An evolution of matrix-valued orthogonal polynomials

Erik Koelink; Pablo Rom\'an; Wadim Zudilin

REVIEW 3 major objections 4 minor 26 references

An evolution of matrix-valued orthogonal polynomials

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read This paper proves explicit two-way connection formulas between matrix-valued and scalar Gegenbauer polynomials, giving access to symmetries, generating functions, zeros, and differential-difference structure.

desk verdict Real new connection formulas with a load-bearing computer-algebra check that needs to be made explicit. read the letter →

arxiv 2411.18362 v2 pith:GD6OZBHC submitted 2024-11-27 math.CA math-phmath.COmath.MPmath.NTmath.RT

Erik Koelink , Pablo Román , Wadim Zudilin This is my paper · ORCID

classification math.CAmath-phmath.COmath.MPmath.NTmath.RT MSC 33C4533C4733E3033F10

keywords matrix-valuedorthogonalpolynomialsGegenbauerconnectionformulascreativetelescopingzerosofgeneratingfunctionsdifferential-differenceoperatorsexperimentalmathematics

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

Matrix-valued Gegenbauer polynomials are families of orthogonal polynomials whose entries are themselves polynomials, but their explicit form has been hard to use. This paper proves that, after a degree-dependent symmetrization, each matrix-valued Gegenbauer polynomial of size $2\ell+1$ is a short linear combination of ordinary scalar Gegenbauer polynomials with at most $2\ell$ terms, with coefficients given explicitly by rational functions and gamma factors. The expansion can be inverted: each scalar Gegenbauer polynomial, multiplied by the identity matrix, is a short combination of the symmetrized matrix polynomials. If correct, these formulas turn the scalar theory of Gegenbauer polynomials into a working tool for the matrix case, yielding symmetries, generating functions, entry-wise zero information, and new differential-difference identities that have no scalar analogue.

What carries the argument

The load-bearing objects are the symmetrized polynomials $\hat P_n^{(\nu)} = D_n^{(\nu)}P_n^{(\nu)}$, with $D_n^{(\nu)}$ a diagonal matrix chosen so that $\hat P_n^{(\nu)}$ is genuinely symmetric, and the two triangular coefficient matrices $F_{k,n}^{(\nu)}$ and $G_{r,m}^{(\nu)}$ defined by the expansions. The proof of the main expansion verifies that the candidate right-hand side satisfies the same three-term recurrence and initial conditions as $\hat P_n^{(\nu)}$; the comparison reduces to a rational identity in gamma functions, which the paper treats with a matrix-level version of creative telescoping. The proliferation of gamma ratios and binomial factors is what makes the number of terms in the expansions depend only on the matrix size parameter $\ell$, not on the polynomial degree $n$.

What would settle it

Evaluate both sides of the coefficient identity displayed inside the proof of Theorem 3.6 at several explicit parameter choices, for example $\nu=2$, $\ell=3$, $n=4$, $i=2$, $j=5$, $k=1$ and $k=3$, to high precision; any mismatch would disprove the expansion. A direct check would compare $\hat P_n^{(\nu)}(x)$ computed from the three-term recurrence with the sum $\sum_k F_{k,n}^{(\nu)}C_{n-k}^{(\nu+2\ell)}(x)$ at a non-rational point such as $x=\sqrt{2}/3$.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.6: for the symmetrized matrix-valued Gegenbauer polynomial $\hat P_n^{(\nu)}(x)$ of size $(2\ell+1)\times(2\ell+1)$, the expansion $$\hat $P_n^{{(\nu)}}$(x) = \sum_{k=0}^{n\wedge 2\ell} F_{k,n}^{(\nu)} C_{n-k}^{(\nu+2\ell)}(x)$$ holds, where each $(i,j)$-entry of $F_{k,n}^{(\nu)}$ is $n!\Gamma(\nu+2\ell)\gamma(\nu+n;i,j,k)/2^n$, and $\gamma$ is an explicitly written product of binomial coefficients and gamma functions that vanishes unless $i+j\equiv k \pmod 2$. The companion Theorem 3.4 gives the inverse expansion of $C_m^{(\nu)}(x)\mathbf{1}$ in the matrix polynomials with coefficients $G_{r,m}^{(\nu)}$ of the same type. From these two expansions the paper derives a matrix hypergeometric summation identity, a family of commutation identities that become differential-difference equations, and closed generating functions, and it uses the expansions to study where individual entries vanish.

Load-bearing premise

The main expansion holds only if the rational identity in gamma functions that the proof of Theorem 3.6 verifies by computer algebra is genuinely true, and the paper supplies no derivation or code to reproduce it.

Editorial extensions

If this is right

Each entry of a symmetrized matrix-valued Gegenbauer polynomial becomes a linear combination of at most $2\ell$ scalar Gegenbauer polynomials, so scalar identities can be imported to the matrix setting.
Generating functions for the matrix polynomials exist in closed form: for each fixed $\ell$ the generating series is a rational expression in $x,t$ times $(1-2xt+t^2)^{-\nu-2\ell-\lfloor\ell\rfloor}$ with polynomial entries.
Entry-wise zeros of the first and second echelons (entries near the matrix border) are real and lie in $(-1,1)$, with interlacing between consecutive degrees in the first echelon.
Symmetry of $\hat P_n^{(\nu)}$ produces mixed differential-difference identities for the coefficient matrices $F_{k,n}^{(\nu)}$ that degenerate in the scalar $1\times1$ case.
Composing the two expansions recovers the known scalar connection formula, yielding a new matrix hypergeometric summation identity that the paper says it cannot yet prove by classical methods.

Reading between the lines

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

A natural next step, not pursued in the paper, is to push the expansion through known asymptotics of scalar Gegenbauer polynomials to obtain strong asymptotic formulas for entries of matrix-valued Gegenbauer polynomials as $n\to\infty$.
The same expansion could be used to test whether zero patterns of entries persist for larger $\ell$ and other symmetrizable matrix orthogonal families, since the structure reduces to zeros of short linear combinations of scalar polynomials.
If the matrix version of creative telescoping became algorithmic, the rational identity currently checked by computer algebra could be proven mechanically and the method extended to other families such as matrix-valued Hermite and Jacobi polynomials.
The generating-function representation may support arithmetic or Pad\'e-type applications, since the paper notes the matrix moments' generating function values at rational points can be studied entry-wise.

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.

Desk Editor's Note

read the letter

Here's the short version: this is a genuinely useful paper for the matrix-valued orthogonal polynomials community, and the main theorems are new. The caveat is that the proof of Theorem 3.6, the expansion of the symmetrized matrix Gegenbauer polynomials into scalar ones, reduces at the end to a rational identity in gamma functions that the authors say is "checked (by computer algebra)" and then give neither the check nor code. That identity is load-bearing: Corollaries 3.8-3.9, the differential-difference identities in Section 4, the generating function in Section 5, and the zero discussion all feed from Theorem 3.6. One wrong normalization in the gamma factors would break the recurrence comparison. So the central result is presently established only conditionally on that unshown computation.

What is good? Theorems 3.4 and 3.6 give explicit bidirectional expansions connecting scalar Gegenbauer polynomials to the matrix-valued ones for arbitrary matrix size, not just the 2x2 case where this was known before. From those expansions the paper gets new symmetries (Corollary 3.7), a matrix analogue of creative telescoping, generating functions for the matrix polynomials, and a genuinely noncommutative differential-difference structure that has no scalar counterpart. The exposition is honest about what is experimental: Section 6 is clearly flagged as observations, and the higher-echelon zero patterns are presented as speculations, not theorems. The paper also builds on [16] in a clean way; there is no circularity.

Soft spots, besides the main one: Lemma 3.3 is proved in a single sentence, and while the statement is plausible from parity and orthogonality, it deserves a few lines. The claimed matrix creative telescoping is manual and no algorithm or artifact is supplied; that is a missed opportunity, not a flaw in the math. The zero-location section is suggestive but should be labeled more firmly as a conjecture for echelons 3 and above.

Who is it for: specialists in matrix orthogonal polynomials, and anyone who works with connection formulas for Gegenbauer/Jacobi families. It deserves a serious referee, with the explicit request that the computer algebra identity in Theorem 3.6 be made reproducible--either a derivation, a Maple/Sage worksheet, or a certificate. If that computation checks out, the paper is publishable roughly as is; if not, the main expansion is unsupported. I would send it to review.

Referee Report

3 major / 4 minor

Summary. The paper establishes explicit connection formulas between scalar Gegenbauer polynomials and the symmetrized matrix-valued Gegenbauer polynomials introduced in [16]. Theorem 3.4 expresses each scalar Gegenbauer polynomial times the identity as a finite sum of matrix-valued Gegenbauer polynomials, and Theorem 3.6 gives the inverse expansion: each symmetrized matrix-valued Gegenbauer polynomial is written as a short sum of scalar Gegenbauer polynomials with explicit rational/gamma coefficients. From these formulas the authors derive symmetry of the matrix-valued polynomials, new differential-difference identities, generating functions for fixed matrix size, and an experimental study of the zero loci of individual entries. Part of the proof is described as a manually performed matrix analogue of creative telescoping, with a final coefficient identity verified only by computer algebra.

Significance. If fully substantiated, the results are valuable: they give the first explicit entrywise bridge between scalar and matrix Gegenbauer polynomials for arbitrary matrix size, yielding new symmetries, finite-sum expansions, matrix-valued generating functions, and differential-difference structures with no scalar analogue. The paper also contains a concrete double-sum identity (Corollary 3.9) and a clearly labelled experimental zero analysis. However, the central expansion Theorem 3.6 currently rests on an unverified computer-algebra check, and one supporting lemma has a logical gap; the claims are therefore conditional at this stage.

major comments (3)

[Section 3, proof of Theorem 3.6] The proof reduces to a six-term rational identity involving gamma-factors γ at shifted values of k, i, and ν, and the text states only that “This rational identity is checked (by computer algebra) to be valid.” No derivation, code, or certificate is supplied. This identity is load-bearing: the expansion in Theorem 3.6 is used in Corollaries 3.7–3.9, in all of Section 4, in the generating function of Section 5, and in the zero-location discussion of Section 6. Because the identity mixes several shifted parameter sets, a single normalization error would break the recurrence comparison. Please provide a human-verifiable derivation of the identity or include the computer algebra code/output as supplementary material.
[Section 3, proof of Lemma 3.2] The proof states that the p-th term vanishes when m + i - p > 4\ell - p - j, i.e. when m > 4\ell - i - j, and then concludes that the integral (3.4) vanishes for m > 2\ell when i + j \le 2\ell. This implication is not justified: for i + j \le 2\ell one has 4\ell - i - j \ge 2\ell, so m > 2\ell does not imply m > 4\ell - i - j. The argument as written proves vanishing only under the stronger condition m > 4\ell - i - j. Since the bound m > 2\ell is used to truncate the sum in Theorem 3.6 at n \wedge 2\ell, this gap must be repaired.
[Section 3, Lemma 3.3] The proof of Lemma 3.3 is a single sentence: “Recall formula (2.7). Using the orthogonality relations (2.5) and the fact that the matrix-valued Gegenbauer polynomials are symmetric, the statement follows.” This is insufficient for a lemma that supplies the zero pattern of the integral in (3.6) and is needed for Theorem 3.4. The proof should show explicitly how (2.7) and (2.5) imply the three restrictions m < j-i, i+j \equiv m \pmod 2, and m > 2\ell; the reference to symmetry of the matrix-valued Gegenbauer polynomials appears premature and is not needed for this step.

minor comments (4)

[Section 2, formula (2.7)] The parenthetical definition “a \vee b, a \wedge b stands for min {a, b} and max {a, b}, respectively” is reversed relative to standard usage and to the actual bounds in the formula; \vee should be max and \wedge should be min.
[Section 3, after Theorem 3.6] The sentence “Since \binom{n}{k} is non-zero only for n,k\in N with 0 \le k \le n” appears to refer to a binomial coefficient that is not present in the surrounding argument; please remove it or rephrase it to match the actual notation.
[Section 4, proof of Proposition 4.2] There is a typo: “Gegenbaure” should be “Gegenbauer”.
[Section 5] The claim that the entries of \tilde F^{(\nu)}_{k,n} are polynomials in \nu + n of degree \lfloor \ell \rfloor is stated without proof; a short justification would help the reader verify the generating-function computation for general \ell.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the connection formulas are proved from previously established recurrence and orthogonality facts, and the only load-bearing gap is an unverified computer-algebra identity, which is a verification concern rather than a circular one.

full rationale

The paper's derivation is self-contained relative to its stated inputs. The matrix-valued Gegenbauer polynomials, their three-term recurrence (2.11), derivative identity (2.10), and weight decomposition are taken from the prior papers [16] and [19]; although the authorship overlaps with the present paper, these are published, explicitly stated results that do not include the target expansion (3.1) or the coefficient formula of Theorem 3.6, so they are legitimate external foundations rather than a self-citation chain. Theorem 3.4 is derived by direct evaluation of the integral (3.6) using orthogonality (2.5), the explicit weight (2.7), and the known norm (H_0), not by assuming the conclusion. Theorem 3.6 is proved by showing that the proposed right-hand side obeys the same three-term recurrence and initial data as the symmetrized polynomials; this reduces the proof to a rational identity in gamma functions. The paper says this identity 'is checked (by computer algebra) to be valid' and that the matrix creative-telescoping calculations 'were manually performed' because no implementation exists. That is a missing certificate or omitted derivation, which is a correctness risk if the identity were false, but it is not circular: the identity is not the theorem, and the proof does not assume the expansion it is meant to establish. The later sections derive differential-difference identities, generating functions, and zero-location statements from Theorems 3.4 and 3.6; they do not feed back into the proof. Corollary 3.9 follows by composing the two connection formulas and comparing with the classical scalar connection formula (2.3); this is an honest derivation of a new-looking identity, not a renaming of an input as the main result. No fitted parameter is relabelled as a prediction, and no uniqueness theorem from the authors' prior work is invoked to force the chosen form. The only flagged limitation is the unreported computer-algebra verification in the proof of Theorem 3.6, which should be assessed under correctness risk, not circularity.

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

The central results rest on classical scalar Gegenbauer identities and on the pre-existing framework of matrix-valued Gegenbauer polynomials from [16]. No new physical or mathematical entities are introduced. The paper introduces the coefficient matrices F and G, but these are derived objects, not axioms.

assumptions (3)

standard math Standard identities for scalar Gegenbauer polynomials: orthogonality (2.5), connection and linearisation formulas (2.2)-(2.4), three-term recurrence (2.6), derivative relation.
These are classical results used throughout the proofs; they are not proved in the paper.
domain assumption Definition and structural properties of the matrix-valued Gegenbauer polynomials from [16]: positive definite weight (2.7), LDU decomposition (2.8), orthogonality (2.9), derivative relation (2.10), three-term recurrence (2.11), and differential operators used in Section 4.
The paper takes these as given and builds its new connection formulas on them. They are prior results by the first two authors and collaborators.
standard math Uniqueness of expansion in a basis of scalar Gegenbauer polynomials of degree at most n.
Used implicitly when comparing coefficients in the proof of Theorem 3.6 and elsewhere.

how reviews work

0 comments

Cite this review

Pith. "Pith review of An evolution of matrix-valued orthogonal polynomials." pith.science (2026). https://pith.science/paper/GD6OZBHC

@misc{pith2026241118362,
  author       = {Pith},
  title        = {Pith review of: An evolution of matrix-valued orthogonal polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GD6OZBHC}},
  note         = {Machine review of arXiv:2411.18362}
}

abstract

We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating functions, distribution of zeros for individual entries of the matrices and new type of differential-difference structure. We further speculate about other potentials of the connection formulas found. Part of our proofs makes use of creative telescoping in a matrix setting$-$the strategy which is not yet developed algorithmically.

Figures

Figures reproduced from arXiv: 2411.18362 by the authors.

**Figure 6.1.** Zeros of the middle entry Pˆ (3) 30 (x) ℓ,ℓ for ℓ = 2 (left), ℓ = 4 (center) and ℓ = 6 (right) when multiple zeros and non-strict interlacing are allowed). This justifies the location of zeros of entries Pb(ν) n (x) i,j with echℓ(i, j) = 2. Remark 6.1. For the middle (1, 1)-entry in the 3 × 3 case (ℓ = 1), Theorem 3.6 gives the following explicit expression [PITH_FULL_IMAGE:figures/full_fig_p019_6_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 25 canonical work pages

[16]

Koelink, A.M

E. Koelink, A.M. de los R ´ ıos, P. Rom´ an,Matrix-valued Gegenbauer-type polynomials, Constr. Approx. 46 (2017), 459–487

work page 2017
[1]

Andrews, R.A

G.E. Andrews, R.A. Askey, R. Roy, Special functions, Encycl. Math. Appl. 71, Cambridge Univ. Press, 1999

work page 1999
[2]

Askey, Orthogonal polynomials and special functions , Regional Conf

R. Askey, Orthogonal polynomials and special functions , Regional Conf. Series Appl. Maths 21, SIAM, 1975

work page 1975
[3]

Bailey, J.M

D.H. Bailey, J.M. Borwein, N.J. Calkin, R. Girgensohn, D.R. Luke, V.H. Moll, Experimental mathematics in action , A K Peters, 2007. AN EVOLUTION OF MATRIX-V ALUED ORTHOGONAL POLYNOMIALS 21

work page 2007
[4]

Beardon, K.A

A.F. Beardon, K.A. Driver, The zeros of linear combinations of orthogonal polynomials, J. Approx. Theory 137 (2005), 179–186

work page 2005
[5]

Cagliero, T.H

L. Cagliero, T.H. Koornwinder, Explicit matrix inverses for lower triangular matrices with entries involv- ing Jacobi polynomials , J. Approx. Theory 193 (2015), 20–38

work page 2015
[6]

Casper, M

W.R. Casper, M. Yakimov, The matrix Bochner problem , Amer. J. Math. 144 (2022), 1009–1065

work page 2022
[7]

Damanik, A

D. Damanik, A. Pushnitski, B. Simon, The analytic theory of matrix orthogonal polynomials , Surv. Ap- prox. Theory 4 (2008), 1–85

work page 2008

Show all 26 references

[8]

Dur´ an,Markov’s theorem for orthogonal matrix polynomials , Canad

A.J. Dur´ an,Markov’s theorem for orthogonal matrix polynomials , Canad. J. Math. 48 (1996), 1180–1195

1996
[9]

Dur´ an,Zeros of linear combinations of orthogonal polynomials , Preprint arxiv.org/2505.11956, 2025

A.J. Dur´ an,Zeros of linear combinations of orthogonal polynomials , Preprint arxiv.org/2505.11956, 2025

2025 arXiv
[10]

Dur´ an, P

A.J. Dur´ an, P. L´ opez-Rodr ´ ıguez,Orthogonal matrix polynomials: zeros and Blumenthal’s theorem , J. Approx. Theory 84 (1996), 96–118

1996
[11]

Dur´ an, P

A.J. Dur´ an, P. L´ opez-Rodr ´ ıguez, E.B. Saff,Zero asymptotic behaviour for orthogonal matrix polynomials , J. Anal. Math. 78 (1999), 37–60

1999
[12]

Groot, A.B.J

A. Groot, A.B.J. Kuijlaars, Matrix-valued orthogonal polynomials related to hexagon tilings , J. Approx. Theory 270 (2021), Paper No. 105619, 36 pp

2021
[13]

Ismail, Classical and quantum orthogonal polynomials in one variable , Encycl

M.E.H. Ismail, Classical and quantum orthogonal polynomials in one variable , Encycl. Mathematics Appl. 98, Cambridge Univ. Press, 2009

2009
[14]

Ismail, E

M.E.H. Ismail, E. Koelink, P. Rom´ an,Matrix valued Hermite polynomials, Burchnall formulas and non- abelian Toda lattice, Adv. in Appl. Math. 110 (2019), 235–269

2019
[15]

Koekoek, P.A

R. Koekoek, P.A. Lesky, R.F. Swarttouw, Hypergeometric orthogonal polynomials and their q-analogues, Springer, 2010

2010
[17]

Koelink, M

E. Koelink, M. van Pruijssen, P. Rom´ an, Matrix-valued orthogonal polynomials related to (SU(2) × SU(2), diag), Int. Math. Res. Not. IMRN 2012 (2012), no. 24, 5673–5730

2012
[18]

Koelink, M

E. Koelink, M. van Pruijssen, P. Rom´ an, Matrix-valued orthogonal polynomials related to (SU(2) × SU(2), diag) II, Publ. Res. Inst. Math. Sci. 49 (2013), 271–312

2013
[19]

Koelink, P

E. Koelink, P. Rom´ an, Orthogonal vs. non-orthogonal reducibility of matrix-valued measures , SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 008, 9 pp

2016
[20]

Koelink, P

E. Koelink, P. Rom´ an, W. Zudilin, A partial-sum deformation for a family of orthogonal polynomials , Indagationes Math. (to appear), 17 pages; doi:10.1016/j.indag.2025.04.007

2025 doi
[21]

Koornwinder, Matrix elements of irreducible representations of SU(2) × SU(2) and vector-valued orthogonal polynomials, SIAM J

T.H. Koornwinder, Matrix elements of irreducible representations of SU(2) × SU(2) and vector-valued orthogonal polynomials, SIAM J. Math. Anal. 16 (1985), 602–613

1985
[22]

Mart ´ ınez-Finkelshtein, R

A. Mart ´ ınez-Finkelshtein, R. Morales, D. Perales,Real roots of hypergeometric polynomials via finite free convolution, Int. Math. Res. Not. IMRN 2024 (2024), no. 16, 11642–11687

2024
[23]

Pacharoni, I

I. Pacharoni, I. Zurri´ an,Matrix Gegenbauer polynomials: the 2 × 2 fundamental cases, Constr. Approx. 43 (2016), 253–271

2016
[24]

Rahman, G

Q.I. Rahman, G. Schmeisser, Analytic theory of polynomials , London Math. Soc. Monogr. (N.S.) 26. Oxford Univ. Press, 2002

2002
[25]

H. S. Wilf, D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and “q”) multi- sum/integral identities, Invent. Math. 108 (1992), 575–633

1992
[26]

Zeilberger, The method of creative telescoping , J

D. Zeilberger, The method of creative telescoping , J. Symbolic Comput. 11 (1991), 195–204. IMAPP, Radboud Universiteit, Nijmegen, The Netherlands Email address : e.koelink@math.ru.nl F aMAF-CIEM, Universidad Nacional de C´ordoba, Argentina Email address : pablo.roman@unc.edu....

1991

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [16]

Koelink, A.M

E. Koelink, A.M. de los R ´ ıos, P. Rom´ an,Matrix-valued Gegenbauer-type polynomials, Constr. Approx. 46 (2017), 459–487

work page 2017

[2] [1]

Andrews, R.A

G.E. Andrews, R.A. Askey, R. Roy, Special functions, Encycl. Math. Appl. 71, Cambridge Univ. Press, 1999

work page 1999

[3] [2]

Askey, Orthogonal polynomials and special functions , Regional Conf

R. Askey, Orthogonal polynomials and special functions , Regional Conf. Series Appl. Maths 21, SIAM, 1975

work page 1975

[4] [3]

Bailey, J.M

D.H. Bailey, J.M. Borwein, N.J. Calkin, R. Girgensohn, D.R. Luke, V.H. Moll, Experimental mathematics in action , A K Peters, 2007. AN EVOLUTION OF MATRIX-V ALUED ORTHOGONAL POLYNOMIALS 21

work page 2007

[5] [4]

Beardon, K.A

A.F. Beardon, K.A. Driver, The zeros of linear combinations of orthogonal polynomials, J. Approx. Theory 137 (2005), 179–186

work page 2005

[6] [5]

Cagliero, T.H

L. Cagliero, T.H. Koornwinder, Explicit matrix inverses for lower triangular matrices with entries involv- ing Jacobi polynomials , J. Approx. Theory 193 (2015), 20–38

work page 2015

[7] [6]

Casper, M

W.R. Casper, M. Yakimov, The matrix Bochner problem , Amer. J. Math. 144 (2022), 1009–1065

work page 2022

[8] [7]

Damanik, A

D. Damanik, A. Pushnitski, B. Simon, The analytic theory of matrix orthogonal polynomials , Surv. Ap- prox. Theory 4 (2008), 1–85

work page 2008