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General Geronimus Perturbations for Mixed Multiple Orthogonal Polynomials

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Pith's one-line read This paper proves that, for Geronimus perturbations defined by a regular matrix polynomial with no zeros on the support, the perturbed mixed multiple orthogonality exists if and only if the associated tau-determinants are all nonzero, and…

desk verdict A useful extension with a real gap in the converse theorem: the tau-tilde definitions don't match the proof's Schur complement. read the letter →

arxiv 2411.16022 v1 pith:DXRY5PDZ submitted 2024-11-25 math.CA math-phmath.MP

Manuel Mañas , Miguel Rojas This is my paper · ORCID
classification math.CAmath-phmath.MP MSC 42C0533C4533C4747B3947B36
keywords mixedmultipleorthogonalpolynomialsGeronimustransformationsChristoffel-typeformulastau-determinantsmatrixJordanchainsMarkov-StieltjesfunctionsJacobi-Piñeiro
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

The paper proves an equivalence for Geronimus perturbations—the inverse operation of Christoffel transformations—in the setting of mixed multiple orthogonal polynomials. For a regular matrix polynomial $R(x)$ that is neither required to be monic nor restricted in the rank of its leading coefficient, and whose eigenvalues avoid the support of the measures, the perturbed orthogonality exists exactly when a family of tau-determinants $\tau_n$ never vanishes. When that condition holds, the paper gives explicit Christoffel-type formulas: the perturbed type I and type II polynomials are written as determinants built from the original polynomials, their Cauchy transforms evaluated at the eigenvalues of $R$, and the free masses that define the perturbation. The same structure is developed for left multiplication, for eigenvalues of arbitrary multiplicity via Jordan chains, and for the Markov–Stieltjes matrix function, which transforms as $\check{F}(z)=(F(z)+S(z))R^{-1}(z)$ with $\deg S=\deg R-1$. A worked three-weight Jacobi–Piñeiro example exhibits the formulas concretely.

What carries the argument

The load-bearing object is the tau-determinant $\tau_n$, the determinant of the $M\times M$ linear system (with $M=Np-r$) that determines the unknown entries of the connection matrix $\Omega=\check{S}S^{-1}$. The columns of this system are the vectors $D^{(i)}_m-W^{(i)}_m$, where $D^{(i)}_m$ are Cauchy transforms of the original type II polynomials evaluated at the eigenvalue $x_i$ of $R(x)$ and $W^{(i)}_m$ are the corresponding contributions of the delta masses; for multiple eigenvalues these become row vectors of length equal to the partial multiplicities, using Jordan chains. Nonzero $\tau_n$ makes the system invertible, yielding $\Omega_{n,n-M}=(-1)^M\tau_n/\tau_{n-1}$ and the other connection coefficients as ratios of tau-determinants, which then feed into determinantal Christoffel-type formulas for $\check{A}$ and $\check{B}$. The companion machinery is the matrix Christoffel–Darboux kernel $K_D^{[n]}$, the mixed kernel $K^{[n],(i)}$, and the banded recurrence matrix $T$, together with the divisibility theory of matrix polynomials for the Jordan-chain generalization.

What would settle it

The cleanest test is the scalar case $p=q=1$ with $R(x)=x-a$ for $a$ outside the support: the Geronimus measure is $d\check{\mu}=d\mu/(x-a)+\xi\,\delta(x-a)$. Compute the leading principal minors of the perturbed moment matrix and compare their vanishing with the tau-determinants $\tau_n$ defined in Definition 2.8 (here $M=1$, so each $\tau_n$ is the single entry $D_n(a)-W_n(a)$ up to normalization). Any mismatch between a zero $\tau_n$ and a singular perturbed moment submatrix would disprove the claimed equivalence; the paper predicts exact coincidence.

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Extended reading notes

Core claim

The central claim, stated as Theorems 2.12, 2.13, 3.1 and 3.5, is that for a Geronimus perturbation $d\check{\mu}(x)=d\mu(x)R^{-1}(x)$ plus delta masses at the eigenvalues of $R$, perturbed mixed multiple orthogonality exists if and only if $\tau_n\ne 0$ for every $n\in\mathbb{N}_0$. The tau-determinants are built from the data $D^{(i)}_n-W^{(i)}_n$: Cauchy transforms of the original polynomials evaluated at the eigenvalues of $R$, corrected by contributions of the added masses. When the determinants are nonzero, the connection matrix $\Omega$ between the perturbed and original Gauss–Borel factorizations is lower unitriangular with entries given by ratios of tau-determinants, and the perturbed type II polynomials are explicit $(Np-r+1)\times(Np-r+1)$ determinants whose last column contains $B^{(b)}_{n-M}(x),\dots,B^{(b)}_n(x)$, while the perturbed type I polynomials are expressed through the mixed Christoffel–Darboux kernels and the inverse of the linear system. The proof of necessity uses a contour-integral argument: a vanishing $\tau_n$ forces a nonzero vector that is annihilated by the eigenvectors, a contradiction. The proof of sufficiency shows that nonzero $\tau_n$, together with the first $Np-r$ orthogonality conditions, propagate the biorthogonality and degree structure for all $n$. For eigenvalues of higher multiplicity, eigenvectors are replaced by canonical sets of Jordan chains, and the left-multiplication version swaps the roles of type I and type II and of left and right eigenvectors.

Load-bearing premise

The matrix polynomial $R(x)$ must be invertible at every point of the support of the original measures, so that the regular part $d\mu R^{-1}$ in the perturbed measure has no poles on the support.

Editorial extensions

If this is right

  • Geronimus-perturbed orthogonality is controlled by an explicit sequence: if any $\tau_n$ vanishes, no orthogonality exists beyond the first $Np-r$ degrees; if none vanish, the whole biorthogonal family exists.
  • The Christoffel-type formulas give a direct algorithm: compute the Cauchy data $D^{(i)}_m$ and mass corrections $W^{(i)}_m$, form the tau-determinants, and obtain the perturbed polynomials by determinant expansions—no need to solve for the full connection matrix.
  • Right and left Geronimus perturbations are dual: formulas for $\check{B}$ under right multiplication become formulas for $\check{A}$ under left multiplication, with left and right eigenvectors exchanged.
  • The Markov–Stieltjes matrix function of the perturbed measure is a matrix linear spectral transformation $\check{F}(z)=(F(z)+S(z))R^{-1}(z)$ with $\deg S=\deg R-1$, extending Zhedanov's scalar rational spectral transformation picture to the mixed multiple setting.
  • In the Jacobi–Piñeiro three-weight case, the general formulas reduce to explicit determinants involving endpoint values of the classical polynomials and two free mass parameters $\xi_0,\xi_1$.

Reading between the lines

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

  • The tau-determinant criterion should compose: applying a Geronimus perturbation followed by a Christoffel perturbation (a Uvarov transformation) should produce a combined criterion combining the tau-determinants of this paper with those of the companion Christoffel paper, since both transformations act on the same Gaussian factorization.
  • The equivalence gives a practical detection tool for spectral algorithms: in numerical constructions of banded recurrence matrices, the index at which a tau-determinant crosses zero marks exactly where the attempted perturbed orthogonal family stops being orthogonal.
  • For scalar weights ($p=q=1$), the paper's machinery should reproduce the classical Geronimus formulas with free parameters, which would provide a simple consistency check for the determinant formulas.
  • The matrix linear spectral transformation form suggests an interpretation of these perturbations as finite-rank updates of the resolvent: $\check{F}(z)R(z)=F(z)+S(z)$ resembles a finite-rank perturbation of the Stieltjes transform, which may connect to operator-theoretic treatments.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops a general theory of Geronimus transformations for mixed multiple orthogonal polynomials. For a regular matrix polynomial R(x) satisfying condition (C2) and with no zeros on the support of the measure, the authors define a Geronimus-perturbed matrix of measures by right (and later left) multiplication, with delta masses carrying free parameters. They derive explicit Christoffel-type formulas expressing the perturbed type I and type II polynomials in terms of the original polynomials and certain tau-determinants, and they claim that the existence of the perturbed orthogonality is equivalent to the non-vanishing of these tau-determinants. The paper also computes the effect on Markov-Stieltjes matrix functions and works out the Jacobi-Pineiro example with three weights.

Significance. If correct, the paper would give a useful and fairly general framework for Geronimus perturbations of mixed multiple orthogonal polynomials, extending the authors' earlier Christoffel paper [40] and avoiding restrictive assumptions such as monicity or rank conditions on the leading coefficient. The derivation is constructive and parameter-free with respect to the mass amplitudes, and the only-if direction (Theorem 2.12) is structurally convincing: the residue argument from a vanishing determinant to a zero connection vector is a genuine nontrivial step. The explicit Jacobi-Pineiro example is a valuable concrete illustration. However, the converse half of the central equivalence, and one of the main Christoffel formulas as printed, have load-bearing defects that must be repaired before the claims can be accepted.

major comments (3)
  1. [Section 3, Definition 3.3 and following paragraph] The printed definition of tilde-tau_n as the 2x2 corner determinant det[[I_{0,0},I_{0,n}],[I_{n,0},I_{n,n}]] is inconsistent with the computation that immediately follows. The displayed identity I_{n,n} - [I_{n,0} ... I_{n,n-1}] [I_{i,l}]_{0<=i,l<=n-1}^{-1} [I_{0,n} ... I_{n-1,n}]^T = tilde-tau_n / tilde-tau_{n-1} is the Schur complement formula and equals the ratio of the leading principal minors of sizes n+1 and n, not the ratio of 2x2 corner determinants. Moreover, for n=0 the printed definition gives tilde-tau_0 = det[[I_{0,0},I_{0,0}],[I_{0,0},I_{0,0}]] = 0 identically, so the ratio is not even meaningful at n=0. Since Theorem 3.5 and hence the 'if' direction of the abstract equivalence rest on this step, the converse is not proven as written. A redefinition of tilde-tau_n as the full leading principal minor det(I_{i,l})_{0<=i,l<=n} appears to repair the argument, but this must be stated explicitly and the subsequent invertibility assertions adjusted accordingly.
  2. [Theorem 2.13, Eq. (9)] The determinant in Eq. (9) is non-square as printed: it has M rows (the M-1 rows indexed n-M+1 through n-1 plus the row containing the K-terms) but M+1 columns (the M columns D^(1),...,D^(M) plus the K-column). A non-square determinant is undefined, and Eq. (9) is one of the two central Christoffel formulas of the paper. The analogous formula in Theorem 2.21, Eq. (11), appears to have the same problem. The intended square determinant should be stated unambiguously, for example by specifying which column or row is replaced in the underlying M x M linear system.
  3. [Proposition 2.7 and Theorem 2.13, Eq. (8)] As printed, the linear system in Proposition 2.7 has incompatible block dimensions. The row block [Omega_{n,n-M} ... Omega_{n,n-1}] is a p x pM block row, since each Omega_{n,r} is p x p by Proposition 2.2, while the matrix with entries D^(i)_r - W^(i)_r is displayed as an M x M scalar matrix. Unless the entries D^(i)_r - W^(i)_r are intended to be p x 1 or p x p blocks, which Definition 2.6 does not state, the product is undefined. Because this system is the origin of the connection coefficients used in Eqs. (8)-(10), the statement needs a consistent block-calibrated formulation.
minor comments (5)
  1. [Theorem 3.1 proof] The proof defines two new families of matrix polynomials but gives both the name tilde-A: 'tilde-A(x)=R(x)A(x), tilde-A(x)=check-A(x)Omega'. The second symbol should be different, otherwise the subsequent argument is ambiguous.
  2. [Proposition 2.11] There is a missing closing parenthesis in the superscript D^{i)}_{n-M}; it should be D^{(i)}_{n-M}.
  3. [Theorem 2.13 proof] The notation Omega_{n.n-M} uses a period instead of a comma between the indices; it should be Omega_{n,n-M}.
  4. [Definition 2.28] The index range n in {M-1, M, M-2, ... } should presumably be {M-1, M, M+1, ... }; as written it contains a typographical reversal.
  5. [Definition 3.2] The displayed formula for I_{i,l} contains a redundant d-check-mu inside the integral and reuses the summation index i in the second term; clarifying the indices would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Geronimus equivalence is derived from the original polynomial data and free mass parameters, not from itself.

full rationale

The paper's central claims—Christoffel-type formulas and the equivalence between non-vanishing tau-determinants and existence of Geronimus-perturbed orthogonality—are self-contained with respect to circularity. The tau-determinants are constructed from the original polynomials, the perturbation matrix R, and the arbitrary mass parameters xi (Definitions 2.8, 2.20, 2.28), not from the perturbed orthogonality being tested. The connection coefficients are solved from explicit linear systems (Proposition 2.7), and the formulas in Theorems 2.13 and 2.29 are algebraic rearrangements of those solutions. The only-if direction (Theorem 2.12, Proposition 3.4) assumes existence and derives non-vanishing determinants; the if-direction (Theorems 3.1 and 3.5) assumes non-vanishing determinants and constructs/verifies biorthogonality via the definitions of I_{i,l} and the matrix Omega. No fitted parameter is relabeled as a prediction, and no result is imported from a self-citation as a substitute for proof. The self-citations ([40], [7], [8]) provide background and the precedent Christoffel case, but the present Geronimus derivation is carried out with explicit computations. One non-circularity concern should be noted: Definition 3.3 defines tau-tilde_n as the 2x2 corner determinant det[[I_{0,0},I_{0,n}],[I_{n,0},I_{n,n}]], while the subsequent Schur-complement computation uses the ratio of full leading principal minors; this is an internal consistency/gap issue in the proof of Theorem 3.5, not a reduction of the claim to its own inputs.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The construction rests on standard orthogonal-polynomial machinery: a normal moment matrix (invertible leading principal submatrices) that yields a Gauss-Borel factorization, and the spectral theory of regular matrix polynomials (Smith form and Jordan chains). The support condition sigma(R) intersect Delta = empty and the structural condition (C2) on leading coefficients are domain assumptions that define the class of allowed Geronimus perturbations. No numbers are fitted to data; the only free inputs are the arbitrary mass amplitudes xi in the delta terms, which are part of the perturbation data, not fitted parameters.

free parameters (1)
  • Geronimus mass amplitudes xi_i
    Arbitrary vector functions (scalars in examples) multiplying the delta masses at the eigenvalues of R or L in the perturbed measure; the tau-determinants and the existence criterion depend on them. They are inputs to the transformation, not fitted constants.
assumptions (4)
  • domain assumption The moment matrix M has an LU factorization, equivalently all leading principal submatrices M[k] are invertible (normal matrix of measures).
    Invoked at the start of Section 1.1; defines the existence of the mixed multiple orthogonal polynomials A_n and B_n and the Christoffel-Darboux kernels. Without it the tau-determinants and the entire construction are undefined.
  • domain assumption The matrix polynomial R(x) is regular and satisfies sigma(R) intersect Delta = empty, so R is invertible on the support of the measure.
    Stated in Section 2.1 before Eq. (2); needed to write d mu_tilde = d mu R^{-1} plus delta masses. If an eigenvalue of R lies in Delta, the regular part of the perturbed measure has a pole on the support.
  • domain assumption The leading and subleading coefficients of R (or L) have the structural form (C2) with identity blocks, after an invertible triangular normalization that preserves orthogonality.
    Section 1.5, Equations (C1)-(C2). This ensures det R has degree Np-r and that R(Lambda^T) is lower banded, which controls the bandwidth of the connection matrix Omega.
  • standard math Canonical sets of Jordan chains for matrix polynomials and the Smith form (from Gohberg-Lancaster-Rodman, ref [34]) describe the local structure at eigenvalues.
    Used in Section 2.2 for eigenvalues of arbitrary multiplicity (Lemma 2.16, Proposition 2.17).

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Pith. "Pith review of General Geronimus Perturbations for Mixed Multiple Orthogonal Polynomials." pith.science (2026). https://pith.science/paper/DXRY5PDZ

@misc{pith2026241116022,
  author       = {Pith},
  title        = {Pith review of: General Geronimus Perturbations for Mixed Multiple Orthogonal Polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DXRY5PDZ}},
  note         = {Machine review of arXiv:2411.16022}
}
abstract

General Geronimus transformations, defined by regular matrix polynomials that are neither required to be monic nor restricted by the rank of their leading coefficients, are applied through both right and left multiplication to a rectangular matrix of measures associated with mixed multiple orthogonal polynomials. These transformations produce Christoffel-type formulas that establish relationships between the perturbed and original polynomials. Moreover, it is proven that the existence of Geronimus-perturbed orthogonality is equivalent to the non-cancellation of certain $\tau$-determinants. The effect of these transformations on the Markov-Stieltjes matrix functions is also determined. As a case study, we examine the Jacobi-Pi\~neiro orthogonal polynomials with three weights.

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Reference graph

Works this paper leans on

49 extracted references · 48 canonical work pages

  1. [17]

    Branquinho, A

    A. Branquinho, A. Foulquié-Moreno, and M. Mañas, Multiple orthogonal polynomials: Pearson equations and Christof- fel formulas, Analysis and Mathematical Physics 12 (2022) paper 129

    work page 2022

  2. [40]

    General Christoffel Perturbations for Mixed Multiple Orthogonal Polynomials

    M.Mañas and M. Rojas, General Christoffel Perturbations for Mixed Multiple Orthogonal Polynomials , arXiv:2405.11630

    work page Pith review arXiv

  3. [1]

    Adler, P

    M. Adler, P. van Moerbeke and P. Vanhaecke,Moment matrices and multi-component KP, with applications to random matrix theory, Communications in Mathematical Physics 286 (2009) 1-38

    work page 2009

  4. [2]

    N. I. Akhiezer, Classical moment problem and some related questions in analysis, Translated by N. Kemmer Hafner Publishing Co., New York 1965

    work page 1965

  5. [3]

    Apéry, Irrationalité de𝜁(2) et𝜁(3)

    R. Apéry, Irrationalité de𝜁(2) et𝜁(3). Astérisque 61 (1979) 11-13

    work page 1979

  6. [4]

    Aptekarev, V

    A. Aptekarev, V. Kaliaguine, G. López Lagomasino, and I. Rocha, On the limit behavior of recurrence coefficients for multiple orthogonal polynomials, Journal of Approximation Theory 139 (2006) 346-370

    work page 2006

  7. [5]

    Álvarez-Fernández, G

    C. Álvarez-Fernández, G. Ariznabarreta, J. C. García-Ardila, M. Mañas, and F. Marcellán, Christoffel transfor- mations for matrix orthogonal polynomials in the real line and the non-Abelian 2D Toda lattice hierarchy , International Mathematics Research Notices 2017 (2017) 128-1341

    work page 2017

  8. [6]

    Álvarez-Fernández, U

    C. Álvarez-Fernández, U. Fidalgo Prieto, and M. Mañas, Multiple orthogonal polynomials of mixed type: Gauss–Borel factorization and the multi-component 2D Toda hierarchy, Advances in Mathematics 227 (2011) 1451-1525

    work page 2011

Show all 49 references
  1. [7]

    Ariznabarreta, J

    G. Ariznabarreta, J. C. García-Ardila, M. Mañas, and F. Marcellán. Non-Abelian integrable hierarchies: matrix biorthogonal polynomials and perturbations, Journal of Physics A: Mathematical & Theoretical 51 (2018) 205204

    2018

  2. [8]

    Ariznabarreta, J

    G. Ariznabarreta, J. C. García-Ardila, M. Mañas, and F. Marcellán,Matrix biorthogonal polynomials on the real line: Geronimus transformations, Bulletin of Mathematical Sciences 9 (2019) 1950007 (68 pages)

    2019

  3. [9]

    Araznibarreta and M

    G. Araznibarreta and M. Mañas, A Jacobi type Christoffel-Darboux formula for multiple orthogonal polynomials of mixed type, Linear Algebra and its Applications 468 (2015) 154–170

    2015

  4. [10]

    Balland and T

    K. Balland and T. Rivoal, Irrationalité d’une infinité de valeurs de la fonction zêta aux entiers impairs , Inventiones mathematicae 146 (2001) 193-207. GERONIMUS PERTURBATIONS FOR MIXED MULTIPLE ORTHOGONALITY 39

    2001

  5. [11]

    P. M. Bleher, A. B.J. Kuijlaars,Random matrices with external source and multiple orthogonal polynomials, International Mathematics Research Notices 2004 (2004) 109–129

    2004

  6. [12]

    Orthogonal Polynomials, Special Functions and Applications – Proceedings of the 16th International Symposium, Montreal, Canada, In honor to Richard Askey

    A. Branquinho, J. E. F. Díaz, A. Foulquié-Moreno, and M. Mañas, Markov Chains and Multiple Orthogonality, To appear in “Orthogonal Polynomials, Special Functions and Applications – Proceedings of the 16th International Symposium, Montreal, Canada, In honor to Richard Askey” in...

    2024

  7. [13]

    Branquinho, J

    A. Branquinho, J. E. F. Díaz, A Foulquié-Moreno, and Manuel Mañas,Finite Markov chains and multiple orthogonal polynomials, 2023, to appear in Journal of Computational and Applied Mathematics, arXiv:2308.00182

    2023 arXiv

  8. [14]

    Branquinho, J

    A. Branquinho, J. E. F. Díaz, A. Foulquié-Moreno, M. Mañas, and C. Álvarez-Fernández, Jacobi–Piñeiro Markov chains, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 118 (15) (2024)

    2024

  9. [15]

    Branquinho, J

    A. Branquinho, J. E. F. Díaz, A. Foulquié-Moreno, and M. Mañas, Hypergeometric multiple orthogonal polynomials and random walks, 2021. arXiv:2107.00770

    2021 arXiv

  10. [16]

    Branquinho, J

    A. Branquinho, J. E. F. Díaz, A. Foulquié-Moreno, and M. Mañas, Classical Multiple Orthogonal Polynomials For Arbitrary Number Of Weights And Their Explicit Representation , 2024. arXiv:2404.13958v2

    2024 arXiv

  11. [18]

    Branquinho, A

    A. Branquinho, A. Foulquié-Moreno, and M. Mañas, Spectral theory for bounded banded matrices with positive bidi- agonal factorization and mixed multiple orthogonal polynomials, Advances in Mathematics 434 (2023) 109313

    2023

  12. [19]

    Branquinho, A

    A. Branquinho, A. Foulquié-Moreno, and M. Mañas, Positive bidiagonal factorization of tetradiagonal Hessenberg matrices, Linear Algebra and its Applications 667 (2023) 132–160

    2023

  13. [20]

    Branquinho, A

    A. Branquinho, A. Foulquié-Moreno, and M. Mañas, Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and random walks, Physica Scripta 98 (2023) 105223

    2023

  14. [21]

    Branquinho, A

    A. Branquinho, A. Foulquié-Moreno, and M. Mañas, Banded matrices and their orthogonality, to appear in Recent Progress in Special Functions, Galina Filipuk, editor. AMS Contemporary Mathematics, 2024

    2024

  15. [22]

    Branquinho, A

    A. Branquinho, A. Foulquié-Moreno, and M. Mañas,Banded totally positive matrices and normality for mixed multiple orthogonal polynomials, 2024. arXiv:2404.13965

    2024 arXiv

  16. [23]

    Brezinski, Padé-type approximation and general orthogonal polynomials , Birkhäuser Verlag, Basel-Boston, Mass

    C. Brezinski, Padé-type approximation and general orthogonal polynomials , Birkhäuser Verlag, Basel-Boston, Mass. 1980

    1980

  17. [24]

    M. I. Bueno and F. Marcellán, Darboux transformation and perturbation of linear functionals, Linear Algebra and its Applications 384 (2004) 215-242

    2004

  18. [25]

    T. S. Chihara, An Introduction to Orthogonal Polynomials. In : Mathematics and its Applications Series, Vol.13. Gordon and Breach Science Publishers, New York-London-Paris, 1978

    1978

  19. [26]

    E. B. Christoffel, Über die Gaussische Quadratur und eine Verallgemeinerung derselben , Journal für die Reine und Angewandte Mathematik 55 (1858) 61-82

  20. [27]

    Daems and A

    E. Daems and A. B. J. Kuijlaars, Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions, Journal of Approximation Theory 146 (2007) 91-114

    2007

  21. [28]

    Darboux, Sur une proposition relative aux équations linéaires , Comptes rendus hebdomadaires des séances de l’Académie des Sciences 94 (1882) 1456-1459

    G. Darboux, Sur une proposition relative aux équations linéaires , Comptes rendus hebdomadaires des séances de l’Académie des Sciences 94 (1882) 1456-1459

  22. [29]

    Derevyagin, J

    M. Derevyagin, J. C. García-Ardila, and F. Marcellán, Multiple Geronimus transformations, Linear Algebra and its Applications 454 (2014) 158-183

    2014

  23. [30]

    Fidalgo Prieto, A

    U. Fidalgo Prieto, A. López-García, G. López-Lagomasino, and V. N. Sorokin, Mixed type multiple orthogonal polynomials for two Nikishin systems, Constructive Approximation 32 (2010) 255-306

    2010

  24. [31]

    Fidalgo Prieto, S

    U. Fidalgo Prieto, S. M. Peralta, J. M. Ceniceros,Mixed type multiple orthogonal polynomials: perfectness and interlacing properties of zeros, Linear Algebra and its Applications 438 (2013) 1229-1239

    2013

  25. [32]

    Gautschi, An algorithmic implementation of the generalized Christoffel theoremin Numerical Integration, edited by G

    W. Gautschi, An algorithmic implementation of the generalized Christoffel theoremin Numerical Integration, edited by G. Hämmerlin, Int. S. Num. M. 57, Birkhäuser, Basel,1982. 89-106

    1982

  26. [33]

    Geronimus, On polynomials orthogonal with regard to a given sequence of numbers , Comm

    J. Geronimus, On polynomials orthogonal with regard to a given sequence of numbers , Comm. Inst. Sci. Math. Mec. Univ. Kharkoff [Zapiski Inst. Mat. Mech.] (4)17 (1940) 3-18

    1940

  27. [34]

    Gohberg, P

    I. Gohberg, P. Lancaster, and L. Rodman, Matrix Polynomials, Classics in Applied Mathematics 58 SIAM, 2009

    2009

  28. [35]

    F. A. Grünbaum and L. Haine,Bispectral Darboux transformations: an extension of the Krall polynomials,International Mathematics Research Notices 8 (1997) 359-392. 40 M MAÑAS AND M ROJAS

    1997

  29. [36]

    F. A. Grünbaum and M. Yakimov,Discrete bispectral Darboux transformations from Jacobi operators, Pacific Journal of Mathematics 204 (2002) 395-431

    2002

  30. [37]

    M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cam- bridge, 2005

    2005

  31. [38]

    Karlberg and H

    L. Karlberg and H. Wallin, Padé type approximants and orthogonal polynomials for Markov-Stieltjes functions, Journal of Computational and Applied Mathematics 32 (1990) 153-157

    1990

  32. [39]

    Mañas, Revisiting biorthogonal polynomials

    M. Mañas, Revisiting biorthogonal polynomials. An LU factorization discussion. EIBPOA: Iberoamerican school on orthogonal polynomials and applications 2018. SEMA SIMAI Springer Series. Edited by E.J. Huertas and F. Marcellán, Springer Verlag, 2021

    2018

  33. [41]

    Maroni, Sur la suite de polynômes orthogonaux associée á la forme 𝑢 = 𝛿𝑐+𝜆(𝑥−𝑐)−1, Periodica Mathematica Hungarica 21 (1990) 223-248

    P. Maroni, Sur la suite de polynômes orthogonaux associée á la forme 𝑢 = 𝛿𝑐+𝜆(𝑥−𝑐)−1, Periodica Mathematica Hungarica 21 (1990) 223-248

    1990

  34. [42]

    Martínez-Finkelshtein and W

    A. Martínez-Finkelshtein and W. Van Assche, What is a multiple orthogonal polynomial? , Notices of the AMS 63 (2016) 1029–1031

    2016

  35. [43]

    V. B. Matveev and M. A. Salle, Differential-difference evolution equations. II: Darboux Transformation for the Toda lattice, Letters in Mathematical Physics 3 (1979) 425-429

    1979

  36. [44]

    Th. F. Moutard, Sur la construction des équations de la forme (1/𝑧)𝜕2𝑧/𝜕𝑥𝜕𝑦 = 𝜆(𝑥,𝑦) qui admettenent une intégrale générale explicite. Journal de l’École Polytechnique 45 (1878) 1–11

  37. [45]

    E. M. Nikishin and V. N. Sorokin,Rational Approximations and Orthogonality, Translations of Mathematical Mono- graphs 92, American Mathematical Society, Providence, 1991

    1991

  38. [46]

    Szegő, Orthogonal Polynomials

    G. Szegő, Orthogonal Polynomials. 4th ed., AMS Colloquium Publication Series, vol 23, AMS., Providence RI, (1975)

    1975

  39. [47]

    V. B. Uvarov, The connection between systems of polynomials that are orthogonal with respect to different distribution functions, USSR Computational and Mathematical Physics 9 (1969) 25-36

    1969

  40. [48]

    Yoon, Darboux transforms and orthogonal polynomials , Bulletin of Korean Mathematical Society 39 (2002) 359- 376

    G. Yoon, Darboux transforms and orthogonal polynomials , Bulletin of Korean Mathematical Society 39 (2002) 359- 376

    2002

  41. [49]

    Zhedanov, Rational spectral transformations and orthogonal polynomials, Journal of Computational Applied Math- ematics 85 (1997) 67-86

    A. Zhedanov, Rational spectral transformations and orthogonal polynomials, Journal of Computational Applied Math- ematics 85 (1997) 67-86

    1997

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