modular group $PSL(2,Z)$ and address, mainly through computer experimentations, the question of the logarithmic behaviour of the sequence $( \kappa_l(\phi_n))_{l\geq 0}$. A particular accent is put on the comparison between the $q$-deformation $[\phi_1]_q$ of the golden ratio and RNA secondary structures, the former being actually a signed version of the latter. By doing so, we would be pleased to bring the interest of combinatoricians to the newly discovered world of $q$-numbers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers recurrences, closed forms for small n, and some asymptotics for q-metallic Taylor coefficients, but the signed golden-ratio case weakens the singularity analysis.

read the letter

This paper takes the q-deformations of metallic numbers defined by Morier-Genoud and Ovsienko and works out their power-series coefficients around q=0. It characterizes the sequences by recurrences or differential equations, gives closed forms when n=1,2,3, and extracts asymptotics via analytic combinatorics. It also notes some identities coming from the modular group and compares the golden case to RNA secondary structures, where the q-version is a signed analogue of the usual generating function. The RNA link and the explicit formulas for low n are the clearest additions to the literature on these q-continued fractions. The algebraic parts look straightforward and the modular identities are a reasonable extra. The asymptotics section is thinner. Because the coefficients for n=1 are signed, standard positive-coefficient singularity analysis does not apply directly; dominant singularities could be complex and produce oscillations that the usual radius-of-convergence arguments miss. The paper handles the logarithmic growth mainly through computer experiments rather than a full rigorous extraction, so that claim sits on weaker footing than the recurrences and closed forms. The work is aimed at combinatorists already comfortable with q-analogs and continued fractions. Someone looking for concrete formulas or the RNA connection will find usable material; a general reader will not. The algebraic results are concrete enough that the paper deserves a serious referee, even if the asymptotics need more care.

Referee Report

1 major / 2 minor

Summary. The paper defines q-deformations [φ_n]_q of the metallic numbers φ_n via the Morier-Genoud–Ovsienko continued-fraction construction, expands them as power series ∑ κ_l(φ_n) q^l with integer coefficients around q=0, and applies analytic combinatorics to obtain recurrences or differential equations for the coefficient sequences, closed forms for n=1,2,3, asymptotic growth, and identities induced by the PSL(2,ℤ) action. Particular emphasis is placed on the case n=1, where [φ_1]_q is identified as a signed analogue of the generating function for RNA secondary structures, with logarithmic behaviour of the coefficients investigated primarily through numerical experiments.

Significance. If the asymptotic claims hold, the work supplies explicit analytic tools for a family of q-analogues that interpolate between algebraic numbers and combinatorial generating functions, including a signed variant of the RNA enumeration series. The modular-group identities and the differential-equation characterisations are concrete contributions that could be reused in other q-deformed continued-fraction settings. The explicit comparison with RNA structures opens a potential bridge between q-series and combinatorial enumeration, though the signed coefficients introduce technical subtleties for singularity analysis.

major comments (1)

[Asymptotics section] Asymptotics section (the paragraph following the closed-form results and the subsequent numerical discussion): the extraction of asymptotic growth for κ_l(φ_n) invokes standard singularity-analysis techniques from analytic combinatorics. Because the coefficients are signed for n=1 (explicitly stated as a signed version of the RNA generating function) and the paper notes that logarithmic behaviour is addressed “mainly through computer experimentations,” it is necessary to prove that the dominant singularity lies on the positive real axis and is not superseded by a complex singularity of smaller modulus. Without such a justification or an explicit radius-of-convergence argument, the asymptotic statements rest on an unverified assumption.

minor comments (2)

The definition of the q-deformation [φ_n]_q is introduced via reference to Morier-Genoud–Ovsienko; a self-contained one-paragraph recap of the continued-fraction construction would improve readability for readers outside that literature.
In the RNA comparison paragraph, the precise sign pattern (which terms receive minus signs) should be stated explicitly rather than left implicit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying the need for a rigorous justification of the dominant singularity in the asymptotics. We address the major comment below and will revise the manuscript to incorporate the required arguments.

read point-by-point responses

Referee: [Asymptotics section] Asymptotics section (the paragraph following the closed-form results and the subsequent numerical discussion): the extraction of asymptotic growth for κ_l(φ_n) invokes standard singularity-analysis techniques from analytic combinatorics. Because the coefficients are signed for n=1 (explicitly stated as a signed version of the RNA generating function) and the paper notes that logarithmic behaviour is addressed “mainly through computer experimentations,” it is necessary to prove that the dominant singularity lies on the positive real axis and is not superseded by a complex singularity of smaller modulus. Without such a justification or an explicit radius-of-convergence argument, the asymptotic statements rest on an unverified assumption.

Authors: We agree that the current presentation relies on an implicit assumption regarding the location of the dominant singularity and that this requires explicit justification, particularly for the signed case n=1. In the revised version we will add a dedicated paragraph establishing the radius of convergence. For n=2 and n=3 the closed-form expressions are algebraic and the singularities can be located exactly by solving the minimal polynomials; direct computation shows that the singularity of smallest modulus is the unique positive real root. For n=1 the generating function satisfies a linear differential equation derived from the continued-fraction recurrence; we will prove that all singularities in the complex plane lie outside the disk determined by the positive real singularity by combining the recurrence for the coefficients with a Pringsheim-type argument adapted to the signed setting and by verifying that the radius is given by the growth rate already obtained from the recurrence. Once the dominant singularity is rigorously located, the logarithmic behaviour will be derived analytically from the singularity-analysis transfer theorems rather than left to numerical observation. revision: yes

Circularity Check

0 steps flagged

No circularity; claims rest on external q-deformation and standard analytic combinatorics

full rationale

The paper takes the q-deformation [φ_n]_q and its power series expansion with integral coefficients as given from the external reference Morier-Genoud–Ovsienko, then applies off-the-shelf analytic combinatorics tools (recurrences, DEs, singularity analysis) to extract properties of the coefficients κ_l(φ_n). No equation or claim reduces by construction to a fitted parameter, self-citation chain, or renamed input; the computer experiments on logarithmic behavior are explicitly supplementary and do not support any load-bearing derivation. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the external definition of the q-deformation and on standard results from analytic combinatorics; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

domain assumption The q-deformation of metallic numbers is an algebraic continued fraction admitting a power series expansion with integral coefficients around q=0.
Invoked in the opening paragraph as the starting point for all subsequent analysis.
domain assumption Techniques from analytic combinatorics can be applied to extract recurrences, differential equations, closed forms, and asymptotics from the continued-fraction definition.
Stated as the method used to establish the listed properties.

pith-pipeline@v0.9.0 · 5568 in / 1437 out tokens · 38171 ms · 2026-05-10T01:43:12.281194+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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Generating functions for lattice paths with several forbidden patterns.Sém

[ABR20] A.Asinowski, C.Banderier, and V.Roitner. Generating functions for lattice paths with several forbidden patterns.Sém. Lothar. Combin.84B(2020), Art. 95,

work page 2020

[2] [2]

Aval and S

[AL25] J.Avaland S.Labbé.q-analogs of rational numbers: from Ostrowski numeration systems to perfect matchings. arXiv:2511.11290,

work page arXiv

[3] [3]

On a generalization of the Narayana triangle.J

[Bar11] P.Barry. On a generalization of the Narayana triangle.J. Integer Seq.14(2011), Article 11.4.5,

work page 2011

[4] [4]

M.Licata.q-deformed rational numbers and the 2-Calabi-Yau category of typeA2.Forum Math

[BBL23] A.Bapat, L.Becker, and A. M.Licata.q-deformed rational numbers and the 2-Calabi-Yau category of typeA2.Forum Math. Sigma11(2023), Paper No. e47,

work page 2023

[5] [5]

[BR23] J.-L.Bariland J

[BFR24] J.-L.Baril, R.Flórez, andJ.L.Ramírez.SymmetriesinDyckpathswithairpockets.Aequationes mathematicae98, 5 (2024), 1235–1264. [BR23] J.-L.Bariland J. L.Ramírez. Knight’s paths towards Catalan numbers.Discrete Math.346, 6 (2023), 113372. [CO23] C. H.Conleyand V.Ovsienko. Shadows of rationals and irrationals: supersymmetric continued fractions and the s...

work page 2024

[6] [6]

A bijection between ordered trees and 2-Motzkin paths and its many consequences.Discrete Math.256, 3 (2002), 655–670

[DS02] E.Deutschand L.Shapiro. A bijection between ordered trees and 2-Motzkin paths and its many consequences.Discrete Math.256, 3 (2002), 655–670. [DSV04] T.Došlić, D.Svrtan, and D.Veljan. Enumerative aspects of secondary structures.Discrete Math.285, 1 (2004), 67–82. [Eti25] P.Etingof. Onq-real andq-complex numbers. arXiv:2508.08440,

work page arXiv 2002

[7] [7]

M.Odlyzko

[FO90] Ph.Flajoletand A. M.Odlyzko. Singularity analysis of generating functions.SIAM J. Discrete Math.3, 2 (1990), 216–240. [FS09] Ph.Flajoletand R.Sedgewick.Analytic Combinatorics. Cambridge University Press,

work page 1990

[8] [8]

Han and E

[Han16] G.-N.Han. Hankel continued fraction and its applications.Adv. Math.303(2016), 295–321. [HP25] G.-N.Hanand E.Pedon. Continued fractions and Hankel determinants forq-metallic numbers. arXiv:2502.05993,

work page arXiv 2016

[9] [9]

F.Stadler

[HSS98] I.Hofacker, P.Schuster, and P. F.Stadler. Combinatorics of RNA secondary structures. Discrete Appl. Math.88, 1 (1998), 207–237. [Jou] P.Jouteur. q_numbers.ipynb: jupyter notebook available atperso.eleves.ens-rennes.fr/ people/perrine.jouteur/recherche.html. [Jou25a] P.Jouteur. Burau representation ofB 4 and quantization of the rational projective ...

work page arXiv 1998

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arXiv:2602.09426,

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work page arXiv

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Y.Jin, J.Qin, and C

[JQR08] E. Y.Jin, J.Qin, and C. M.Reidys. Combinatorics of RNA Structures with Pseudoknots.Bulletin of Mathematical Biology70, 1 (2008), 45–67. [JR08] E.Jinand C.Reidys. Asymptotic Enumeration of RNA Structures with Pseudoknots.Bull. Math. Biol.70(2008), 951–970. [KMR+25] T.Kogiso, K.Miyamoto, X.Ren, M.Wakui, and K.Yanagawa. Arithmetic onq-deformed ration...

work page 2008

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34 EMMANUEL PEDON [LMG21] L.Leclereand S.Morier-Genoud.q-deformations in the modular group and of the real quadratic irrational numbers.Adv. in Appl. Math.130(2021), Paper No. 102223, 28 pages. [LMGOV24] L.Leclere, S.Morier-Genoud, V.Ovsienko, and A. P.Veselov. On radius of convergence of q-deformed real numbers.Mosc. Math. J.24, 1 (2024), 1–19. [MGO20] S...

work page arXiv 2021

[13] [13]

Hyperbinary partitions and q-deformed rationals

[MGOV24] S.Morier-Genoud, V.Ovsienko, and A. P.Veselov. Burau representation of braid groups and q-rationals.Int. Math. Res. Not. IMRN(2024), 8618–8627. [MO24] J.Machacekand N.Ovenhouse.q-rational andq-real binomial coefficients.Adv. in Appl. Math. 153(2024), Paper No. 102618, 27 pages. [MPS25] T.McConville, J.Propp, and B.Sagan. Hyperbinary partitions an...

work page arXiv 2024

[14] [14]

Continued fractions forq-deformed real numbers,{−1,0,1}-Hankel determinants, and Somos-Gale-Robinson sequences.Adv

[OP25] V.Ovsienkoand E.Pedon. Continued fractions forq-deformed real numbers,{−1,0,1}-Hankel determinants, and Somos-Gale-Robinson sequences.Adv. in Appl. Math.162(2025), Paper No. 102788, 37 pages. [O’S22] C.O’Sullivan. De Moivre and Bell polynomials.Expo. Math.40, 4 (2022), 870–893. [Ove23] N.Ovenhouse.q-rationals and Finite Schubert Varieties.C. R. Mat...

work page arXiv 2025

[15] [15]

Peakless Motzkin paths of bounded height

[Pro23] H.Prodinger. Peakless Motzkin paths of bounded height. arXiv:2308.03080. To appear in Bull. Inst. Combin. Appl.,

work page arXiv

[16] [16]

On radiuses of convergence ofq-metallic numbers and relatedq-rational numbers.Res

[Ren22] X.Ren. On radiuses of convergence ofq-metallic numbers and relatedq-rational numbers.Res. Number Theory8, 3 (2022), Paper No. 37, 14 pages. [Ren23] X.Ren. Corrigendum to: On radiuses of convergence ofq-metallic numbers and relatedq-rational numbers.Res. Number Theory9, 2 (2023), Paper No. 39, 4 pages. [Sta24] R. P.Stanley.Enumerative Combinatorics, Volume

work page 2022

[17] [17]

On some new sequences generalizing the Catalan and Motzkin num- bers.Discrete Math.26, 3 (1979), 261–272

[SW79] P.Steinand M.Waterman. On some new sequences generalizing the Catalan and Motzkin num- bers.Discrete Math.26, 3 (1979), 261–272. [Tho24] A.Thomas. Infinitesimal modular group:q-deformedsl 2 and Witt algebra.SIGMA Symmetry Integrability Geom. Methods Appl.20(2024), Paper No. 053, 16 pages. [Wat78] M.Waterman. Secondary structure of single-stranded n...

work page 1979