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arxiv: 1907.04280 · v1 · pith:UFV3WAXUnew · submitted 2019-07-09 · 🧮 math.CA · math-ph· math.MP

Revisiting Biorthogonal Polynomials. An LU factorization discussion

Manuel Ma\~nas This is my paper

Pith reviewed 2026-05-24 23:49 UTC · model grok-4.3

classification 🧮 math.CA math-phmath.MP
keywords biorthogonal polynomialsLU factorizationGram matricesChristoffel-Darboux kernelbilinear formsHankel determinantsGauss quadratureorthogonal polynomials
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The pith

The LU factorization of Gram matrices of bilinear forms constructs biorthogonal polynomial families, their kernels, and spectral matrices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the Gauss-Borel or LU factorization of the Gram matrix associated to a bilinear form provides the foundation for defining biorthogonal polynomials and their second-kind functions. From this factorization one obtains the spectral matrices that represent multiplication by the variable x, along with the Christoffel-Darboux kernel and its projection properties. The same construction recovers the classical three-term recurrence, Heine formulas, Gauss quadrature, and Christoffel-Darboux identity when the bilinear form is of Hankel type, and it characterizes the Hermite, Laguerre, and Jacobi orthogonal polynomials inside this framework. Finally the approach yields explicit Christoffel-type formulas for general Christoffel and Geronimus perturbations of the original bilinear form.

Core claim

The Gauss-Borel or LU factorization of Gram matrices of bilinear forms is the pivotal element in the discussion of the theory of biorthogonal polynomials. The construction of biorthogonal families of polynomials and its second kind functions, of the spectral matrices modeling the multiplication by the independent variable x, the Christoffel-Darboux kernel and its projection properties, are discussed from this point of view. Then, the Hankel case is presented and different properties, specific of this case, as the three terms relations, Heine formulas, Gauss quadrature and the Christoffel-Darboux formula are given. The classical orthogonal polynomial of Hermite, Laguerre and Jacobi type are 4

What carries the argument

The LU (Gauss-Borel) factorization of the Gram matrix of the bilinear form, which decomposes the moment matrix to define the biorthogonal polynomials, second-kind functions, and kernels.

If this is right

  • Biorthogonal families and second-kind functions are obtained directly from the factors of the Gram matrix.
  • Spectral matrices for multiplication by x arise as products of the factorization factors.
  • The Christoffel-Darboux kernel and its projection properties follow from the same factorization.
  • In the Hankel case the three-term recurrence, Heine formulas, and Gauss quadrature are recovered as direct consequences.
  • Christoffel formulas for Christoffel and Geronimus perturbations of the bilinear form are derived explicitly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The factorization perspective may extend naturally to matrix-valued or non-Hermitian bilinear forms where classical moment-matrix methods become cumbersome.
  • Numerical construction of the polynomials could proceed by stable LU routines on the moment matrix rather than by recurrence relations alone.
  • The same decomposition might illuminate connections between biorthogonal systems and structured random-matrix ensembles that share the same moment data.

Load-bearing premise

The Gram matrices associated with the bilinear forms admit an LU factorization that can be used to construct the polynomial families and kernels without additional regularity conditions beyond those stated for the classical cases.

What would settle it

A concrete bilinear form whose Gram matrix does not admit an LU factorization yet still possesses a well-defined biorthogonal polynomial system with the usual projection and recurrence properties.

read the original abstract

The Gauss-Borel or $LU$ factorization of Gram matrices of bilinear forms is the pivotal element in the discussion of the theory of biorthogonal polynomials. The construction of biorthogonal families of polynomials and its second kind functions, of the spectral matrices modeling the multiplication by the independent variable $x$, the Christoffel-Darboux kernel and its projection properties, are discussed from this point of view. Then, the Hankel case is presented and different properties, specific of this case, as the three terms relations, Heine formulas, Gauss quadrature and the Christoffel-Darboux formula are given. The classical orthogonal polynomial of Hermite, Laguerre and Jacobi type are discussed and characterized within this scheme. Finally, it is shown who this approach is instrumental in the derivation of Christoffel formulas for general Christoffel and Geronimus perturbations of the bilinear forms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper argues that the Gauss-Borel (LU) factorization of Gram matrices associated to bilinear forms serves as the central organizing principle for the theory of biorthogonal polynomials. It uses this factorization to construct the biorthogonal families, second-kind functions, spectral matrices for multiplication by x, and the Christoffel-Darboux kernel with its projection properties; specializes the framework to the Hankel case to recover three-term recurrences, Heine formulas, Gauss quadrature, and the CD formula; characterizes the classical Hermite, Laguerre, and Jacobi cases; and derives Christoffel-type formulas for general Christoffel and Geronimus perturbations of the bilinear forms.

Significance. If the constructions hold, the LU-factorization viewpoint supplies a coherent linear-algebraic route to standard results on biorthogonal polynomials and their kernels, while making the treatment of perturbations more systematic. The explicit recovery of the classical orthogonal-polynomial cases and the perturbation formulas constitute concrete evidence of utility.

major comments (1)
  1. [Abstract / general bilinear-form setup] The weakest assumption—that the Gram matrices of the bilinear forms admit an LU factorization without further regularity conditions—is load-bearing for every subsequent construction (abstract and the opening discussion of the general case). The manuscript should state the precise hypotheses (e.g., non-vanishing leading principal minors or positivity requirements) that guarantee the factorization exists for the bilinear forms under consideration.
minor comments (2)
  1. Notation for the bilinear form and its Gram matrix should be introduced once and used consistently to avoid confusion with ordinary inner-product notation.
  2. A short table or diagram summarizing the objects obtained from the LU factors (polynomials, kernels, spectral matrices) would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for major revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / general bilinear-form setup] The weakest assumption—that the Gram matrices of the bilinear forms admit an LU factorization without further regularity conditions—is load-bearing for every subsequent construction (abstract and the opening discussion of the general case). The manuscript should state the precise hypotheses (e.g., non-vanishing leading principal minors or positivity requirements) that guarantee the factorization exists for the bilinear forms under consideration.

    Authors: We agree that the existence of the LU factorization is a foundational assumption that should be stated with precision. In the revised manuscript we will add an explicit hypothesis in both the abstract and the opening discussion of the general bilinear-form case: the Gram matrices are assumed to have non-vanishing leading principal minors. This is the standard algebraic condition that guarantees the existence of the LU factorization without pivoting and is the minimal regularity needed for all subsequent constructions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper frames the Gauss-Borel LU factorization of Gram matrices as the central device for constructing biorthogonal polynomial families, kernels, and spectral matrices. This is a direct application of standard linear-algebra factorization to bilinear-form matrices under the stated existence assumption. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the derivations of three-term relations, Christoffel-Darboux formulas, and perturbation formulas follow from the matrix factorization without circular reduction to the inputs. The approach is self-contained against external linear-algebra benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central constructions rest on the existence of LU factorizations for Gram matrices of bilinear forms and on standard properties of Hankel matrices and classical weights; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Gram matrices of the bilinear forms admit LU (Gauss-Borel) factorization
    Stated as the pivotal element enabling all subsequent constructions.
  • standard math The Hankel case satisfies the three-term recurrence and Heine formulas under the LU framework
    Invoked when specializing to the Hankel case.

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Works this paper leans on

48 extracted references · 48 canonical work pages · 1 internal anchor

  1. [1]

    I/n.sc/t.sc/r.sc/o.sc/d.sc/u.sc/c.sc/t.sc/i.sc/o.sc/n.sc These notes correspond to the orthogonal polynomial part of the five lectures I delivered during the VII Iberoamerican Workshop in Orthogonal Polynomials and Applications (Seventh EIBPOA). As such, more than presenting new original material, they are intended to give an alternative, but consistent an...

    work page internal anchor Pith review Pith/arXiv arXiv 1907

  2. [2]

    we gave a complete description for the Christoffel formulas corresponding to Christoffel perturbations for univariate CMV Laurent orthogonal polynomials. We also mention recent developments on multivariate orthogonal polynomials in real spaces (MVOPR), the corresponding Christoffel formula and the interplay with algebraic geometry [14, 15]. Similar multivari...

    work page

  3. [3]

    LDU factorization

    LU /f.sc /a.sc/c.sc/t.sc/o.sc/r.sc/i.sc/z.sc/a.sc /t.sc/i.sc/o.sc/n.sc /a.sc/n.sc/d.sc G/r.sc/a.sc/m.sc /m.sc/a.sc /t.sc/r.sc/i.sc/x.sc Given a square complex matrix A∈ CN×N , an LU factorization refers to the factorization of A into a lower triangular matrix L and an upper triangular matrix U 2.1. LDU factorization. An LDU decomposition is a decompositio...

    work page

  4. [4]

    We say that a bilinear form⟨·,·⟩ is quasi-de/f_inite whenever its Gram matrix has all its leading principal minors different from zero

    O/r.sc/t.sc/h.sc/o.sc/g.sc/o.sc/n.sc/a.sc/l.sc /p.sc/o.sc/l.sc/y.sc/n.sc/o.sc/m.sc/i.sc/a.sc/l.sc/s.sc De/f_inition 1(Quasi-definite bilinear forms). We say that a bilinear form⟨·,·⟩ is quasi-de/f_inite whenever its Gram matrix has all its leading principal minors different from zero. Proposition 3 (Quasi-definiteness and LDU factorization). The Gram matrix ...

    work page 1920

  5. [5]

    We will consider in this section some properties that appear in this situation and not in the general scheme

    S/t.sc /a.sc/n.sc/d.sc /a.sc/r.sc/d.sc /o.sc/r.sc/t.sc/h.sc/o.sc/g.sc/o.sc/n.sc/a.sc/l.sc/i.sc/t.sc/y.sc: H/a.sc/n.sc/k.sc/e.sc/l.sc /r.sc/e.sc/d.sc/u.sc/c.sc/t.sc/i.sc/o.sc/n.sc Recall that for bilinear forms associated to a Borel measure or a linear functional the Gram matrix is a Hankel matrix, Gi, j+1 = Gi+1, j. We will consider in this section some p...

    work page

  6. [6]

    Neither Bessel polynomials nor Laguerre and Jacobi polynomials with right parameters leading to quasi definite linear functionals are considered

    V/e.sc/r.sc/y.sc /c.sc/l.sc/a.sc/s.sc/s.sc/i.sc/c.sc/a.sc/l.sc /o.sc/r.sc/t.sc/h.sc/o.sc/g.sc/o.sc/n.sc/a.sc/l.sc /p.sc/o.sc/l.sc/y.sc/n.sc/o.sc/m.sc/i.sc/a.sc/l.sc/s.sc: H/e.sc/r.sc/m.sc/i.sc/t.sc/e.sc, L/a.sc/g.sc/u.sc/e.sc/r.sc/r.sc/e.sc /a.sc/n.sc/d.sc J/a.sc/c.sc/o.sc/b.sc/i.sc /p.sc/o.sc/l.sc/y.sc/n.sc/o.sc/m.sc/i.sc/a.sc/l.sc/s.sc Here we study the...

    work page

  7. [7]

    Some history

    C/h.sc/r.sc/i.sc/s.sc/t.sc/o.sc/f.sc/f.sc/e.sc/l.sc /a.sc/n.sc/d.sc G/e.sc/r.sc/o.sc/n.sc/i.sc/m.sc/u.sc/s.sc /t.sc/r.sc/a.sc/n.sc/s.sc/f.sc/o.sc/r.sc/m.sc/a.sc /t.sc/i.sc/o.sc/n.sc/s.sc 6.1. Some history. Three perturbations have attracted the interest of the researchers. Christoffel perturbations, that appear when you consider a new functional ˆu = p(x)u...

    work page 1969

  8. [8]

    Adler and P

    M. Adler and P. van Moerbeke, Group factorization, moment matrices and T oda lattices , International Mathematics Research Notices 12 (1997) 556-572

    work page 1997

  9. [9]

    Adler and P

    M. Adler and P. van Moerbeke, Generalized orthogonal polynomials, discrete KP and Riemann–Hilbert problems , Communications in Mathematical Physics 207 (1999) 589-620

    work page 1999

  10. [10]

    Adler and P

    M. Adler and P. van Moerbeke, Darboux transforms on band matrices, weights and associated polynomials , International Mathematics Research Notices 18 (2001) 935-984

    work page 2001

  11. [11]

    Adler, P

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

    work page 2009

  12. [12]

    Álvarez-Fernández, G

    C. Álvarez-Fernández, G. Ariznabarreta, J. C. García-Ardila, M. Mañas, and F. Marcellán, Christoffel T ransformations for Matrix Orthogonal Polynomials in the Real Line and the non-Abelian 2D T oda Lattice Hierarchy , International Mathematics Research Notices 2016 1–57

    work page 2016

  13. [13]

    Álvarez-Fernández and M

    C. Álvarez-Fernández and M. Mañas, Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D T oda type integrable hierarchies, Advances in Mathematics 240 (2013) 132-193

    work page 2013

  14. [14]

    Álvarez-Fernández and M

    C. Álvarez-Fernández and M. Mañas, On the Christoffel–Darboux formula for generalized matrix orthogonal polynomials , Journal of Mathematical Analysis and Applications 418 (2014) 238-247

    work page 2014

  15. [15]

    Álvarez-Fernández, U

    C. Álvarez-Fernández, U. Fidalgo Prieto, and M. Mañas, The multicomponent 2D T oda hierarchy: generalized matrix orthogonal polyno- mials, multiple orthogonal polynomials and Riemann-Hilbert problems , Inverse Problems 26 (2010) 055009 (15pp)

    work page 2010

  16. [16]

    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 T oda hierarchy, Advances in Mathematics 227 (2011) 1451-1525

    work page 2011

  17. [17]

    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 (2018) https:/ /doi.org/10.1007/s13373-018-0128-y

    work page doi:10.1007/s13373-018-0128-y 2018

  18. [18]

    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 and Theoretical, 51 (2018) 205204

    work page 2018

  19. [19]

    Ariznabarreta and M

    G. Ariznabarreta and M. Mañas, Matrix orthogonal Laurent polynomials on the unit circle and T oda type integrable systems , Advances in Mathematics 264 (2014) 396-463

    work page 2014

  20. [20]

    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. ORTHOGONAL POL YNOMIALS AND LU F ACTORIZA TION 23

    work page 2015

  21. [21]

    Ariznabarreta and M

    G. Ariznabarreta and M. Mañas, Multivariate orthogonal polynomials and integrable systems , Advances in Mathematics 302 (2016) 628–739

    work page 2016

  22. [22]

    Ariznabarreta and M

    G. Ariznabarreta and M. Mañas, Christoffel transformations for multivariate orthogonal polynomials , Journal of Approximation Theory 225 (2018) 242–283

    work page 2018

  23. [23]

    Ariznabarreta and M

    G. Ariznabarreta and M. Mañas, Multivariate orthogonal Laurent polynomials and integrable systems , Publications of the Research Institute for Mathematical Sciences (K yoto University) To appear

    work page

  24. [24]

    Ariznabarreta, M

    G. Ariznabarreta, M. Mañas, and A. Toledano, CMV biorthogonal Laurent polynomials: perturbations and Christoffel formulas , Studies in Applied Mathematics 140 (2018) 333–400

    work page 2018

  25. [25]

    Ariznabarreta, M

    G. Ariznabarreta, M. Mañaas, Matrix orthogonal matrix polynomials on the unit circle and T oda type integrable systems , Advances in Mathematics 264, 396-463 (2014)

    work page 2014

  26. [26]

    M. J. Bergvelt and A. P. E. ten Kroode, Partitions, V ertex Operators Constructions and Multi-Component KP Equations , Pacific Journal of Mathematics 171 (1995) 23-88

    work page 1995

  27. [27]

    Bochner, Über Sturm-Liouvillesche Polynomsysteme, Mathematische Zeitschrift 29 (1929) 730–736

    S. Bochner, Über Sturm-Liouvillesche Polynomsysteme, Mathematische Zeitschrift 29 (1929) 730–736

    work page 1929

  28. [28]

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

    work page 2004

  29. [29]

    M. I. Bueno and F. Marcellán, Polynomial perturbations of bilinear functionals and Hessenberg matrices , Linear Algebra and its Appli- cations 414 (2006) 64-83

    work page 2006

  30. [30]

    E. B. Christoffel, Über die Gaussische Quadratur und eine V erallgemeinerung derselben , Journal für die Reine und Angewandte Mathe- matik (Crelle’s journal) 55, 61-82 (1858) (in German)

    work page

  31. [31]

    E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, T ransformation groups for soliton equations. euclidean Lie algebras and reduction of the KP hierarchy, Publications of the Research Institute for Mathematical Sciences, 18 (1982) 1077–1110

    work page 1982

  32. [32]

    Gautschi, Orthogonal Polynomials:computation and approximation , Oxford University Press, 2004

    W. Gautschi, Orthogonal Polynomials:computation and approximation , Oxford University Press, 2004

    work page 2004

  33. [33]

    I. M. Gel’fand, S. Gel’fand, V . S. Retakh, and R. Wilson, Quasideterminants , Advances in Mathematics 193 (2005), 56–141

    work page 2005

  34. [34]

    Geronimus, On polynomials orthogonal with regard to a given sequence of numbers and a theorem by W

    J. Geronimus, On polynomials orthogonal with regard to a given sequence of numbers and a theorem by W. Hahn, Izvestiya Akademii Nauk SSSR 4, 215–228 (1940) (in Russian)

    work page 1940

  35. [35]

    Golinskii, On the scientific legacy of Y a

    L. Golinskii, On the scientific legacy of Y a. L. Geronimus (to the hundredth anniversary), in Self-Similar Systems (Proceedings of the International Workshop ( July 30 - August 7, Dubna, Russia, 1998)), 273-281, Edited by V .B. Priezzhev and V . P. Spiridonov, Publishing Department, Joint Institute for Nuclear Research, Moscow Region, Dubna

    work page 1998

  36. [36]

    Sym- metries and Integrability of Difference Equations (Estérel, 1994)

    F. A. Grünbaum and L. Haine, Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation , in: “Sym- metries and Integrability of Difference Equations (Estérel, 1994)”, CRM Proceedings Lecture Notes 9, American Mathematical Society , Providence, 1996, pp. 143 –154

    work page 1994

  37. [37]

    Kac and J

    V .G. Kac and J. W. van de Leur, The n-component KP hierarchy and representation theory , Journal of Mathematical Physics 44 (2003) 3245-3293

    work page 2003

  38. [38]

    Mañas and L

    M. Mañas and L. Martínez-Alonso, The multicomponent 2D T oda hierarchy: dispersionless limit , Inverse Problems 25 (2009) 115020

    work page 2009

  39. [39]

    Mañas, L

    M. Mañas, L. Martínez-Alonso, and C. Álvarez-Fernández, The multicomponent 2d T oda hierarchy: discrete /f_lows and string equations, Inverse Problems 25 (2009) 065007

    work page 2009

  40. [40]

    Mulase, Complete integrability of the Kadomtsev–Petviashvili equation , Advances in Mathematics 54 (1984) 57-66

    M. Mulase, Complete integrability of the Kadomtsev–Petviashvili equation , Advances in Mathematics 54 (1984) 57-66

    work page 1984

  41. [41]

    P. J. Olver, On multivariate interpolation , Studies in Applied Mathematics 116 (2006) 201-240

    work page 2006

  42. [42]

    M. Sato, Soliton equations as dynamical systems on in/f_inite dimensional Grassmann manifolds (random systems and dynamical systems) , Research Institute for Mathematical Sciences Kokyuroku 439 (1981) 30-46

    work page 1981

  43. [43]

    Simon The Christoffel–Darboux kernel , Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A V olume in Honor of Vladimir G

    B. Simon The Christoffel–Darboux kernel , Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A V olume in Honor of Vladimir G. Maz’ya’s 70th Birthday , Proc. Sympos. Pure Math. 79 (2008) 295-336

    work page 2008

  44. [44]

    Schwartz, Théorie des noyaux , Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950), vol

    L. Schwartz, Théorie des noyaux , Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950), vol. 1 p. 220-230, American Mathematical Society , Providence, RI, 1952

    work page 1950

  45. [45]

    Szegő, Orthogonal Polynomials, vol

    G. Szegő, Orthogonal Polynomials, vol. XXIII of American Mathematical Society Colloquium Publications, American Mathematical Society , 1939

    work page 1939

  46. [46]

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

    work page 1969

  47. [47]

    Zhang (editor), The Schur Complement and its Applications , Springer, New Y ork, 2005

    F. Zhang (editor), The Schur Complement and its Applications , Springer, New Y ork, 2005

    work page 2005

  48. [48]

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

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

    work page 1997