Zeros of classical orthogonal polynomials in two variables Lidia - - PowerPoint PPT Presentation

Zeros of classical orthogonal polynomials in two variables

Lidia Fern´ andez

Joint work with Antonia M. Delgado, Teresa E. P´ erez & Miguel A. Pi˜ nar

Optimal points configuration and orthogonal polynomials April 2017, Castro Urdiales

Background Orthogonal polynomials in two variables

Orthogonal polynomials in two variables (Dunkl-Xu)

Let ·, · be an inner product defined on the space of polynomials of two variables f, g =

• D

f(x, y) g(x, y) w(x, y) dx dy,

Background Orthogonal polynomials in two variables

Orthogonal polynomials in two variables (Dunkl-Xu)

Let ·, · be an inner product defined on the space of polynomials of two variables f, g =

• D

f(x, y) g(x, y) w(x, y) dx dy, A polynomial p is orthogonal with respect to ·, · if p, q = 0, deg(q) ≤ deg(p)

Background Orthogonal polynomials in two variables

Orthogonal polynomials in two variables (Dunkl-Xu)

Let ·, · be an inner product defined on the space of polynomials of two variables f, g =

• D

f(x, y) g(x, y) w(x, y) dx dy, A polynomial p is orthogonal with respect to ·, · if p, q = 0, deg(q) ≤ deg(p) Vn the space of orthogonal polynomials of total degree n. dim Vn = n + 1.

Background Orthogonal polynomials in two variables

Basis of orthogonal polynomials in two variables

Pn = {Pn,k : 0 ≤ k ≤ n} denotes a basis of Vn.

Background Orthogonal polynomials in two variables

Basis of orthogonal polynomials in two variables

Pn = {Pn,k : 0 ≤ k ≤ n} denotes a basis of Vn. The basis is said to be mutually orthogonal if Pn,k, Pn,j = 0, k = j

Background Orthogonal polynomials in two variables

Basis of orthogonal polynomials in two variables

Pn = {Pn,k : 0 ≤ k ≤ n} denotes a basis of Vn. The basis is said to be mutually orthogonal if Pn,k, Pn,j = 0, k = j If, further, Pn,k, Pn,k = 1 for 0 ≤ k ≤ n then the basis is said to be orthonormal.

Background Orthogonal polynomials in two variables

Basis of orthogonal polynomials in two variables

Pn = {Pn,k : 0 ≤ k ≤ n} denotes a basis of Vn. The basis is said to be mutually orthogonal if Pn,k, Pn,j = 0, k = j If, further, Pn,k, Pn,k = 1 for 0 ≤ k ≤ n then the basis is said to be orthonormal. If Pn is a basis of Vn and M is a non-singular matrix of order n + 1, then M Pn is another basis.

Background Orthogonal polynomials in two variables

Basis of orthogonal polynomials in two variables

Pn = {Pn,k : 0 ≤ k ≤ n} denotes a basis of Vn. The basis is said to be mutually orthogonal if Pn,k, Pn,j = 0, k = j If, further, Pn,k, Pn,k = 1 for 0 ≤ k ≤ n then the basis is said to be orthonormal. If Pn is a basis of Vn and M is a non-singular matrix of order n + 1, then M Pn is another basis. Let Pn = {Pn,k : 0 ≤ k ≤ n} and Qn = {Qn,k : 0 ≤ k ≤ n} be two bases of Vn. They are biorthogonal bases if Pn,k, Qn,j = 0, k = j

Background Zeros of orthogonal polynomials in one variable

Zeros of orthogonal polynomials in one variable

{pn(x)}n≥0 orthogonal polynomials w. r. t. dµ in [a, b].

Background Zeros of orthogonal polynomials in one variable

Zeros of orthogonal polynomials in one variable

{pn(x)}n≥0 orthogonal polynomials w. r. t. dµ in [a, b]. What about the zeros of the polynomial pn?

Background Zeros of orthogonal polynomials in one variable

Zeros of orthogonal polynomials in one variable

{pn(x)}n≥0 orthogonal polynomials w. r. t. dµ in [a, b]. What about the zeros of the polynomial pn? Localization: They are real, simple and are located in (a, b).

Background Zeros of orthogonal polynomials in one variable

Zeros of orthogonal polynomials in one variable

{pn(x)}n≥0 orthogonal polynomials w. r. t. dµ in [a, b]. What about the zeros of the polynomial pn? Localization: They are real, simple and are located in (a, b). Interlacing property: x1 < · · · < xn zeros of pn y1 < · · · < yn−1 zeros of pn−1 x1 < y1 < x2 < · · · < yn−1 < xn

Background Zeros of orthogonal polynomials in one variable

Zeros of orthogonal polynomials in one variable

{pn(x)}n≥0 orthogonal polynomials w. r. t. dµ in [a, b]. What about the zeros of the polynomial pn? Localization: They are real, simple and are located in (a, b). Interlacing property: x1 < · · · < xn zeros of pn y1 < · · · < yn−1 zeros of pn−1 x1 < y1 < x2 < · · · < yn−1 < xn Eigenvalues of a truncated Jacobi matrix

Background Zeros of orthogonal polynomials in one variable

Zeros of orthogonal polynomials in one variable

{pn(x)}n≥0 orthogonal polynomials w. r. t. dµ in [a, b]. What about the zeros of the polynomial pn? Localization: They are real, simple and are located in (a, b). Interlacing property: x1 < · · · < xn zeros of pn y1 < · · · < yn−1 zeros of pn−1 x1 < y1 < x2 < · · · < yn−1 < xn Eigenvalues of a truncated Jacobi matrix Density

Background Zeros of orthogonal polynomials in one variable

Zeros of orthogonal polynomials in one variable

{pn(x)}n≥0 orthogonal polynomials w. r. t. dµ in [a, b]. What about the zeros of the polynomial pn? Localization: They are real, simple and are located in (a, b). Interlacing property: x1 < · · · < xn zeros of pn y1 < · · · < yn−1 zeros of pn−1 x1 < y1 < x2 < · · · < yn−1 < xn Eigenvalues of a truncated Jacobi matrix Density Electrostatic interpretation

Background Zeros of orthogonal polynomials in two variables

Zeros of orthogonal polynomials in two variables

Several difficulties when we deal with zeros in two variables:

Background Zeros of orthogonal polynomials in two variables

Zeros of orthogonal polynomials in two variables

Several difficulties when we deal with zeros in two variables: The zeros are points or curves in the plane.

Background Zeros of orthogonal polynomials in two variables

Zeros of orthogonal polynomials in two variables

Several difficulties when we deal with zeros in two variables: The zeros are points or curves in the plane. They depend on the basis.

Background Zeros of orthogonal polynomials in two variables

Zeros of orthogonal polynomials in two variables

Several difficulties when we deal with zeros in two variables: The zeros are points or curves in the plane. They depend on the basis. Some of the proofs in one variable are based on the factorization

• f the polynomials and now this is not possible in general.

Background Zeros of orthogonal polynomials in two variables

Zeros of orthogonal polynomials in two variables

Several difficulties when we deal with zeros in two variables: The zeros are points or curves in the plane. They depend on the basis. Some of the proofs in one variable are based on the factorization

• f the polynomials and now this is not possible in general.

Some of these difficulties can be solved talking about common zeros.

Background Zeros of orthogonal polynomials in two variables

Common zeros of orthogonal polynomials in two variables (Y. Xu)

Let Pn = {Pn,k : 0 ≤ k ≤ n} denotes a basis of Vn.

Background Zeros of orthogonal polynomials in two variables

Common zeros of orthogonal polynomials in two variables (Y. Xu)

Let Pn = {Pn,k : 0 ≤ k ≤ n} denotes a basis of Vn. A common zero of Pn is a zero for every polynomial Pn,k, 0 ≤ k ≤ n.

Background Zeros of orthogonal polynomials in two variables

Common zeros of orthogonal polynomials in two variables (Y. Xu)

Let Pn = {Pn,k : 0 ≤ k ≤ n} denotes a basis of Vn. A common zero of Pn is a zero for every polynomial Pn,k, 0 ≤ k ≤ n. It can be considered as a zero of the subspace Vn.

Background Zeros of orthogonal polynomials in two variables

Common zeros of orthogonal polynomials in two variables (Y. Xu)

Let Pn = {Pn,k : 0 ≤ k ≤ n} denotes a basis of Vn. A common zero of Pn is a zero for every polynomial Pn,k, 0 ≤ k ≤ n. It can be considered as a zero of the subspace Vn. Properties: All common zeros of Pn are points in R2.

Background Zeros of orthogonal polynomials in two variables

Common zeros of orthogonal polynomials in two variables (Y. Xu)

Let Pn = {Pn,k : 0 ≤ k ≤ n} denotes a basis of Vn. A common zero of Pn is a zero for every polynomial Pn,k, 0 ≤ k ≤ n. It can be considered as a zero of the subspace Vn. Properties: All common zeros of Pn are points in R2. All zeros of Pn are distinct and simple.

Background Zeros of orthogonal polynomials in two variables

Common zeros of orthogonal polynomials in two variables (Y. Xu)

Let Pn = {Pn,k : 0 ≤ k ≤ n} denotes a basis of Vn. A common zero of Pn is a zero for every polynomial Pn,k, 0 ≤ k ≤ n. It can be considered as a zero of the subspace Vn. Properties: All common zeros of Pn are points in R2. All zeros of Pn are distinct and simple. Pn and Pn−1 do not have common zeros.

Background Zeros of orthogonal polynomials in two variables

Common zeros of orthogonal polynomials in two variables (Y. Xu)

Let Pn = {Pn,k : 0 ≤ k ≤ n} denotes a basis of Vn. A common zero of Pn is a zero for every polynomial Pn,k, 0 ≤ k ≤ n. It can be considered as a zero of the subspace Vn. Properties: All common zeros of Pn are points in R2. All zeros of Pn are distinct and simple. Pn and Pn−1 do not have common zeros. They are related with the joint eigenvalues of some truncated block Jacobi matrices.

Background Zeros of orthogonal polynomials in two variables

Common zeros of orthogonal polynomials in two variables (Y. Xu)

Let Pn = {Pn,k : 0 ≤ k ≤ n} denotes a basis of Vn. A common zero of Pn is a zero for every polynomial Pn,k, 0 ≤ k ≤ n. It can be considered as a zero of the subspace Vn. Properties: All common zeros of Pn are points in R2. All zeros of Pn are distinct and simple. Pn and Pn−1 do not have common zeros. They are related with the joint eigenvalues of some truncated block Jacobi matrices. Not every inner product has orthogonal polynomials with common

• zeros. In fact, there are not common zeros in centrally symmetric inner

products such as on the unit disk.

Zeros of orthogonal polynomial on the disk

Orthogonal polynomials on the disk: Hermite 1865

• Ch. Hermite, Sur quelques dev´

eloppments en s´ erie de fonctions de plusieurs variables, Comptes Rendus de l’Acad´ emie des Sciences, 1865.

Zeros of orthogonal polynomial on the disk

Orthogonal polynomials on the disk: Hermite 1865

Zeros of orthogonal polynomial on the disk

Orthogonal polynomials on the disk: Hermite 1865

Zeros of orthogonal polynomial on the disk

Orthogonal polynomials on the disk: Hermite 1865

Zeros of orthogonal polynomial on the disk

Orthogonal polynomials on the disk

• Ch. Hermite, Sur quelques dev´

eloppments en s´ erie de fonctions de plusieurs variables, Comptes Rendus de l’Acad´ emie des Sciences, 1865.

Zeros of orthogonal polynomial on the disk

Orthogonal polynomials on the disk

• Ch. Hermite, Sur quelques dev´

eloppments en s´ erie de fonctions de plusieurs variables, Comptes Rendus de l’Acad´ emie des Sciences, 1865.

• W. Tramm, Geometrische Diskussion des Hermite’schen

Polynoms, Inaugural disertation, Zurich, 1908.

• Ch. Willigens, Sur les polynˆ
• mes Um,n, Nouvelles Annales de

Math´ ematiques, 1911.

Zeros of orthogonal polynomial on the disk

Orthogonal polynomials on the disk

• Ch. Hermite, Sur quelques dev´

eloppments en s´ erie de fonctions de plusieurs variables, Comptes Rendus de l’Acad´ emie des Sciences, 1865.

• W. Tramm, Geometrische Diskussion des Hermite’schen

Polynoms, Inaugural disertation, Zurich, 1908.

• Ch. Willigens, Sur les polynˆ
• mes Um,n, Nouvelles Annales de

Math´ ematiques, 1911. P . Appell and J. Kamp´ e de F´ eriet, Fonctions Hyperg´ eom´ etriques et HyperSph´

• eriques. Polynomes d’Hermite, Gauthier-Villars,

Paris, 1926.

Zeros of orthogonal polynomial on the disk

Orthogonal polynomials on the disk

Inner product on B2 := {(x, y) ∈ R2 : x2 + y2 1} f, gµ = 1 ωµ

• B2 f(x, y) g(x, y) (1 − x2 − y2)µ dx,

µ > −1

Zeros of orthogonal polynomial on the disk

Orthogonal polynomials on the disk

Inner product on B2 := {(x, y) ∈ R2 : x2 + y2 1} f, gµ = 1 ωµ

• B2 f(x, y) g(x, y) (1 − x2 − y2)µ dx,

µ > −1 Four different bases:

Zeros of orthogonal polynomial on the disk

Orthogonal polynomials on the disk

Inner product on B2 := {(x, y) ∈ R2 : x2 + y2 1} f, gµ = 1 ωµ

• B2 f(x, y) g(x, y) (1 − x2 − y2)µ dx,

µ > −1 Four different bases: Um,n Orthogonal basis obtained by the Rodrigues formula.

Zeros of orthogonal polynomial on the disk

Orthogonal polynomials on the disk

Inner product on B2 := {(x, y) ∈ R2 : x2 + y2 1} f, gµ = 1 ωµ

• B2 f(x, y) g(x, y) (1 − x2 − y2)µ dx,

µ > −1 Four different bases: Um,n Orthogonal basis obtained by the Rodrigues formula. Pn,k Orthogonal basis obtained by Koornwinder construction.

Zeros of orthogonal polynomial on the disk

Orthogonal polynomials on the disk

Inner product on B2 := {(x, y) ∈ R2 : x2 + y2 1} f, gµ = 1 ωµ

• B2 f(x, y) g(x, y) (1 − x2 − y2)µ dx,

µ > −1 Four different bases: Um,n Orthogonal basis obtained by the Rodrigues formula. Pn,k Orthogonal basis obtained by Koornwinder construction. Pn

j,i

Orthogonal basis in polar coordinates.

Zeros of orthogonal polynomial on the disk

Orthogonal polynomials on the disk

Inner product on B2 := {(x, y) ∈ R2 : x2 + y2 1} f, gµ = 1 ωµ

• B2 f(x, y) g(x, y) (1 − x2 − y2)µ dx,

µ > −1 Four different bases: Um,n Orthogonal basis obtained by the Rodrigues formula. Pn,k Orthogonal basis obtained by Koornwinder construction. Pn

j,i

Orthogonal basis in polar coordinates. Vm,n Orthogonal basis biorthogonal with Um,n.

Zeros of orthogonal polynomial on the disk Rodrigues formula

Basis Um,n

For m ≥ 0, n ≥ 0 Um,n(x, y) = 1 (1 − x2 − y2)µ ∂m+n ∂xm∂yn

• (1 − x2 − y2)m+n+µ

Zeros of orthogonal polynomial on the disk Rodrigues formula

Basis Um,n

For m ≥ 0, n ≥ 0 Um,n(x, y) = 1 (1 − x2 − y2)µ ∂m+n ∂xm∂yn

• (1 − x2 − y2)m+n+µ

deg(Um,n) = m + n

Zeros of orthogonal polynomial on the disk Rodrigues formula

Basis Um,n

For m ≥ 0, n ≥ 0 Um,n(x, y) = 1 (1 − x2 − y2)µ ∂m+n ∂xm∂yn

• (1 − x2 − y2)m+n+µ

deg(Um,n) = m + n {Un−k,k(x, y) : 0 ≤ k ≤ n} basis not mutually orthogonal of Vn.

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n, for µ = 0

Um,n(x, y) = 1 (1 − x2 − y2)µ ∂m+n ∂xm∂yn

• (1 − x2 − y2)m+n+µ



• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5



Degree 1

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n, for µ = 0

Um,n(x, y) = 1 (1 − x2 − y2)µ ∂m+n ∂xm∂yn

• (1 − x2 − y2)m+n+µ



• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5



Degree 2

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n, for µ = 0

Um,n(x, y) = 1 (1 − x2 − y2)µ ∂m+n ∂xm∂yn

• (1 − x2 − y2)m+n+µ



• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5



Degree 3

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n, for µ = 0

Um,n(x, y) = 1 (1 − x2 − y2)µ ∂m+n ∂xm∂yn

• (1 − x2 − y2)m+n+µ



• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5



Degree 4

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n, for µ = 0

Um,n(x, y) = 1 (1 − x2 − y2)µ ∂m+n ∂xm∂yn

• (1 − x2 − y2)m+n+µ



• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5



Degree 5

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n, for µ = 0



• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5



Degree 15 (Some of them)

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n

Um,n = 0 is generally composed of multiple curves.

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n

Um,n = 0 is generally composed of multiple curves. The coordinate axes are going to be one of the branches of Um,n = 0 when m or n are odd.

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n

Um,n = 0 is generally composed of multiple curves. The coordinate axes are going to be one of the branches of Um,n = 0 when m or n are odd. The curves are inside the unit disk (except the axes).

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n

Um,n = 0 is generally composed of multiple curves. The coordinate axes are going to be one of the branches of Um,n = 0 when m or n are odd. The curves are inside the unit disk (except the axes). Every variety Um,n = 0 is composed of closed curves symmetric with respect to the axes, centered on the origin and concentric.

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n

Um,n = 0 is generally composed of multiple curves. The coordinate axes are going to be one of the branches of Um,n = 0 when m or n are odd. The curves are inside the unit disk (except the axes). Every variety Um,n = 0 is composed of closed curves symmetric with respect to the axes, centered on the origin and concentric. The only multiple points (except for the axes) coming from the intersection of several branches are (0, −1), (0, 1), (−1, 0) y (1, 0) and they only appear for n ≥ 4 or m ≥ 4.

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n

U5,4 = 0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n

Um,0 = 0 is decomposed in [ m

2 ] ellipses whose axis of symmetry

are the coordinates axes.

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n

Um,0 = 0 is decomposed in [ m

2 ] ellipses whose axis of symmetry

are the coordinates axes.

The semi-major axis is of size 1 and it is on the axis OY.

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n

Um,0 = 0 is decomposed in [ m

2 ] ellipses whose axis of symmetry

are the coordinates axes.

The semi-major axis is of size 1 and it is on the axis OY. The semi-minor axis is on OX and its size is given by one of the roots of the Gegenbauer polynomial P(µ,µ)

m

.

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n

Um,0 = 0 is decomposed in [ m

2 ] ellipses whose axis of symmetry

are the coordinates axes.

The semi-major axis is of size 1 and it is on the axis OY. The semi-minor axis is on OX and its size is given by one of the roots of the Gegenbauer polynomial P(µ,µ)

m

.

U8,0 = 0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n

Um,n = 0 intersects a line parallel to OX in m points and a line parallel to OY in n points. U5,4 = 0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n

Every straight line (not the axes) passing through the origin intersects Um,n = 0 in n + m real points. U5,4 = 0 U5,5 = 0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n: Interlacing

U5,0=0 U4,0=0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n: Interlacing

U5,5=0 U5,4=0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

U5,5=0 U4,5=0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Rodrigues formula

Zeros of Um,n: Density

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Koornwinder basis

Basis Pn,k

For 0 ≤ k ≤ n Pn,k(x, y) = P(k+µ+1,k+µ+1)

n−k

(x) (1 − x2)k/2 P(µ+1/2,µ+1/2)

k

• y

√ 1 − x2

Zeros of orthogonal polynomial on the disk Koornwinder basis

Basis Pn,k

For 0 ≤ k ≤ n Pn,k(x, y) = P(k+µ+1,k+µ+1)

n−k

(x) (1 − x2)k/2 P(µ+1/2,µ+1/2)

k

• y

√ 1 − x2

Zeros of orthogonal polynomial on the disk Koornwinder basis

Basis Pn,k

For 0 ≤ k ≤ n Pn,k(x, y) = P(k+µ+1,k+µ+1)

n−k

(x) (1 − x2)k/2 P(µ+1/2,µ+1/2)

k

• y

√ 1 − x2

Zeros of orthogonal polynomial on the disk Koornwinder basis

Basis Pn,k

For 0 ≤ k ≤ n Pn,k(x, y) = P(k+µ+1,k+µ+1)

n−k

(x) (1 − x2)k/2 P(µ+1/2,µ+1/2)

k

• y

√ 1 − x2

• deg(Pn,k) = n

Zeros of orthogonal polynomial on the disk Koornwinder basis

Basis Pn,k

For 0 ≤ k ≤ n Pn,k(x, y) = P(k+µ+1,k+µ+1)

n−k

(x) (1 − x2)k/2 P(µ+1/2,µ+1/2)

k

• y

√ 1 − x2

• deg(Pn,k) = n

{Pn,k(x, y) : 0 ≤ k ≤ n} basis mutually orthogonal of Vn.

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

For 0 ≤ k ≤ n Pn,k(x, y) = P(k+µ+1,k+µ+1)

n−k

(x) (1 − x2)k/2 P(µ+1/2,µ+1/2)

k

• y

√ 1 − x2

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

For 0 ≤ k ≤ n Pn,k(x, y) = P(k+µ+1,k+µ+1)

n−k

(x) (1 − x2)k/2 P(µ+1/2,µ+1/2)

k

• y

√ 1 − x2

• n − k lines x = a where a is a zero of P(k+µ+1,k+µ+1)

n−k

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

For 0 ≤ k ≤ n Pn,k(x, y) = P(k+µ+1,k+µ+1)

n−k

(x) (1 − x2)k/2 P(µ+1/2,µ+1/2)

k

• y

√ 1 − x2

• n − k lines x = a where a is a zero of P(k+µ+1,k+µ+1)

n−k

k ellipses (for k ≥ 2) a2x2 + y2 = a2 where a is a positive zero of P(µ+1/2,µ+1/2)

k

. For k = 1 we obtain y = 0.

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

For 0 ≤ k ≤ n Pn,k(x, y) = P(k+µ+1,k+µ+1)

n−k

(x) (1 − x2)k/2 P(µ+1/2,µ+1/2)

k

• y

√ 1 − x2

• n − k lines x = a where a is a zero of P(k+µ+1,k+µ+1)

n−k

k ellipses (for k ≥ 2) a2x2 + y2 = a2 where a is a positive zero of P(µ+1/2,µ+1/2)

k

. For k = 1 we obtain y = 0. The ellipses are centered at the origin, oriented on OX, with semi-major axis equal to 1 and semi-minor axis equal to a.

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

Degree 5



• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

1 ,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

2 ,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

3 ,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

4 ,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

5 

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

Degree 6



• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

1 ,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

2 ,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

3 ,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

4 ,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

5 ,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

6 

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

Pn,k = 0 is generally composed of multiple curves (ellipses and straight lines).

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

Pn,k = 0 is generally composed of multiple curves (ellipses and straight lines). The coordinate axes are going to be one of the branches of Pn,k = 0 when n − k or k are odd.

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

Pn,k = 0 is generally composed of multiple curves (ellipses and straight lines). The coordinate axes are going to be one of the branches of Pn,k = 0 when n − k or k are odd. The ellipses are inside the unit disk but not the lines.

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

Pn,k = 0 is generally composed of multiple curves (ellipses and straight lines). The coordinate axes are going to be one of the branches of Pn,k = 0 when n − k or k are odd. The ellipses are inside the unit disk but not the lines. The ellipses in Pn,k = 0 are symmetric with respect to the axes, centered on the origin and concentric.

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

Pn,k = 0 is generally composed of multiple curves (ellipses and straight lines). The coordinate axes are going to be one of the branches of Pn,k = 0 when n − k or k are odd. The ellipses are inside the unit disk but not the lines. The ellipses in Pn,k = 0 are symmetric with respect to the axes, centered on the origin and concentric. The straight lines in Pn,k = 0 are symmetric with respect to the axis OY and they are inside (−1, 1) × R.

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

Pn,k = 0 is generally composed of multiple curves (ellipses and straight lines). The coordinate axes are going to be one of the branches of Pn,k = 0 when n − k or k are odd. The ellipses are inside the unit disk but not the lines. The ellipses in Pn,k = 0 are symmetric with respect to the axes, centered on the origin and concentric. The straight lines in Pn,k = 0 are symmetric with respect to the axis OY and they are inside (−1, 1) × R. There are multiple points coming from the intersection of several branches.

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

Pn,0 = 0 is decomposed in n straight lines. P8,0 = 0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

Pn,n = 0 is decomposed in [ n

2] ellipses whose axis of symmetry

are the coordinates axes.

The semi-major axis is of size 1 and it is on the axis OX The semi-minor axis is on OY and its size is given by one of the roots of the Gegenbauer polynomial P(µ+1/2,µ+1/2)

n

.

P8,8 = 0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

Pn,k = 0 intersects a line parallel to OX in n − k points and a line parallel to OY in k points. P9,4 = 0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k

Every straight line passing through the origin intersects Pn,k = 0 in n real points. P9,4 = 0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k: Interlacing

P9,4 P9,5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k: Interlacing

P9,4 P8,4

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k: Density

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Koornwinder basis

Zeros of Pn,k: Density

For n = k

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Polar coordinates

Basis Pn

j,1, Pn j,2

Polar coordinates (x, y) = (r cos(θ), r sin(θ)), with 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π

Zeros of orthogonal polynomial on the disk Polar coordinates

Basis Pn

j,1, Pn j,2

Polar coordinates (x, y) = (r cos(θ), r sin(θ)), with 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π For 0 ≤ j ≤ n/2, Pn

j,1(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j cos((n − 2j)θ) Pn

j,2(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j sin((n − 2j)θ)

Zeros of orthogonal polynomial on the disk Polar coordinates

Basis Pn

j,1, Pn j,2

Polar coordinates (x, y) = (r cos(θ), r sin(θ)), with 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π For 0 ≤ j ≤ n/2, Pn

j,1(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j cos((n − 2j)θ) Pn

j,2(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j sin((n − 2j)θ) deg(Pn

j,i) = n for i = 1, 2

Zeros of orthogonal polynomial on the disk Polar coordinates

Basis Pn

j,1, Pn j,2

Polar coordinates (x, y) = (r cos(θ), r sin(θ)), with 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π For 0 ≤ j ≤ n/2, Pn

j,1(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j cos((n − 2j)θ) Pn

j,2(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j sin((n − 2j)θ) deg(Pn

j,i) = n for i = 1, 2

{Pn

j,i(x, y) : 0 ≤ j ≤ n/2, i = 1, 2} basis mutually orthogonal of Vn.

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

Pn

j,1(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j cos((n − 2j)θ) = 0

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

Pn

j,1(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j cos((n − 2j)θ) = 0

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

Pn

j,1(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j cos((n − 2j)θ) = 0 j circles centered on (0, 0) with radio r = a+1

2

< 1 where a is a zero of P(µ,n−2j)

j

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

Pn

j,1(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j cos((n − 2j)θ) = 0 j circles centered on (0, 0) with radio r = a+1

2

< 1 where a is a zero of P(µ,n−2j)

j

n − 2j lines passing through the origin with angle θ =

2k+1 2(n−2j)π,

k ∈ Z.

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

Pn

j,1(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j cos((n − 2j)θ) = 0 j circles centered on (0, 0) with radio r = a+1

2

< 1 n − 2j lines passing through the origin.

• 1.0
• 0.5

0.5 1.0

• 1.0
• 0.5

0.5 1.0

P2,1

6

 0

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

Pn

j,2(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j sin((n − 2j)θ) = 0

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

Pn

j,2(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j sin((n − 2j)θ) = 0 j circles centered on (0, 0) with radio r = a+1

2

< 1 where a is a zero of P(µ,n−2j)

j

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

Pn

j,2(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j sin((n − 2j)θ) = 0 j circles centered on (0, 0) with radio r = a+1

2

< 1 where a is a zero of P(µ,n−2j)

j

n − 2j lines passing through the origin with angle θ =

k n−2j π, k ∈ Z.

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

Pn

j,2(x, y) = P(µ,n−2j) j

(2r 2 − 1) r n−2j sin((n − 2j)θ) = 0 j circles centered on (0, 0) with radio r = a+1

2

< 1 n − 2j lines passing through the origin.

• 1.0
• 0.5

0.5 1.0

• 1.0
• 0.5

0.5 1.0

P2,2

6

 0

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

Pn

j,i = 0 is generally composed of multiple curves (circles and

straight lines passing through the origin).

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

Pn

j,i = 0 is generally composed of multiple curves (circles and

straight lines passing through the origin). The circles are inside the unit disk but not the lines.

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

Pn

j,i = 0 is generally composed of multiple curves (circles and

straight lines passing through the origin). The circles are inside the unit disk but not the lines. The circles in Pn

j,i = 0 are centered on the origin and concentric.

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

Pn

j,i = 0 is generally composed of multiple curves (circles and

straight lines passing through the origin). The circles are inside the unit disk but not the lines. The circles in Pn

j,i = 0 are centered on the origin and concentric.

There are multiple points coming from the intersection of several branches.

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2

The properties related with horizontal and vertical lines do not hold. Every straight line passing through the origin intersects Pn

j,i = 0 in

n real points. P11

4,1 = 0

• 1.0
• 0.5

0.5 1.0

• 1.0
• 0.5

0.5 1.0

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2: Interlacing property

Pn

j,i = 0 and Pn+1 j,i

= 0: the straight lines and the circles ”interlace”. Pn

j,i = 0 and Pn+1 j+1,i = 0: the straight lines and the circles ”interlace”.

P11

4,1 and P10 4,1

P11

4,1 and P10 3,1

• 1.0
• 0.5

0.5 1.0

• 1.0
• 0.5

0.5 1.0

• 1.0
• 0.5

0.5 1.0

• 1.0
• 0.5

0.5 1.0

Zeros of orthogonal polynomial on the disk Polar coordinates

Zeros of Pn

j,1, Pn j,2: Density

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Basis Vm,n

For m ≥ 0, n ≥ 0 Vm,n(x, y) =

[m/2]

• i=0

[n/2]

• j=0

(−1)k+j (−m)2i (−n)2j (µ + 1)m+n−i−j 22i+2j i! j! (µ + 1)m+n xm−2iyn−2j

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Basis Vm,n

For m ≥ 0, n ≥ 0 Vm,n(x, y) =

[m/2]

• i=0

[n/2]

• j=0

(−1)k+j (−m)2i (−n)2j (µ + 1)m+n−i−j 22i+2j i! j! (µ + 1)m+n xm−2iyn−2j deg(Vm,n) = m + n.

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Basis Vm,n

For m ≥ 0, n ≥ 0 Vm,n(x, y) =

[m/2]

• i=0

[n/2]

• j=0

(−1)k+j (−m)2i (−n)2j (µ + 1)m+n−i−j 22i+2j i! j! (µ + 1)m+n xm−2iyn−2j deg(Vm,n) = m + n. {Vn−k,k(x, y) : 0 ≤ k ≤ n} basis not mutually orthogonal of Vn.

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Basis Vm,n

For m ≥ 0, n ≥ 0 Vm,n(x, y) =

[m/2]

• i=0

[n/2]

• j=0

(−1)k+j (−m)2i (−n)2j (µ + 1)m+n−i−j 22i+2j i! j! (µ + 1)m+n xm−2iyn−2j deg(Vm,n) = m + n. {Vn−k,k(x, y) : 0 ≤ k ≤ n} basis not mutually orthogonal of Vn. Biorthogonal to the polynomials Um,n: Un−k,k, Vn−j,jµ = kn,j δj,k

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n

Degree 7



• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5



Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n

Degree 8



• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

,

• 1.5 -1.0 -0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5



Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n

Vm,n = 0 is generally composed of multiple curves but now they are not closed curves and straight lines.

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n

Vm,n = 0 is generally composed of multiple curves but now they are not closed curves and straight lines. The coordinate axes are going to be one of the branches of Vm,n = 0 when m or n are odd.

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n

Vm,n = 0 is generally composed of multiple curves but now they are not closed curves and straight lines. The coordinate axes are going to be one of the branches of Vm,n = 0 when m or n are odd. The curves are not inside the unit disk.

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n

Vm,n = 0 is generally composed of multiple curves but now they are not closed curves and straight lines. The coordinate axes are going to be one of the branches of Vm,n = 0 when m or n are odd. The curves are not inside the unit disk. The curves in Vm,n = 0 are inside (−1, 1) × R or R × (−1, 1).

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n

Vm,n = 0 is generally composed of multiple curves but now they are not closed curves and straight lines. The coordinate axes are going to be one of the branches of Vm,n = 0 when m or n are odd. The curves are not inside the unit disk. The curves in Vm,n = 0 are inside (−1, 1) × R or R × (−1, 1). There are not multiple points coming from the intersection of several branches (except for the axes).

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n

Vm,n = 0 is generally composed of multiple curves but now they are not closed curves and straight lines. The coordinate axes are going to be one of the branches of Vm,n = 0 when m or n are odd. The curves are not inside the unit disk. The curves in Vm,n = 0 are inside (−1, 1) × R or R × (−1, 1). There are not multiple points coming from the intersection of several branches (except for the axes). For m = 0 or n = 0 Vm,n = 0 is composed of parallel straight lines.

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n

Vm,n = 0 intersects a line parallel to OX in m points and a line parallel to OY in n points. V5,4 = 0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n

Not every straight line passing through the origin intersects Vm,n = 0 in m + n real points. V5,4 = 0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n: Interlacing

m=0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5 Degree 6 Degree 5

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n: Interlacing

n=0

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5 Degree 6 Degree 5

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n: Interlacing

m=3

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5 V3,6 V3,5

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n: Interlacing

m=3

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5 V6,3 V5,3

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Zeros of Vm,n: Density

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Open problems

Are there invariants?

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Open problems

Are there invariants? Relation between zeros and Jacobi matrices.

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Open problems

Are there invariants? Relation between zeros and Jacobi matrices. Electrostatic interpretation

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Open problems

Are there invariants? Relation between zeros and Jacobi matrices. Electrostatic interpretation ...

Zeros of orthogonal polynomial on the disk Biorthogonal basis

Thanks for your attention!