Elliptical multipoles for gravitational lenses

Daniel Gilman (UChicago); Hadrien Paugnat (UCLA)

REVIEW 2 major objections 6 minor 1 cited by

Elliptical multipoles for gravitational lenses

T0 review · 2 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The standard circular multipole parameterization of lens perturbations is biased for flattened galaxies, and elliptic multipoles defined in eccentric-anomaly coordinates fix it.

desk verdict The analytic potentials are a real, careful contribution that deserves referee time, but the claim that elliptical multipoles are the physically correct choice rests on an untested light-to-mass transfer that the paper itself hedges on. read the letter →

arxiv 2502.03530 v2 pith:OBDIQWPD submitted 2025-02-05 astro-ph.CO astro-ph.GA

Hadrien Paugnat (UCLA) , Daniel Gilman (UChicago) This is my paper · ORCID

classification astro-ph.COastro-ph.GA

keywords stronggravitationallensingmultipoleexpansionellipticalmultipolespotentialsflux-ratioanomaliesisophotalshapesisothermallensmodelsgalaxystructure

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

Gravitational lens models of galaxies routinely add angular perturbations as multipole expansions defined around a circle, even though the underlying density profile is an ellipse. This paper argues that those 'circular multipoles' produce perturbation patterns that depend on the axis ratio and disagree with observed galaxy isophotal shapes whenever the lens is not near-circular. The authors propose 'elliptical multipoles,' defined with respect to the eccentric anomaly of the ellipse, and show that these preserve the expected physical interpretation (skewness, boxiness/diskyness) for any axis ratio. They derive analytic lensing potentials for these elliptical multipoles under a near-isothermal reference profile, implement them in a lens modeling code, and demonstrate that circular multipoles overestimate flux-ratio perturbations in mock quadruply imaged quasars.

What carries the argument

The load-bearing object is the coordinate change from polar coordinates to elliptical coordinates: $R=\sqrt{q x^2+y^2/q}$ and $\varphi\equiv\arctan(q x,y)$, the eccentric anomaly along an isocontour. The convergence shape function of an elliptical multipole is $G_m(\phi;q,\varphi_m) = a_m\sqrt{q}\cos[m(\varphi(\phi;q)-\varphi_m)] r/R$, which decomposes through Chebyshev polynomials into two components $G_m^{(1)}$ and $G_m^{(2)}$. The Poisson equation $\nabla^2\psi=2\kappa$ then becomes an ordinary differential equation $F+F''=G$ for the angular part of the potential $\psi=rF(\phi)$; the paper solves this equation analytically, using a $r\ln r$ correction term for odd $m$ (including the previously undetermined $m=1$ circular multipole) to enforce periodicity and symmetry without introducing deflection-angle discontinuities.

What would settle it

Compare the azimuthal shape of isodensity contours in a sample of strongly lensed galaxies with $q\lesssim0.6$ against both multipole prescriptions: if real perturbations follow the eccentric anomaly, the circular prescription should require significantly larger amplitudes or fail to reproduce the observed boxy/disky shapes. A direct numerical check would be to generate convergence maps from hydrodynamical simulations of flattened early-type galaxies, decompose them into $m=4$ multipole moments in both coordinate systems, and see which basis gives amplitudes that are constant with radius and aligned with the isophotes.

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Extended reading notes

Core claim

The central claim is that the standard circular multipole parameterization, in which one adds $a_m\cos(m(\phi-\phi_m))$ to a radius expressed in polar angle, is only physically meaningful for near-circular lenses. For an elliptical reference profile with axis ratio $q$, these perturbations create patterns that depend on $q$ and, for $m=4$ with $q\lesssim0.6$, turn expected boxy/disky distortions into 'peanut' and 'spinny-top' shapes. The paper replaces this with elliptical multipoles defined by $R\mapsto R + a_m\cos(m(\varphi-\varphi_m))$, where $R=\sqrt{q x^2+y^2/q}$ is the elliptical radius and $\varphi=\arctan(q x, y)$ is the eccentric anomaly; the amplitude $a_m$ then directly measures a fractional change in elliptical radius on every isodensity contour. For a near-isothermal reference ($\gamma=2$), the Poisson equation for these shape functions has analytic solutions for all $m$, with explicit potentials for $m=1,3,4$ and a general construction for arbitrary $m$ in the appendices. Using these solutions in mock lensed systems, the paper shows that circular multipoles produce larger perturbations to quasar image flux ratios than elliptical multipoles of the same amplitude, with the discrepancy growing as the axis ratio decreases.

Load-bearing premise

The argument depends on the assumption that deviations of the lens galaxy's total mass density from an ellipse are well described by perturbations defined in the eccentric anomaly of the isophotes, so that real astrophysical multipoles follow the elliptical rather than the circular coordinate system.

Editorial extensions

If this is right

Flux-ratio anomaly analyses that use circular multipoles overestimate the flux perturbation attributable to azimuthal structure in flattened lenses, so dark-matter substructure constraints may shift when the elliptical formulation is adopted.
For lenses with axis ratio $q\lesssim0.7$, fitted circular multipole amplitudes and orientations cannot be interpreted as physical skewness or boxiness; those parameters will be biased when the true perturbation is elliptical.
The explicit analytic potentials allow any $m$-order elliptical multipole to be included in lens models without numerical Poisson solvers, making tests of azimuthal structure in large lens samples feasible.
Because the isothermal approximation introduces only a logarithmic radial error of order $|2-\gamma|\ln(R/\theta_E)$, the elliptical formulation is still preferable to slope-matched circular multipoles for realistic near-isothermal lenses.

Reading between the lines

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

If lens mass multipoles track light isophotes, existing circular-multipole fits to highly flattened lenses should be re-examined; the paper's mock experiments suggest that circular multipoles can mimic disky features while biasing recovered flux ratios by roughly 10 percent.
The same eccentric-anomaly construction could be extended to non-isothermal elliptical multipoles, though the paper argues the gain is modest; a testable prediction is that flux-ratio statistics from a full substructure forward model will shift toward weaker multipole perturbations when elliptical multipoles are used.
The analytic solutions provide a clean test of the axis-ratio dependence: comparing the $m=4$ pattern of isophotes in real galaxies with $q\sim0.5$ against the circular-multipole prediction would show the peanut/spinny-top artifacts claimed.
For time-delay cosmography, the choice of multipole parametrization may change inferred $H_0$ in systems with strong azimuthal structure, since the multipoles absorb shear-like complexity differently.

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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 core math is the real news here. The authors solve Poisson's equation for elliptical multipole perturbations of any order in the isothermal limit, with explicit potentials for m=1, 3, 4 and general constructions in the appendices. They handle the awkward m=1 and odd-m cases by subtracting a circular m=1 piece, which is a nice trick that preserves symmetry and differentiability. The q->1 limits, periodicity checks, and the determinant condition for the linear systems all look right. Shipping the implementations in lenstronomy with deflection fields spelled out makes this immediately usable. That is a genuine, citable contribution independent of any interpretive debate.

Where the paper is softer is the physical branch of the argument. The claim that circular multipoles are 'wrong' for flattened lenses assumes that real deviations from ellipticity in the total mass distribution (stars plus dark matter) follow a Fourier series in the eccentric anomaly, with amplitudes like those measured in light isophotes. The paper never tests that; Section V A just imports light-based Gaussian priors on the amplitudes. The dark matter halo is rounder and smoother than the stars, so the transfer is not obvious. That said, the authors are more careful than the abstract suggests. Section VI explicitly says the elliptical expansion is 'merely one that is more appropriate', not the correct one, and Appendix B quantifies the isothermal approximation. The stress-test note overreaches slightly if it reads the abstract as a definitive claim of physical superiority; the body of the paper is hedged appropriately.

Two minor but real issues: the abstract says the m=1 circular potential was 'previously undetermined', yet Ref. [69] already proposed it and the paper acknowledges this in the text. That should be reconciled. And the flux-ratio comparison is based on two mock geometries, so the 30-80% numbers are illustrative, not statistical conclusions. Neither changes the value of the analytic work.

Net: the potentials are solid, the implementation is a service to the community, and the caveats are mostly stated. The paper deserves peer review. I would cite it for the potentials and the code, and I'd want the authors to soften the abstract and add a sentence explicitly limiting the astrophysical claim until mass-side data can test it.

Referee Report

2 major / 6 minor

Summary. The manuscript argues that the standard 'circular multipole' perturbations used in lens modeling, defined by Fourier terms in the polar angle, produce perturbation patterns that depend on the reference ellipse axis ratio and do not match the isophotal-shape language normally used to describe deviations from ellipticity. It proposes 'elliptical multipoles' defined in eccentric-anomaly coordinates, derives analytic lensing potentials for them under an isothermal reference profile (explicit for m=1, 3, 4 and general m in the Appendices), gives a systematic treatment of the circular m=1 multipole, implements the new models in lenstronomy, and compares flux-ratio perturbations in mock quadruply lensed quasars. The empirical conclusion is that elliptical multipoles produce smaller flux-ratio perturbations than circular multipoles at comparable amplitudes, with the difference growing for more flattened lenses.

Significance. The analytic derivations are the main strength of the paper. The solutions enforce the required symmetries, avoid jump discontinuities by using the r ln r circular m=1 potential, match the q->1 circular limits, and the matrix systems in Appendix D have non-zero determinants for q != 1. The explicit treatment of the circular m=1 multipole, including the warning about divergent current near-isothermal implementations, is practically useful. Appendix B provides a concrete criterion for when the isothermal multipole approximation is adequate. The lenstronomy implementation and the mock experiments make the work reproducible and testable. The physical interpretation and the flux-ratio conclusions, however, rest on the assumption that total-mass multipoles follow the angular structure and amplitude distribution measured in light isophotes; this assumption is acknowledged but not tested.

major comments (2)

[III A, V A] Section V A, Eq. (41): the headline quantitative result (μ_circ/μ_ell ≈ 1.3 for q=0.73 and ≈ 1.8 for q=0.58) is obtained by assigning identical Gaussian priors (σ=0.005, 0.005, 0.01) to a_circ_m and a_m. However, Section III A defines a_circ_m with a separate normalization convention and explicitly warns that a_circ_m is not a fractional change of the semi-major axis, whereas a_m is defined as a fractional change of the elliptical radius. The paper does not demonstrate that equal numerical amplitudes correspond to comparable physical perturbation strengths in the two formulations. Please state the exact amplitude conventions implemented in lenstronomy for both cases and either justify that the priors are matched or rerun the comparison under an explicit matched-amplitude criterion.
[Abstract, V A, VI] The conclusion that elliptical multipoles are more physical than circular multipoles, and the associated statement that they 'typically produce smaller flux-ratio perturbations', are only established under the assumption that total-mass multipoles (stars plus dark matter) follow the same eccentric-anomaly structure and amplitude distribution as galaxy light isophotes. The light-based priors from Refs. [36,48] are imported into mass multipoles without testing this light-to-mass transfer. If the dark matter halo is rounder or has a different angular structure, the amplitude priors and the flux-ratio shift would change. Section VI acknowledges that real perturbations can be more complex, but the abstract does not carry this caveat. Please qualify the abstract and Section V A conclusions as conditional on light-traces-mass, or include a robustness test with modified amplitude priors or a rounder dark-matter component.

minor comments (6)

[III A] The rescaling described after Eq. (5) is called normalization by the semi-major axis, but the factor θ_E√q is the semi-minor axis of the κ=1/2 isocontour (whose semi-major axis is θ_E/√q). Please clarify the convention, since this affects the interpretation of all a_circ_m values.
[IV B] The sentence after Eq. (24) refers to B^(1)_3(q); this should be B^(1)_1(q).
[Appendix D 1] Eq. (D15) gives λ_m(q) as 'verified at least for 1 ≤ k < 10' rather than proven for all odd m. Since the appendix claims a general solution for all odd m, please either provide a proof by induction or explicitly label this as a numerically verified conjecture and restrict the generality claim accordingly.
[Appendix D 1, Eq. (D11)] The stated determinant det[M_m(q)] appears to have a sign that can be negative for k=1, whereas the formula as written is positive; since only the property of being non-zero is used, this is cosmetic, but please correct the expression or state it up to sign.
[Abstract and Section IV A] The abstract says the m=1 circular multipole potential was 'previously undetermined', yet Section IV A and Appendix A 2 b cite Ref. [69] for an equivalent r ln r potential. Please rephrase to indicate that the paper provides a systematic derivation and regularizes the near-isothermal limit.
[Figures 8-11] There are several typographical slips: 'dosh-dotted' in the Figure 8 caption should be 'dash-dotted', 'substraction' in the Figures 10-11 captions should be 'subtraction', and 'Unsuprisingly' in Section V B should be 'Unsurprisingly'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the multipole potentials are solved from the Poisson equation with constants fixed by symmetry and q→1 limits; the flux-ratio conclusions are forward-model outputs, not fitted inputs.

full rationale

The paper's central derivation is self-contained and does not reduce to its inputs. In Section III and Section IV, the elliptical multipole convergence shape functions are defined through Eq. (6) and Eq. (11), and the lensing potentials are obtained by solving the differential equation F''+F=G, i.e., the Poisson equation in the isothermal family of Eq. (2). The undetermined constants A(q) and B(q) are fixed by symmetry requirements and by demanding convergence to the known circular multipole solutions in the q→1 limit (e.g., Eqs. (28), (34), and (40)), not by fitting the flux-ratio statistics that the paper later reports. The m=1 circular multipole potential is independently justified in Appendix A through the isothermal limit of the non-isothermal solution and through direct complex-plane integration; neither route assumes the result it is meant to establish. Section V A is a forward simulation: multipole amplitudes are sampled from external isophotal priors, image positions are held fixed, and the resulting flux-ratio distributions are compared. The finding that circular multipoles produce larger perturbations (μcirc_pert > μell_pert) is a computed outcome, not an enforced one. Section V B likewise generates mock data with elliptical multipoles and fits them with circular multipoles; the reported flux-ratio biases are outputs of the fitting procedure. The only premise that is arguably close to circularity is the physical transfer of isophotal eccentric-anomaly multipoles to total mass density. However, the paper explicitly presents this as an adopted, physically motivated parametrization ('we simply adopt the definition that was originally used in isophote shape studies', Section III B) and in Section VI disclaims that the elliptical multipole expansion is 'not the correct parametrization, merely one that is more appropriate'. That is an external empirical assumption, not a conclusion derived from the target result. Self-citations to lenstronomy and to prior flux-ratio analyses are implementation and context references; they are not load-bearing for the Poisson solutions or for the claimed flux-ratio shift. No uniqueness theorem is imported from the authors' prior work, and no fitted parameter is renamed as a prediction. The derivation chain is therefore free of self-definitional or fitted-input circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The potential derivations introduce no fitted constants: ODE integration constants are fixed by symmetry and by matching the known circular multipole limit. The flux-ratio comparison, however, depends on hand-chosen priors and mock parameters listed as free parameters. The isothermal-reference assumption and the isophote-to-mass transfer are domain assumptions that the paper acknowledges but cannot independently justify.

free parameters (3)

r_E normalization radius for m=1 multipole potentials = theta_E
Introduced in Eq. (21) as the scale in r ln(r/r_E); it enters the potential through a term that the prismatic degeneracy makes unobservable, and the authors set r_E = theta_E by convention.
multipole amplitude priors in Section V A = sigma(m=1)=0.005, sigma(m=3)=0.005, sigma(m=4)=0.01
Chosen from isophotal shape studies; the comparison of circular versus elliptical flux-ratio perturbation distributions depends on these prior widths and on the two mock lens geometries.
mock disky multipole amplitude a4 in Section V B = 0.035
Hand-picked as significant but physically realistic; it sets the size of the true perturbation that the circular-multipole fit attempts to mimic.

assumptions (5)

domain assumption The reference lens profile is isothermal (gamma=2), so the potential has the form psi = r F(phi) and convergence kappa = G(phi)/(2r).
Stated in Section II; the analytic elliptical multipole potentials are derived exactly only for isothermal SIE profiles. For EPL profiles with gamma near 2, the error scales as |2-gamma| ln(R/theta_E) as quantified in Appendix B.
domain assumption Multipole perturbations should be parametrized in eccentric-anomaly coordinates following isophotal shape studies.
Adopted in Section III B as the definition of elliptical multipoles. The claim that these are physically more appropriate than circular multipoles relies on the assumption that galaxy mass-density deviations follow the same angular language as light isophotes.
domain assumption The lensing potential must be twice differentiable on R^2 excluding 0 and its deflection field must be continuous.
Used in Section IV A and Appendix A 1 to reject piecewise symmetrized potentials, following the lensing degeneracy and image-contiguity discussion in Ref. [57].
standard math The prismatic degeneracy allows adding constant deflection fields without changing observables.
Standard strong lensing degeneracy from Gorenstein et al. 1988, used throughout Section IV to set integration constants such as A(q) and B(q) and to remove the r_E dependence.
standard math Poisson's equation, Chebyshev polynomial expansions, and the stated symmetry conditions determine the ODE solutions up to physically irrelevant constants.
The derivation of F(phi) from G(phi) via F'' + F = G is the backbone of Section IV and Appendices C and D; the linear-algebra construction for general m relies on these standard tools.

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Cite this review

Pith. "Pith review of Elliptical multipoles for gravitational lenses." pith.science (2026). https://pith.science/paper/OBDIQWPD

@misc{pith2026250203530,
  author       = {Pith},
  title        = {Pith review of: Elliptical multipoles for gravitational lenses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OBDIQWPD}},
  note         = {Machine review of arXiv:2502.03530}
}

abstract

Gravitational lensing galaxies are commonly modeled with elliptical density profiles, to which angular complexity is sometimes added through a multipole expansion - encoding deformations of the elliptical iso-density contours. The formalism that is widely used in current studies and software packages, however, employs perturbations that are defined with respect to a circle. In this work, we show that this popular formulation (the "circular multipoles") leads to perturbation patterns that depend on the axis ratio and do not agree with physical expectations (from studies of galaxy isophotal shapes) when applied to profiles that are not near-circular. We propose a more appropriate formulation, the "elliptical multipoles", representing deviations from ellipticity suited for any axis ratio. We solve for the lensing potentials associated with the $m=1$ circular multipole (previously undetermined in the isothermal case), as well as the elliptical multipoles of any order $m$, assuming that the reference profile is near-isothermal. We implement these solutions into the lens modeling package $\mathtt{lenstronomy}$, and assess the importance of the multipole formulation by comparing flux-ratio perturbations in mock lensed systems with quadruply imaged quasars: we show that elliptical multipoles typically produce smaller flux-ratio perturbations than their circular counterparts.

Figures

Figures reproduced from arXiv: 2502.03530 by the authors.

**Figure 2.** FIG. 2. Top panel: Impact of a [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3. Same as Figure 2, but for a [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (14 more)

**Figure 4.** Figure 4: FIG. 4. Same as Figure 2, but for a [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: FIG. 5. Example of lensing potential [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

**Figure 6.** Figure 6: FIG. 6. Same as Figure 5, but for a [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

**Figure 7.** Figure 7: FIG. 7. Same as Figure 5, but for a [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]

**Figure 8.** Figure 8: FIG. 8. Distribution of flux ratios calculated from 10000 EPL + shear + [PITH_FULL_IMAGE:figures/full_fig_p028_8.png]

**Figure 9.** Figure 9: FIG. 9. Same as Figure 8, but for lens models with axis ratio [PITH_FULL_IMAGE:figures/full_fig_p029_9.png]

**Figure 10.** Figure 10: FIG. 10. Comparison of the simulated imaging data (top left) for the [PITH_FULL_IMAGE:figures/full_fig_p034_10.png]

**Figure 11.** Figure 11: FIG. 11. Same as Figure 10, but for the mock lens with axis ratio [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]

**Figure 12.** Figure 12: FIG. 12. Posterior distribution for the flux ratios (in red) predicted by the EPL + shear + circular [PITH_FULL_IMAGE:figures/full_fig_p035_12.png]

**Figure 13.** Figure 13: FIG. 13. Same as Figure 12, but for the mock lens with axis ratio [PITH_FULL_IMAGE:figures/full_fig_p035_13.png]

**Figure 14.** Figure 14: FIG. 14. Illustration of the jump discontinuity in deflection angles appearing at the [PITH_FULL_IMAGE:figures/full_fig_p040_14.png]

**Figure 15.** Figure 15: FIG. 15. Impact of a [PITH_FULL_IMAGE:figures/full_fig_p045_15.png]

**Figure 16.** Figure 16: FIG. 16. Top panel: Impact of a [PITH_FULL_IMAGE:figures/full_fig_p054_16.png]

**Figure 17.** Figure 17: FIG. 17. Same as Figure 16, but for a [PITH_FULL_IMAGE:figures/full_fig_p058_17.png]

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Forward citations

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astro-ph.CO 2025-11 conditional novelty 6.0 of 10

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Problem with the ∝ r potential We have seen in Section IV A that the solution ˜F circ 1 (ϕ) to differential Equation (2) is not 2π-periodic and does not have the expected ϕ − ϕ1 7→ π − (ϕ − ϕ1) antisymmetry. We can attempt to symmetrize the corresponding potential “manually”, starting by choosing a principal value for the polar angle in order to impose th...
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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

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work page 2015

[2] [1]

manually

Problem with the ∝ r potential We have seen in Section IV A that the solution ˜F circ 1 (ϕ) to differential Equation (2) is not 2π-periodic and does not have the expected ϕ − ϕ1 7→ π − (ϕ − ϕ1) antisymmetry. We can attempt to symmetrize the corresponding potential “manually”, starting by choosing a principal value for the polar angle in order to impose th...

[3] [2]

spinny-top

Justifications for the ∝ r ln r potential a. Isothermal limit of the non-isothermal solution The circular multipoles can easily be generalized to non-isothermal slopes [26, 67], i.e., convergence profiles with a radial dependence κ(r) ∝ r1−γ where γ is a free parameter (1 < γ <3 for galaxy models, with γ = 2 the isothermal case). Defining β = 3 − γ, we ca...

[4] [3]

The m = 1 elliptical multipole deflection field To describe the deflection field of the elliptical m = 1 multipole, we can start by the φ1 = 0 component, taking the derivatives of the lensing potential from Equation (26):    α(1) x,1(r, ϕ; q) = ˆF (1) 1 (ϕ, q) cos(ϕ) − ˆF (1)′ 1 (ϕ, q) sin(ϕ) + λ1(q) · αcirc x,1 (r, ϕ) ϕ1=0,acirc 1 =1 α(1) y,1(r, ϕ; ...

[5] [4]

The m = 3 elliptical multipole deflection field For the deflection field of the elliptical m = 3, we follow the same logic as in Section C 1, first taking the derivatives of the φ3 = 0 component, i.e., ψ(1) 3 from Equation (31):    α(1) x,3(r, ϕ; q) = ˆF (1) 3 (ϕ, q) cos(ϕ) − ˆF (1)′ 3 (ϕ, q) sin(ϕ) + λ3(q) · αcirc x,3 (r, ϕ) ϕ3=0,acirc 3 =1 α(1) y,3...

[6] [5]

simplified

The m = 4 elliptical multipole deflection field In the case of the m = 4 elliptical multipole, the lensing potential belongs to the family of models of Equation (1), so we can simply apply Equation (3) to get the deflection angles:    αx,4(r, ϕ) = F4(ϕ) cosϕ − F ′ 4(ϕ) sinϕ αy,4(r, ϕ) = F4(ϕ) sinϕ + F ′ 4(ϕ) cosϕ (C7) Following Equation (14), the...

[7] [6]

Lensing potential for elliptical multipoles with m odd In this section, we assume that m ≥ 3 is odd, i.e., that we can write m = 2 k + 1 for k ∈ N∗. We have the following explicit expression for the Chebyshev polynomials of the first kind: Tm(x) = m 2 ⌊m/2⌋X p=0 (−1)p (m − p − 1)! p!(m − 2p)! (2x)m−2p = m 2 kX l=0 (−1)k−l 1 k + l + 1 k + l + 1 2l + 1 (2x)...

[8] [7]

(D15) We saw that B(1) m (q) = 0 was imposed from the expected symmetries, but the other remaining coefficient A(1) m (q) cannot be determined in such manner. Instead, using a similar approach to the m = 3 case (see Section IV C), we write the Taylor expansion of ˜F (1) m (ϕ; q) when q → 1 for ϕ = 0: ˜F (1) m (ϕ = 0; q) = q→1 A(1) m (1 − q2)k+1 + k+1X j=0...