Pith. sign in

REVIEW 2 major objections 5 minor 15 references

A Tau function for $q$-Painlev\'e VI as a Fredholm determinant

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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

Pith's one-line read This paper constructs the first tau function for q-Painlevé VI as a Fredholm determinant and proves its zeros mark exactly where the associated Riemann–Hilbert problem stops being solvable.

desk verdict First Fredholm-determinant tau function for qPVI, with a real proof gap in half of the main theorem. read the letter →

arxiv 2608.03345 v1 pith:RUILEU4J submitted 2026-08-04 math-ph math.CAmath.MPnlin.SI

Harini Desiraju , Pieter Roffelsen This is my paper
classification math-phmath.CAmath.MPnlin.SI MSC 39A1334M5534M5647B35
keywords q-PainlevéVItaufunctionFredholmdeterminantRiemann–HilbertproblemWidomconstantq-differenceequationsexceptionallinesasymptoticexpansion
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 establishes that the $q$-difference sixth Painlevé equation ($q$PVI) has a tau function $\tau_W(t)$ given by a Fredholm determinant. The tau function, built from the Riemann–Hilbert problem for $q$PVI after recasting it on a circle, is analytic on its time domain $T$ and vanishes at exactly those times where the associated Riemann–Hilbert problem fails to be solvable. Equivalently, a zero of $\tau_W$ (or of one of three shifted copies) marks the moment the $q$PVI solution $(f,g)$ lands on a specific exceptional line in its initial value space. This gives the first Fredholm determinant representation of a $q$PVI tau function, and it yields explicit rational expressions for the $q$PVI variables in terms of tau functions and a complete asymptotic expansion near $t=0$. A sympathetic reader would care because Fredholm determinant formulas for Painlevé tau functions have historically unlocked connection problems and conformal block proofs, and this opens the discrete/$q$-deformed analogue.

What carries the argument

The load-bearing object is the jump matrix $J(z,t)=\Psi^i_\infty(z/t)(-t)^{\sigma_{0t}\sigma_3}\operatorname{diag}(r_{0t},1)\Psi^e_0(z)^{-1}$ on the circle $\gamma_{0t}$, together with its factorization $J=\widehat{\Phi}_-\widehat{\Phi}_+^{-1}$ into an explicit dual solution built from the interior and exterior $q$-hypergeometric parametrices. The Widom constant $\tau_W=\det_{\mathcal{H}_+}[T_J T_J^{-1}]$ is the tau function; because $\widehat{\Phi}_\pm$ are explicit, $\tau_W$ can be computed as a Fredholm determinant of an operator $1+U$ with kernels given by equation (3.8). The shift matrices arising from symmetries of the $q$-hypergeometric parametrices then connect the four tau functions

What would settle it

Choose parameters satisfying the non-resonance conditions, compute $\tau_W$ numerically from the explicit jump matrix on a circle, and independently solve the $q$PVI system near a predicted exceptional-line crossing; if $\tau_W$ fails to vanish at a time where $(f,g)$ lies on the corresponding exceptional line, or vanishes where the Riemann–Hilbert problem is solvable, the zero-locus theorem is false.

Watch

Extended reading notes

Core claim

The central discovery is that the Widom constant $\tau_W(t)=\det_{\mathcal{H}_+}[T_J T_J^{-1}]$ attached to the jump matrix $J(z,t)$ of Riemann–Hilbert Problem III is a tau function for $q$PVI. Here $T_J$ is the Toeplitz operator on the circle obtained by multiplication by $J$ followed by projection onto the Hardy space $\mathcal{H}_+$. The paper proves $\tau_W$ is analytic on $T$ and that $\tau_W(t_*)=0$ at $t_*\in T$ if and only if RHP III has no solution there (Proposition 3.5). Through the Mano decomposition of the connection matrix into interior and exterior $q$-hypergeometric parametrices, the paper also proves that the shifted copies $\tau_2,\tau_3,\tau_4$ vanish exactly when the solu

Load-bearing premise

The formulas and zero-locus theorem depend on the connection matrix being decomposable into the explicit $q$-hypergeometric pieces on the whole non-resonant time domain; if the parameters hit the excluded special values in (1.3) or (1.5), the construction of the tau function and its exceptional-line description breaks down.

Editorial extensions

If this is right

  • The zero set of $\tau_W$ and of its three shifted copies gives a concrete analytic characterization of where $q$PVI solutions hit exceptional lines in the initial value space.
  • The $q$PVI dependent variables $f$ and $g$ are expressed rationally in terms of $\tau_1,\dots,\tau_4$, providing a discrete analogue of classical tau-function identities for Painlevé equations.
  • The Fredholm determinant structure yields an asymptotic expansion of $\tau$ near $t=0$ to any order, with coefficients depending only on the parameters $\theta,\sigma_{0t}$ and the twist entering through $r_{0t}=c_{0t}s_{0t}$.
  • The small-$t$ expansion reproduces the known Painlevé VI tau asymptotics in the $q\to1$ limit, up to an overall power of $t$, connecting the discrete and continuous theories.
  • Because $\tau_W$ is a Fredholm determinant, the same Toeplitz/Widom machinery can be used to study the divisor and asymptotics of tau functions for other discrete Painlevé equations with analogous Riemann–Hilbert problems.

Reading between the lines

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

  • Beyond the paper: the explicit Fredholm determinant opens a concrete route to the large-$t$ and connection problem for $q$PVI tau functions by expanding the same determinant in the opposite regime; this is suggested but not carried out here.
  • Beyond the paper: the natural extension to four further tau functions $\tau_5,\dots,\tau_8$ corresponding to the remaining exceptional lines $E_5,\dots,E_8$ is left open in the paper, and the parametrix-quotient method used here could settle it.
  • Beyond the paper: the relation to the earlier conformal-block tau function remains unverified beyond leading terms; a decisive check would be to compare the full small-$t$ minor expansion against the conjectured product-of-$q$-gamma prefactor, connecting this analytic construction to the CFT side.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper constructs a tau function for the q-difference sixth Painlevé equation (qPVI) as a Fredholm determinant, specifically as the Widom constant associated with a Riemann-Hilbert problem recast on a circle. The construction uses the Mano decomposition of the connection matrix and explicit exterior/interior parametrices built from q-hypergeometric functions. The main results are: (i) analyticity of the tau function on the time domain T and the equivalence of its zeros with non-solvability of the RHP (Proposition 3.5); (ii) Theorem 1.1, asserting that zeros of four shifted tau functions correspond to the associated (f,g) lying on four exceptional lines of the initial value space; (iii) Theorems 1.2 and 1.3, expressing g and f in terms of ratios of shifted tau functions; (iv) Theorem 1.4, giving a small-t asymptotic expansion with explicit first coefficients. The proof follows the Fredholm-determinant/Widom-constant framework of Cafasso–Gavrylenko–Lisovyy and adapts it to q-difference Painlevé equations.

Significance. If the results are correct, this is a significant advance: it provides the first Fredholm determinant representation of a tau function for qPVI, extending the continuous Painlevé tau-function theory to the q-difference setting. The explicit construction from the RHP, the analytic continuation, the parameter-shift relations, and the asymptotic expansion are valuable tools for connection problems, for comparison with the Jimbo–Nagoya–Sakai conformal block tau function, and for the conjectural bilinear relations. The paper is careful in stating its reliance on prior work [Rof24, JR23] and gives concrete formulas for the first asymptotic coefficients. However, the proof of Theorem 1.1 is incomplete for two of the four cases, and the main zero-locus claim is therefore not fully established.

major comments (2)
  1. [Section 4.2, proof of Theorem 1.1] The proof treats j=1 (via Corollary 4.3) and j=3 in detail, but for j=2,4 it only states 'The remaining cases, j=2,4, are proven similarly.' The corresponding Bäcklund transformations—θ∞ → θ∞−1 for τ_2, and (σ_0t, θ_0, θ∞) → (1/2−σ_0t, θ_0−1/2, θ∞−1/2) for τ_4—are not written down, and the mapping of the exceptional lines E_2, E_4 to E_1 is not verified. The ratio formulas in Sections 4.3–4.4 (Eqs. (4.10), (4.24), (4.28)) express τ_j/τ_1 but do not by themselves establish the required zero equivalence. Since Theorem 1.1 is the central claim that the Widom constants are tau functions with the stated exceptional-line characterization, this gap must be filled.
  2. [Section 3.3, Proposition 3.5] The proof is a single sentence citing [Ber+17, Theorem 2.2]. The hypotheses of that theorem (e.g., invertibility of the Toeplitz operators, trace-class properties of the relevant compositions, and the connection between invertibility and RHP solvability) are not verified in the text. The winding-number computation in Section 3.2 is a necessary ingredient, but the step from that computation to the zero/non-solvability equivalence is asserted. Because Corollary 4.3 and hence Theorem 1.1 rest on this proposition, the proof should be expanded or the applicability of the cited theorem should be checked explicitly.
minor comments (5)
  1. [Section 2.2, Eq. (2.8)] There is a typo in the determinant display: '|Ψ∞(z,t) = 1+O(z^{-1})' should read '|Ψ∞(z,t)| = 1+O(z^{-1})'. The determinant formula for |Ψ_0(z,t)| also appears to be missing a closing bracket or a factor; please check.
  2. [Definition 3.2] The parameter list has a dangling comma: 'τ_W = τ_W(t; θ_0, θ_t, θ_1, θ_∞, σ_0t, s_0t,)' should have the comma after s_0t removed.
  3. [Section 4.2, proof of Theorem 1.1] In the detailed j=3 part, the text reads 'using local coordinates (u_1, v_1) around E_3'; this should be (u_3, v_3) to match the definition in (4.5).
  4. [Section 4.4, Eq. (4.24)] The notation 'κ_1(t)' in the denominator does not correspond to a previously defined function; κ_1 is a constant parameter. This is likely a typo (possibly 'τ_1(t)' or just 'κ_1'), and it makes the formula unreadable as printed.
  5. [Section 4.1] The end of the section states that 'The parameter transformations, underlying the tau functions τ_j, j=2,4, correspond to Bäcklund transformations in a similar way.' This is the same omission flagged in the major comments; at minimum, a reference to a detailed derivation or a supplementary file would be helpful.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity found in the derivation chain: the tau function is constructed from the RHP and its properties are derived, not assumed. Self-citations to [JR23] and [Rof24] are load-bearing but constitute independent published support, not an equivalence to the paper's inputs.

full rationale

The central object τ_W(t) is defined as the Widom constant det(T_J T_J^{-1}) (Definition 3.2) for a jump matrix J built from explicit parametrices; its zero-locus characterization (Proposition 3.5) is imported from the independent Toeplitz/RHP theory [Ber+17; Wid76; CGL19], not from the claimed conclusion. The tau-function property (vanishing iff the RHP is not solvable) is therefore proved, not assumed. The Painlevé-tau formulas (1.8) and (1.9) are derived by computing ratios τ_j/τ_1 through shift matrices and known relations, e.g. (4.10), (4.24), (4.28); they do not insert f or g as inputs. The asymptotic expansion (1.4) comes from a minor expansion of the Fredholm determinant with coefficients computed from q-hypergeometric kernels, with no fitted constants. The paper does rely on [JR23] and [Rof24], both involving coauthor Roffelsen, for the RHP setup, the Mano decomposition (2.3), and solvability/geometry input. However, these are published, independently stated theorems—not machine-checked, but not equivalent to the tau-function claim—and the paper does not invoke them to forbid alternatives. Thus self-citation is present but does not constitute a circular reduction. The proof of Theorem 1.1 for j=2,4 is explicitly left as 'The remaining cases, j=2,4, are proven similarly' (Section 4.2); this is an incompleteness/correctness risk, not a circular step. Likewise, the unverified relation to the JNS17 tau function is honestly labelled a conjecture in Section 5. Overall score 2: no significant circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new entities are postulated. The derivation rests on the RHP framework of Joshi-Roffelsen, the Mano decomposition, and the standard theory of Toeplitz and Widom determinants. The only free data are the solution parameters sigma_0t and s_0t, which are integration constants rather than fitted values.

free parameters (2)
  • sigma_0t (intermediate exponent)
    Integration constant parametrizing the general qPVI solution; not fitted to data. Requires 0 < Re sigma_0t < 1/2 for the construction.
  • s_0t (twist parameter)
    Non-zero complex integration constant parametrizing the general solution; not fitted to data.
assumptions (5)
  • domain assumption The connection matrix C(z,t) admits the Mano decomposition (2.3) into interior and exterior hypergeometric connection matrices on the domain T.
    Taken from [Rof24, Section 4.3] and [ORS20]; all subsequent parametrix and circle RHP constructions rely on this factorization.
  • domain assumption RHP I has at most one solution and generically a meromorphic solution; solvability at t or qt for each t in T.
    From [JR23, Theorem 2.12] and the analytic Fredholm alternative, used in Proposition 2.2 and in the zero-locus theorem.
  • standard math T_J T_{J^{-1}} - I is trace class and the Widom constant equals det(1+U) with U of the stated form.
    Cited [CGL19], [Wid74], [Ber+17]; used in Definition 3.2 and Proposition 3.4.
  • standard math [Ber+17, Theorem 2.2]: vanishing of the Widom constant iff the associated RHP is not solvable.
    Used for Proposition 3.5, the key tau-function property.
  • domain assumption The exterior and interior parametrices in (2.15) and (2.22) solve the model RHPs on gamma_e and gamma_i.
    These are constructed from q-hypergeometric functions; their validity depends on the non-resonance conditions (1.3) and (1.5).

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Tau function for $q$-Painlev\'e VI as a Fredholm determinant." pith.science (2026). https://pith.science/paper/RUILEU4J

@misc{pith2026260803345,
  author       = {Pith},
  title        = {Pith review of: A Tau function for $q$-Painlev\'e VI as a Fredholm determinant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RUILEU4J}},
  note         = {Machine review of arXiv:2608.03345}
}
abstract

We give an analytic construction of a tau function for the $q$-difference sixth Painlev\'e equation ($q$PVI) as a Fredholm determinant through the general Riemann-Hilbert problem associated with it. We show that the tau function is an analytic function on its domain of definition that vanishes at a particular point if and only if the corresponding Riemann-Hilbert problem is point-wise not solvable there. We express the corresponding $q$PVI transcendents in terms of the tau function as well as three copies of it with some of the parameters shifted. Then the vanishing of any of these four tau functions corresponds to the transcendents taking value in a specific corresponding exceptional line on the initial value space of $q$PVI. Finally, we derive an asymptotic expansion of the tau function for small times $t$.

Figures

Figures reproduced from arXiv: 2608.03345 by the authors.

Figure 2.1
Figure 2.1. Topological representation of Jordan curves γe, γ0t and γi relative to each other, the origin and points in the discrete q-lines q Z · x, x ∈ {q ±θt t, q±θ1 }. For the sake of simplicity, all the contours are displayed as circles, but we emphasise that γe and γi need not be. We put together the matrix functions Ψ0(z, t), Ψ0t(z, t) and Ψ∞(z, t) into one piece-wise analytic matrix function, Ψ(z, t) =    Ψ0(z, t) … view at source ↗
Figure 2.2
Figure 2.2. Topological representation of Jordan curve γ0t with respect to the origin and points in the discrete half q-lines q Z≤0 · q ±θ1 and q Z<0 · q ±θt t. The t-domain is restricted to Tδ so that q Z<0 · q ±θt t always lies within the inside of γ0t . (ii) Φ( b z, t) satisfies the following jump condition on γ0t, Φb−(z, t) = J(z, t)Φb+(z, t), (2.30) where the jump matrix is given in equation (2.28). (iii) Φ( b z, t) is nor… view at source ↗
Figure 4.1
Figure 4.1. Configuration of base points We have the following fundamental relation between the initial value space and solvability of the Riemann-Hilbert problems introduced. Proposition 4.2. For any t∗ ∈ T , the following are equivalent. (1) The point (f(t∗), g(t∗)) ∈ Xt lies on the exceptional line E1, (2) RHP I does not have a solution at t = t∗, (3) RHP II does not have a solution at t = t∗, (4) RHP III does not have a sol… view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

15 extracted references · 7 canonical work pages

  1. [1]

    Harold Widom’s work in Toeplitz operators

    [BBE22] E. Basor, A. B¨ ottcher, and T. Ehrhardt. “Harold Widom’s work in Toeplitz operators”. In:Bulletin of the American Mathematical Society59.2 (2022), pp. 175–190. [BDT21] G. Bonelli, F. Del Monte, and A. Tanzini. “BPS Quivers of Five-Dimensional SCFTs, Topological Strings and q-Painlev´ e Equations”. In:Annales Henri Poincar´ e22.8 (Mar. 2021), pp. ...

    work page doi:10.1007/s00023-021-01034-3.url:http: 2022

  2. [12]

    Schlesinger transformations of Painlev´ e II–V

    arXiv:2508.14030 [math-ph].url:https: //arxiv.org/abs/2508.14030. 31 [MF92] U. Mu˘ gan and A. S. Fokas. “Schlesinger transformations of Painlev´ e II–V”. In:J. Math. Phys.33.6 (1992), pp. 2031–2045. [Miw81] T. Miwa. “Painlev´ e property of monodromy preserving deformation equations and the analyticity ofτfunctions”. In:Publ. Res. Inst. Math. Sci.17.2 (198...

    arXiv 1992

  3. [16]

    q-Painlev´ e VI equation arising fromq-UC hierarchy

    arXiv:2509 . 10938 [math-ph]. url:https://arxiv.org/abs/2509.10938. [TM06] T. Tsuda and T. Masuda. “q-Painlev´ e VI equation arising fromq-UC hierarchy”. In: Comm. Math. Phys.262.3 (2006), pp. 595–609.doi:10.1007/s00220-005-1461-z.url: https://doi.org/10.1007/s00220-005-1461-z. [Wid74] H. Widom. “Asymptotic behavior of block Toeplitz matrices and determin...

    work page arXiv 2006

  4. [22]

    The space of monodromy data for the Jimbo-Sakai family ofq-difference equations

    [ORS20] Y. Ohyama, J. Ramis, and J. Sauloy. “The space of monodromy data for the Jimbo-Sakai family ofq-difference equations”. In:Ann. Fac. Sci. Toulouse Math. (6)29.5 (2020), pp. 1119–1250. [Pal99] J. Palmer. “Zeros of the Jimbo, Miwa, Ueno tau function”. In:J. Math. Phys.40.12 (1999), pp. 6638–6681.doi:10.1063/1.533112.url:https://doi.org/10.1063/1.5331...

    work page doi:10.1063/1.533112.url:https://doi.org/10.1063/1.533112 2020

  5. [27]

    On the Riemann-Hilbert problem for aq-difference Painlev´ e equation

    [JR21] N. Joshi and P. Roffelsen. “On the Riemann-Hilbert problem for aq-difference Painlev´ e equation”. In:Comm. Math. Phys.384.1 (2021), pp. 549–585.doi:10.1007/s00220- 021-04024-y.url:https://doi.org/10.1007/s00220-021-04024-y. [JR23] N. Joshi and P. Roffelsen. “On the Monodromy Manifold ofq-Painlev´ e VI and Its Rie- mann–Hilbert Problem.” In:Commun....

    doi:10.1007/s00220- 2021

  6. [38]

    On solutions of the Fuji-Suzuki-Tsuda sys- tem

    [GIL18] P. Gavrylenko, N. Iorgov, and O. Lisovyy. “On solutions of the Fuji-Suzuki-Tsuda sys- tem”. In:SIGMA14 (2018), p. 123.doi:10.3842/SIGMA.2018.123. [GIL19] P. Gavrylenko, N. Iorgov, and O. Lisovyy. “Higher rank isomonodromic deformations and W-algebras”. In:Lett. Math. Phys.110.2 (2019), pp. 327–364.doi:10.1007/s11005- 019-01207-6. [GL18a] P. Gavryl...

    work page doi:10.3842/sigma.2018.123 2018

  7. [116]

    Rational surfaces associated with affine root systems and geometry of the Painlev´ e equations

    [Sak01] H. Sakai. “Rational surfaces associated with affine root systems and geometry of the Painlev´ e equations”. In:Comm. Math. Phys.220.1 (2001), pp. 165–229. [Sto25] A. Stoyan.Blowup relations andq-Painlev´ e VI

    work page 2001

  8. [128]

    On theτ-function of the Painlev´ e equations

    Mathematical Surveys and Mono- graphs. The Riemann-Hilbert approach. American Mathematical Society, Providence, RI, 2006, pp. xii+553. [Oka81] K. Okamoto. “On theτ-function of the Painlev´ e equations”. In:Phys. D2.3 (1981), pp. 525–535.doi:10.1016/0167-2789(81)90026-9.url:https://doi.org/10.1016/ 0167-2789(81)90026-9. [Orm11] C. M. Ormerod. “The lattice ...

    work page doi:10.1016/0167-2789(81)90026-9.url:https://doi.org/10.1016/ 2006

Show all 15 references
  1. [2012]

    Isomonodromic Tau Functions on a Torus as Fredholm Determinants, and Charged Partitions

    4Note the correspondence (f,g) = (g TM, fTM) between the notation for dependent variables in [TM06] and ours. 30 [DDG23a] F. Del Monte, H. Desiraju, and P. Gavrylenko. “Isomonodromic Tau Functions on a Torus as Fredholm Determinants, and Charged Partitions”. In:Commun. Math. P...

    2023 arXiv

  2. [2013]

    Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory andτ-function

    [JMU81] M. Jimbo, T. Miwa, and K. Ueno. “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory andτ-function”. In: Physica D: Nonlinear Phenomena2.2 (1981), pp. 306–352. [JNS17] M. Jimbo, H. Nagoya, and H. Saka...

    1981

  3. [2018]

    The Generalized Riemann Problem for Linear Differential Equations and the Allied Problems for Linear Difference and q-Difference Equations

    arXiv:1710.11603 [hep-th].url:https://arxiv.org/abs/1710.11603. [Bir13] G. D. Birkhoff. “The Generalized Riemann Problem for Linear Differential Equations and the Allied Problems for Linear Difference and q-Difference Equations”. In:Proceedings of the American Academy of Arts ...

    1913 arXiv

  4. [2021]

    Monodromy dependence and connection formulae for isomonodromic tau functions

    [ILP18] A. Its, O. Lisovyy, and A. Prokhorov. “Monodromy dependence and connection formulae for isomonodromic tau functions”. In:Duke Math. J.167.7 (2018), pp. 1347–1432.doi: 10.1215/00127094-2017-0055. arXiv:1604.03082 [math-ph]. [Iwa+13] K. Iwasaki, H. Kimura, S. Shimemura, ...

    2018 arXiv

  5. [2024]

    [BG12] E

    arXiv:2411.00325 [nlin.SI].url:https: //arxiv.org/abs/2411.00325. [BG12] E. L. Basor and I. Gohberg.Toeplitz Operators and Related Topics: The Harold Widom Anniversary Volume Workshop on Toeplitz and Wiener-Hopf Operators, Santa Cruz, Cal- ifornia, September 20–22,

    arXiv

  6. [2025]

    Conformal field theory of Painlev´ e VI

    arXiv:2501.01419 [math-ph].url:https://arxiv.org/abs/2501.01419. [GIL12] O. Gamayun, N. Iorgov, and O. Lisovyy. “Conformal field theory of Painlev´ e VI”. In: JHEP10 (2012). [Erratum: JHEP 10, 183 (2012)], p. 038.doi:10.1007/JHEP10(2012)

    2012 arXiv

  7. [6507]

    Painlev´ e/CFT correspondence on a torus

    [Des22] H. Desiraju. “Painlev´ e/CFT correspondence on a torus”. In:Journal of Mathematical Physics63.8 (2022). arXiv:2305.04240. [Gav25] P. Gavrylenko.Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the generalq-...

    2022 arXiv

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.