Bispectrality of the sieved Jacobi polynomials

Alexei Zhedanov; Luc Vinet

REVIEW 3 major objections 5 minor 1 cited by

Bispectrality of the sieved Jacobi polynomials

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The sieved Jacobi polynomials are bispectral: their CMV Laurent polynomials satisfy a first-order Dunkl eigenvalue equation, which yields second-order Dunkl eigenvalue equations for the real-line sieved Jacobi polynomials of both kinds.

desk verdict Supplies the missing eigenvalue equations for sieved Jacobi polynomials with genuinely new multi-reflection Dunkl operators, but the key proof is sketched rather than fully shown. read the letter →

arxiv 2501.12806 v1 pith:X3JW4GC3 submitted 2025-01-22 math.CA math-phmath.MP

Luc Vinet , Alexei Zhedanov This is my paper · ORCID

classification math.CAmath-phmath.MP MSC 33C4542C05

keywords bispectralitysievedJacobipolynomialsDunkloperatorsorthogonalontheunitcircleCMVmatricesrootsofunityultrasphericalreflection

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 supplies the missing eigenvalue problem for the sieved Jacobi polynomials. It proves that the CMV Laurent polynomials associated with the sieved Jacobi polynomials on the unit circle are eigenfunctions of a first-order differential operator of Dunkl type, built from reflections $z \mapsto q^k/z$ with $q$ a primitive $N$-th root of unity. From this circle result it derives explicit second-order Dunkl-type eigenvalue equations for the sieved Jacobi polynomials of the first and second kind on the real line, and specializes them to the sieved ultraspherical polynomials. The conclusion is that these families are bispectral, meaning they satisfy dual eigenvalue equations in both the degree index and the argument variable. This closes a gap for polynomial families that arise as limits of $q$-ultraspherical polynomials at roots of unity and were expected to admit such equations.

What carries the argument

The load-bearing object is the first-order Dunkl operator $L(N)=z\partial_z + \sum_{k=0}^{N-1}A_k(z;N)(R_k-I)$, in which $R_k$ is the reflection $f(z)\mapsto f(q^k/z)$ and $q$ is a primitive $N$-th root of unity; the coefficients $A_k(z;N)$ are displayed separately for even and odd $N$. The argument works by transporting the known Dunkl eigenvalue equation $K\psi_k=\mu_k\psi_k$ for the ordinary Jacobi OPUC through the sieving relations (4.5)-(4.7), so that every sieved eigenfunction is expressed in terms of unsieved ones and the reflection sums are evaluated by the root-of-unity residue identity $\sum_{l=0}^{N-1} q^{l(h+1)}/(q^l-z)=N z^h/(1-z^N)$. The second-order operators $H(N)$ and $\tilde H(N)$ are then obtained as a quadratic expression in $L(N)$ and its similarity transform by $z-z^{-1}$, which is exactly what makes the symmetric combinations $P_n=\psi_{2n}+(1+a_{2n-1})\psi_{2n-1}$ and their companions $Q_n$ eigenfunctions.

What would settle it

Take a small case not shown in the paper, for instance $N=3$ with $n$ odd (so $n=3k+1$ or $3k+2$), and evaluate $L(3)\psi_n(z;3)-\lambda_n(3)\psi_n(z;3)$ at several generic complex points $z$ using the displayed formulas for $A_k(z;3)$ and the sieving relations (4.5)-(4.7); equivalently, compute the residuals $F_n^{(1)}$ and $F_n^{(2)}$ of (4.22) symbolically. If any such residual fails to vanish identically, the eigenvalue equation (4.15) is false and the claimed bispectrality collapses with it.

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Extended reading notes

Core claim

Fix a positive integer $N$, and let $\Phi_n(z;N)$ be the sieved Jacobi OPUC, whose Verblunsky parameters are those of the Jacobi OPUC at the indices $Nk-1$ and zero at all other indices. The paper's central claim is that the associated CMV Laurent polynomials $\psi_n(z;N)$ obey $L(N)\psi_n = \lambda_n(N)\psi_n$, where $L(N)=z\partial_z + \sum_{k=0}^{N-1} A_k(z;N)(R_k-I)$, $R_k f(z)=f(q^k/z)$, $q=e^{2\pi i/N}$, and the $A_k$ are explicit rational functions depending on $\alpha,\beta$ and the parity of $N$. The eigenvalues are $-n/2$ for even $n$ and $(n+1)/2+(\alpha+\beta+1)N$ for odd $n$. From this, the real-line sieved Jacobi polynomials of the first and second kind are shown to be eigenfunctions of the second-order Dunkl-type operators $H(N)=L(N)^2-N(\alpha+\beta+1)L(N)$ and $\tilde H(N)=(z-z^{-1})^{-1}H(N)(z-z^{-1})$, with quadratic spectra $n(n+N(\alpha+\beta+1))$ and $(n+1)(n+1+N(\alpha+\beta+1))$ respectively. The sieved ultraspherical polynomials of both kinds are obtained as the special case $\alpha=\beta$.

Load-bearing premise

The load-bearing step is the unverified parity case: the proof of Proposition 1 assumes that the residual coefficients $F_n^{(1)}(z)$ and $F_n^{(2)}(z)$ in equation (4.22) vanish identically for every parity combination of $n$, $N$, and $j$, while the paper displays the calculation only when $n$, $N$, and $j$ are all even and asserts that the other cases are analogous.

Editorial extensions

If this is right

The sieved Jacobi polynomials on the real line now have explicit second-order Dunkl-type eigenvalue equations, (6.6) and (6.7), which were previously unknown.
The sieved ultraspherical polynomials of the first and second kind satisfy explicit equations with spectra $n(n+(2\alpha+1)N)$ and $n(n+(2\alpha+1)N+2)$, respectively.
For $N=2$, the operator reproduces the Dunkl-type differential equation for the generalized ultraspherical polynomials, giving an independent derivation of that known result.
Because $H(N)$ commutes with the discrete operators $Y_m=T_m+T_{-m}$, one can form deformed operators $H(N)+\sum_m \tau_m Y_m$ that still have the sieved Jacobi polynomials as eigenfunctions, with eigenvalues depending on the residue class of the degree.
The algebraic relations of Section 5 define a generalized circle Jacobi algebra; if the central claim holds, this algebra supports the sieved polynomials and may yield their Verblunsky parameters from its representations.

Reading between the lines

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

The same sieving mechanism should apply to any CMV-bispectral family whose unsieved Laurent polynomials are Dunkl eigenfunctions with a single reflection: replacing one reflection by the $N$ root-of-unity reflections gives a candidate operator for the sieved family, provided the parity-dependent coefficient cancellations are verified.
The vanishing of the shift coefficients $E_k(z)=0$ in equation (6.12) is a nontrivial rational identity; treating it as a separate condition could yield a sufficient criterion for bispectrality of sieved versions of other OPUC families.
One plausible reading is that the missing eigenvalue operator for sieved ultraspherical polynomials is not a $q$-difference operator but a Dunkl operator with reflections at roots of unity; this may sharpen the expected root-of-unity limit of the corresponding $q$-ultraspherical difference operator.
For $N>2$ and $\alpha=\beta$, the operators (8.9)-(8.10) could be compared numerically against the sieved ultraspherical recurrence to confirm the spectra term by term, extending the $N=2$ check in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The headline is that this paper fills a recognized gap: eigenvalue equations for sieved Jacobi polynomials, on both the unit circle and the real line, were missing, and this manuscript constructs them. The operators L(N) and H(N) with N reflection terms are new in this area of one-variable orthogonal polynomials. That alone is a real contribution.

What it does well: the strategy is clean. It lifts the known N=1 Jacobi OPUC eigenvalue equation from the authors' earlier work [22] and uses the sieved-OPUC relations (4.5)-(4.7) to pull the result up to arbitrary N. The N=2 check reproduces the known generalized ultraspherical equation, which is a reassuring sanity test. The algebraic relations in Section 5 and the commuting finite-difference operators in Section 7 are useful additions, not padding.

The soft spot is exactly where the reader's report puts it: Proposition 1 is the load-bearing step, and its proof shows only the case where n, N, and j are all even. The remaining parity cases are asserted to be analogous. Given that the coefficients A_k(z;N) and the powers of z both change with parity, that is a nontrivial set of omissions. I did not see a sign or exponent error lurking, and the contour-integral identities (4.30)-(4.32) are exactly the machinery needed for the other cases, so the conclusion is plausible. But the paper as written asks the referee to take the most important computation on faith. The dependence on the self-cited [22] base result is acceptable in principle, but it does mean the preprint is not fully self-contained.

There are minor typos: (4.22) should have ψ_k not ψ_n in the arguments, and some exponents in the displayed sums are easy to misread. Cosmetic, but they do not help confidence.

Overall: the central argument holds up as a plausible construction, the result is significant for the special-functions community, and the paper deserves a serious referee. My recommendation is to send it to peer review and ask the authors to either write out the remaining parity cases or supply a machine-checkable verification of the coefficient identities. With that filled in, it would be a solid contribution.

Referee Report

3 major / 5 minor

Summary. The paper claims that the CMV Laurent polynomials associated with the sieved Jacobi polynomials on the unit circle satisfy an eigenvalue equation for a first-order Dunkl-type differential operator L(N), and that the real-line sieved Jacobi polynomials of the first and second kind are eigenfunctions of second-order Dunkl-type operators H(N) and \tilde H(N). The proof proceeds by reducing to the authors' earlier result for ordinary Jacobi OPUC, verifying Proposition 1 by a direct but only partially displayed calculation, and then deriving the real-line operators via the Szegő map. Special cases treated include the generalized ultraspherical polynomials (N=2) and the sieved ultraspherical polynomials of first and second kind for arbitrary N.

Significance. If the result is correct, it fills a genuine gap: no eigenvalue equation was known for the sieved Jacobi and sieved ultraspherical polynomials. The construction of explicit Dunkl-type operators with cyclic reflections is novel and the paper correctly identifies the eigenvalue formulas and their special cases, including agreement with the known Dunkl-type equation for generalized ultraspherical polynomials. The paper also gives explicit algebraic relations for the operators L(N), R_j, T_j. The main weakness is that the central verification in Proposition 1 is incomplete: only one parity case is shown, and several algebraic identities in Section 6 are asserted without proof.

major comments (3)

[Section 4 (Proposition 1 proof)] The proof of Proposition 1 verifies Eq. (4.22) only in the case where n, N, and j are all even; the text then states that the remaining cases 'can be treated analogously' and that the results 'have been validated in all these situations.' This is load-bearing because Proposition 1 supplies the eigenvalues λ_n(N) for all n, and Proposition 3 and all special cases are derived from it. Since the reduction formulas (4.5)-(4.7) and the coefficients A_k(z;N) in (4.17)-(4.18) change with the parities of n, N, and j, an unchecked sign or exponent error in any omitted case would change the eigenvalue or the operator form and propagate to the bispectrality conclusion. The manuscript should provide the complete case analysis, for example a table of all parity cases with the corresponding F_n^(1)(z) and F_n^(2)(z), or a reproducible computer-algebra verification.
[Section 3, Eq. (3.4)] Equation (3.4), the eigenvalue equation for the ordinary Jacobi OPUC, is the starting point of the proof of Proposition 1, but it is imported from the authors' preprint [22], whose proof is not reproduced. Because [22] is a self-cited preprint, the current paper is not self-contained at this load-bearing point. I recommend stating (3.4) as a lemma and proving it in the paper or in an appendix, or otherwise clearly indicating that it is a known result with a proof available in a published source.
[Section 6, Eqs. (6.11)-(6.17)] The derivation of the explicit form of H(N) in Proposition 4 is asserted rather than shown: the text says 'The calculations then show' that E_k(z)=0 in (6.14), and the identities (6.15) and (6.16) are stated without proof. These identities are needed for the two forms of H(N) in Proposition 4 and hence for the explicit eigenvalue equations in Propositions 6 and 7. A detailed algebraic derivation, or a verifiable supplementary computation, should be supplied. Additionally, Remark 6.1 notes that the two forms are not equivalent on all Laurent polynomials, so the phrase 'equivalent expressions' in Proposition 4 should be qualified to make this clear.

minor comments (5)

[Section 4, Eq. (4.22)] In Eq. (4.22), the functions multiplying F_n^(1) and F_n^(2) are written as ψ_n(z^N) and ψ_n(z^{-N}); from the surrounding text and the reduction (4.20), they should be ψ_k(z^N) and ψ_k(z^{-N}).
[Section 4, Eq. (4.26)] The displayed definition of A_l(z;N) in Eq. (4.26) omits the factor z^2/(q^l-z^2) that appears in the general definition (4.17); as written it is inconsistent with the subsequent use of the sums (4.31) and (4.32).
[Section 4, Eq. (4.29)] The auxiliary limit in Eq. (4.29) is incorrect: lim_{w→q^l} (w-q^l)/∏_{k=0}^{N-1}(w-q^k) equals q^l/N, not N/q^l. The final summation formula (4.30) is nevertheless correct, but the displayed identity should be fixed.
[Section 7, Remark 7.1] The sentence 'Since according to Proposition 5, the operators H and Y_m can be diagonalized simultaneously, we may conclude that they commute among themselves' reverses the usual implication. The common eigenbasis established for P_n gives commutation directly; the wording should be corrected.
[Throughout] There are several typographical issues: 'Hovever' in Remark 6.1, the missing subscript n in Eq. (6.6), the inconsistent use of Z and z in Eq. (4.25), and the duplicated author name in reference [3]. These should be corrected in the revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the sieved eigenvalue equations are verified from the independent N=1 Jacobi OPUC result of [22]; the acknowledged unshown parity cases are a proof gap, not a circular reduction.

full rationale

The derivation chain is not circular. Proposition 1 takes the ordinary Jacobi OPUC eigenvalue equation K psi_k = mu_k psi_k from the authors' prior work [22] (Eqs. (3.4)-(3.8)) as an input and verifies the proposed sieved equation (4.15) by substituting the relations (4.5)-(4.7). This is a reduction to the N=1 case, not an assumption of the sieved result: the coefficients A_k(z;N) and eigenvalues lambda_n(N) are written down explicitly, not fitted to the claimed eigenfunctions, and the verification is a direct computation. Propositions 3-7 then follow algebraically from Proposition 1 by defining H(N)=L(N)^2-N(alpha+beta+1)L(N) and using the identity lambda_{2n}=lambda_{2n-1}=Lambda_n; no target eigenvalue is used as an input. The self-citation [22] is load-bearing but independent and externally checkable. The manuscript itself flags a proof gap in Proposition 1: 'While the results have been validated in all these situations, it would be tedious to go through all of them here.' That is an omitted-proof/correctness concern, not circularity, because nothing in the written argument assumes the conclusion (4.15).

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof depends on two external results: the eigenvalue equation for ordinary Jacobi OPUC from the authors' earlier work [22], and the structural relations (4.5)-(4.7) from [15]. The residue sum formula is standard mathematics. There are no fitted free parameters; α, β, and N are the natural parameters of the polynomial family.

assumptions (4)

domain assumption Eigenvalue equation (3.4) for ordinary Jacobi OPUC, Kψ_n = μ_n ψ_n, from [22].
Used as the base case for the sieved construction; the present paper does not reprove this result and relies on the cited paper by the same authors.
domain assumption Relations (4.5)-(4.7) expressing sieved Laurent polynomials in terms of ordinary ones, from [15].
Used to reduce the sieved eigenvalue equation to the base case; taken from Ismail and Li.
standard math Residue sum formula (4.30) for sums over roots of unity.
Used to simplify the sums A_k(z;N) and to prove the vanishings in the verification of Proposition 1.
domain assumption Positivity of the Jacobi weight (3.3) for the parameter range where the orthogonality holds.
Ensures the polynomials and the Hilbert space setup are well-defined, though the eigenvalue equations are algebraic and may extend by continuation.

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Cite this review

Pith. "Pith review of Bispectrality of the sieved Jacobi polynomials." pith.science (2026). https://pith.science/paper/X3JW4GC3

@misc{pith2026250112806,
  author       = {Pith},
  title        = {Pith review of: Bispectrality of the sieved Jacobi polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X3JW4GC3}},
  note         = {Machine review of arXiv:2501.12806}
}

read the original abstract

It is shown that the CMV Laurent polynomials associated to the sieved Jacobi polynomials on the unit circle satisfy an eigenvalue equation with respect to a first order differential operator of Dunkl type. Using this result, the sieved Jacobi polynomials on the real line are found to be eigenfunctions of a Dunkl differential operator of second order. Eigenvalue equations for the sieved ultraspherical polynomials of the first and second kind are obtained as special cases. These results mean that the sieved Jacobi polynomials (either on the unit circle or on the real line) are bispectral.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Eigenvalue equations for sieved polynomials or proving Askey right again
math.CA 2025-07 conditional novelty 4.0 of 10

Sieved Jacobi polynomials are eigenfunctions of an explicit Dunkl-type operator with cyclic reflections, confirming Askey's conjecture and establishing their bispectrality.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [22]

L.Vinet and A.Zhedanov, The CMV bispectrality of the Jacobi polynomials on the unit c ircle, arXiv:2412.11031v1

work page Pith review arXiv

[2] [1]

B.H.Aldana, J.A.Charris and O.Mora-Valbuena, On Block Recursions, Askey’s Sieved Jacobi Polynomials and two related Systems , Colloqium Mathematicum 78 1998, 57–91

work page 1998

[3] [2]

W.Al-Salam, W. R. Allaway and R.Askey, Sieved ultraspherical polynomials , Trans. Amer. Math. Soc. 284 (1984), 39–55

work page 1984

[4] [3]

Askey, Orthogonal polynomials old and new, and some combi natorial connec- tions , in: Enumer- ation and Design, D

R.Askey, R. Askey, Orthogonal polynomials old and new, and some combi natorial connec- tions , in: Enumer- ation and Design, D. M. Jackson and S. A. Vanstone (eds.), Aca demic Press, Toronto, Ont., 1984, 67–84

work page 1984

[5] [4]

V.M.Badkov, Systems of orthogonal polynomials explicitly represented by the Jacobi polynomials , Mathematical Notes of the Academy of Sciences of the USSR 42, 858–863 (1987)

work page 1987

[6] [5]

Belmehdi, Generalized Gegenbauer polynomials, J

S. Belmehdi, Generalized Gegenbauer polynomials, J. Comput. Appl Math. 133 (2001), 195–205

work page 2001

[7] [6]

Bustoz, M

J. Bustoz, M. E. H. Ismail, and J. Wimp, On sieved orthogonal polynomials VI: diﬀerential equation s, Diﬀer- ential and Integral Equation 3 (1990) 757–76

work page 1990

[8] [7]

Ben Cheikh and M.Gaied, Characterization of the Dunkl-classical symmetric orthog onal polynomials, Appl

Y. Ben Cheikh and M.Gaied, Characterization of the Dunkl-classical symmetric orthog onal polynomials, Appl. Math. and Comput. 187, (2007) 105–114

work page 2007