Orthogonal polynomials, zeros and electrostatics F. Marcell an - - PowerPoint PPT Presentation

Orthogonal polynomials, zeros and electrostatics

• F. Marcell´

an Universidad Carlos III de Madrid (UC3M) and Instituto de Ciencias Matem´ aticas (ICMAT)

OPCOP2017 - Universidad de Cantabria

April 19-22, 2017 - (CIEM) Castro Urdiales

(OPCOP2017) OP , zeros and electrostatics UC 2017 1 / 32

Outline

1

From classical to semiclassical orthogonal polynomials Electrostatic model for semiclassical OP

2

Electrostatic model for zeros of POPUC Orthogonal and Paraorthogonal polynomials on the unit circle Differential properties for POPUC Electrostatic interpretation Finding the electric field generators: an algorithmic approach Examples: Lebesgue and Chebyshev polynomials

3

Zeros of exceptional orthogonal polynomials Exceptional orthogonal polynomials Xm-Laguerre-(I) polynomials Xm-Laguerre-(II) polynomials

4

Zeros of Freud-Sobolev type orthogonal polynomials Freud-Sobolev type orthogonal polynomials Electrostatic models for the “even” Q2n and “odd” Q2n+1 subsequences

(OPCOP2017) OP , zeros and electrostatics UC 2017 2 / 32

From classical to semiclassical OP

Let {pn(x)}n≥0 be a sequence of polynomials orthonormal (OPRL) with respect to a weight function w(x) = exp(−v(x)) supported on an interval [c,d] ⊂ R, finite or infinite:

d

c pm(x)pn(x)w(x)dx = δm,n .

Then {pn}n≥0 satisfies a three-term recurrence relation: xpn(x) = an+1pn+1(x)+bnpn(x)+anpn−1(x), n ≥ 0, an > 0, p−1 = 0, p0 = 1. Under certain assumptions on w, the orthonormal polynomials pn also satisfy a difference-differential relation p′

n(x) = A(x;n)pn−1(x)−B(x;n)pn(x),

where A(x,n), B(x;n) are given in terms of w, an-s, and the values pn(c) and pn(d). A direct consequence of the above is that pn satisfies also the second order linear differential equation p′′

n(x)−2R(x;n)p′ n(x)+S(x;n)pn(x) = 0,

with R(x;n) = v′(x)

2

+ A′(x;n)

2A(x;n) ,

S(x;n) = B′(x;n)−B(x;n) A′(x;n)

A(x;n) −B(x;n)[v′(x)+B(x;n)]+ an an−1 A(x;n)A(x;n −1).

(OPCOP2017) OP , zeros and electrostatics UC 2017 3 / 32

From classical to semiclassical OP

M.E.H. Ismail, An electrostatic model for zeros of general orthogonal polynomials, Pacific J. Math. 193 (2000), 355–369. Ismail considers ϕ′(x) = R(x;n), where ϕ(x) is the external field ϕ(x) = v(x) 2 + ln(knA(x;n)) 2 = ϕlong(x)+ϕshort(x) ϕ(x) has two components: ϕlong(x) comes from the orthogonality weight w(x) = exp(−v(x)), and Ismail called it the long range potential. The other ϕshort(x) is called short range potential, allows to give a further generalization of the electrostatic interpretation. Ismail proves that, under certain assumptions, the total energy of the system has a unique point of global minimum, which is located at the vector constituted by the zeros of the orthogonal polynomial pn. A(x;n) is responsible of the creation of “ghost” movable charges, as it was shown first in: F.A. Gr¨ unbaum, Variations on a theme of Heine and Stieltjes: An electrostatic interpretation of the zeros of certain polynomials, J. Comput. Appl. Math. 99 (1998), 189–194.

(OPCOP2017) OP , zeros and electrostatics UC 2017 4 / 32

Semiclassical OPRL

Let us consider here a little bit more general situation. A generalized weight function (or linear functional) is semiclassical if it satisfies the Pearson equation D(φw) = ψw , where φ, ψ are polynomials, with degree of ψ ≥ 1, and D is the “derivative”

• perator (in the usual, but also possibly in a distributional sense).

It is well known that for such a weight the corresponding orthogonal polynomials (called also semiclassical) satisfy a differential equation of the type referred above, where the coefficients R(x;n) and S(x;n) are rational functions. The classical-type orthogonal polynomials considered by Gr¨ unbaum in the above work are an example of a semiclassical family, but there are many more.

• F. Marcell´

an, A. Mart´ ınez-Finkelshtein, P . Mart´ ınez-Gonz´ alez, Electrostatic models for zeros of polynomials: old, new, and some open problems, J. Comput. Appl.

• Math. 207 (2007), no. 2, 258–272.

(OPCOP2017) OP , zeros and electrostatics UC 2017 5 / 32

Outline

1

From classical to semiclassical orthogonal polynomials Electrostatic model for semiclassical OP

2

Electrostatic model for zeros of POPUC Orthogonal and Paraorthogonal polynomials on the unit circle Differential properties for POPUC Electrostatic interpretation Finding the electric field generators: an algorithmic approach Examples: Lebesgue and Chebyshev polynomials

3

Zeros of exceptional orthogonal polynomials Exceptional orthogonal polynomials Xm-Laguerre-(I) polynomials Xm-Laguerre-(II) polynomials

4

Zeros of Freud-Sobolev type orthogonal polynomials Freud-Sobolev type orthogonal polynomials Electrostatic models for the “even” Q2n and “odd” Q2n+1 subsequences

(OPCOP2017) OP , zeros and electrostatics UC 2017 6 / 32

Orthogonal and Paraorthogonal polynomials on the unit circle

Given an infinitely supported probability measure µ on the unit circle, one defines the OPUC {Φn(z;µ)}n≥0 as the monic sequence of monic polynomials satisfying the Szeg˝

• recursion:

Φn+1(z;µ) = zΦn(z;µ)− ¯ αnΦ∗

n(z;µ),

where αn ∈ D := {z : |z| < 1} and Φ∗

n(z) = znΦn(1/¯

z). To each µ on the unit circle, we can associate a sequence {αn}n≥0 of corresponding Verblunsky coefficients. For the above µ, and a complex number β of modulus 1, one can defines paraorthogonal polynomials (POPUC) {Φn(z;β;µ)}n≥0 as the sequence of polynomials given by Φn(z;β;µ) := zΦn−1(z;µ)− ¯ βΦ∗

n−1(z;µ).

All the zeros of Φn(z;β;µ) are simple and lie on the unit circle.

(OPCOP2017) OP , zeros and electrostatics UC 2017 7 / 32

Paraorthogonal polynomials on the unit circle

If τ = β are distinct complex numbers of modulus 1, then the zeros of Φn(z;β;µ) and Φn(z;τ;µ) strictly interlace on the unit circle, i.e.if x and y are two zeros of Φn(z;β;µ) and [x,y] is the arc of the unit circle that runs from x to y in the counter-clockwise direction, then [x,y]\{x,y} contains a zero of Φn(z;τ;µ). Paraorthogonal polynomials are not orthogonal polynomials, but they often serve as an appropriate analog of OPRL in settings where the real line is replaced by the unit circle. The basic reference in this section will be:

• B. Simanek, An electrostatic interpretation of the zeros of paraorthogonal

polynomials on the unit circle. SIAM J. Math. Anal. 48 (3) (2016), 2250–2268.

(OPCOP2017) OP , zeros and electrostatics UC 2017 8 / 32

Theorem 1 (B. Simanek, 2016)

Suppose dµ(θ) = w(θ) dθ

2π is a probability measure on the unit circle, where w is

continuous on [0,2π] (mod 2π) and differentiable on (0,2π) and let {αn}∞

n=0 be the

corresponding sequence of Verblunsky coefficients. If β ∈ C, then the POPUC polynomial y(z) = Φn(z;β) defined above solves the following differential equation on any domain including infinity or zero on which the coefficients are meromorphic: 0 = y′′(z)+ 1−n z − h′

n(z;β;β)

hn(z;β;β)

• y′(z)

+ W[hn(z;β;β),hn(z;−β;β)] 2¯ βzhn(z;β;β) − 1 z ((n +zGn(z))Gn(z)+Jn(z)(Dn(z)−nαn−1))

• y(z),

where Gn(z) := i

2π

|ϕ∗

n−1(eiθ )|2w′(θ)

(z−eiθ ) dθ 2π ,

Dn(z) := −iz

2π

ϕ∗

n−1(eiθ )2w′(θ)

(z−eiθ )einθ dθ 2π ,

Jn(z) := i

2π

ϕn−1(eiθ )2w′(θ) (z−eiθ )ei(n−2)θ dθ 2π ,

hn(z;x;y) := ¯ x(n(1− ¯ yαn−1)+zGn(z)+ ¯ yDn(z))−z(Jn(z)− ¯ yGn(z)), and W[f,g] denotes the Wronskian of f and g.

(OPCOP2017) OP , zeros and electrostatics UC 2017 9 / 32

Theorem 2 (B. Simanek, 2016)

Suppose µ is as in the above Theorem and τ = β are complex numbers. The polynomials u(z) := Φn(z;β,µ) and υ(z) := Φn(z;τ,µ) solve the following system of differential equations on any domain containing infinity or zero where the coefficients are meromorphic: u′(z) = υ(z) hn(z;β;β) z(¯ β − ¯ τ)

• −u(z)

hn(z;τ;β) z(¯ β − ¯ τ)

• υ′(z) = υ(z)

hn(z;β;τ) z(¯ β − ¯ τ)

• −u(z)

hn(z;τ;τ) z(¯ β − ¯ τ)

• .

More precisely, we need to know exactly the coefficient of n−d/2 to estimate the above expressions correctly. The main restriction in the applicability of the above two Theorems is the requirement that the measure is absolutely continuous and the weight is a continuous and differentiable function.

(OPCOP2017) OP , zeros and electrostatics UC 2017 10 / 32

Electrostatic interpretation

The main application of the above results is to prove that the zeros of certain families of POPUC are points satisfying an electrostatic equilibrium. Simanek considers the problem of creating an electric field that will keep identical charges at fixed points on the unit circle stationary. More precisely, assume a set {x1,...,xn} ⊆ ∂D. He proves a way to find a number m, a collection of points {ai}m

i=1 ⊆ C\{x1,...,xn}, and a set of real charges {qi}m i=1 so that if a particle of

charge +1 is placed at each xj (j = 1,...,n) and a particle of charge qi is placed at ai (i = 1,...,m), then the total force on the particle at each xj is zero (j = 1,...,n).

(OPCOP2017) OP , zeros and electrostatics UC 2017 11 / 32

Electrostatic interpretation

In other words, if we have a collection of m identically charged particles all lying on a concentric circle, it can be showed a way to construct an electric field that will keep these particles stationary. The points {ai}m

i=1 and charges {qi}m i=1 comprise the so called a set of electric

field generators. The charged particles at {xj}n

j=1 would be called mobile charges

and the charged particles at {ai}m

i=1 would be called impurity charges.

It is important to keep in mind that the charged particles at the points {xj}n

j=1

interact with each other as well as with the electric field generators. With our motivation now clearly stated, we provide the following definitions:

(OPCOP2017) OP , zeros and electrostatics UC 2017 12 / 32

Definition:

(i) Given a set of electric field generators {ai}m

i=1 and {qi}m i=1, a set of points {xj}n i=1

located on a smooth curve Γ is in Γ-normal electrostatic equilibrium if for each j = 1,2,...,n, the force at xj is normal to Γ at xj. (ii) Given a set of electric fields generators {ai}m

i=1 and {qi}m i=1, a set of points

{xj}n

i=1 ⊂ ∂D is in total electrostatic equilibrium if for each j = 1,2,...,n

∑

k=1,k=j

1 xj −xk +∑ qi xj −ai = 0. If Γ ⊂ ∂D, then the Γ-normal electrostatic equilibrium can be rewritten as Im

• xj
• ∑

k=1,k=j

1 xj −xk +∑ qi xj −ai

• = 0,

j = 1,2,...,n. Total electrostatic equilibrium = ⇒ ∂D-normal electrostatic equilibrium, but the converse is not true in general. Indeed, n particles of identical nonzero charge located at the nth roots of i¡unit and subject to no external force are in ∂D-normal electrostatic equilibrium but are not in total electrostatic equilibrium.

(OPCOP2017) OP , zeros and electrostatics UC 2017 13 / 32

Main result: Theorem 3 (Simanek, 2016)

Given any collection of n ≥ 2 distinct points {x1,...,xn} ⊆ ∂D, there exists a set of electric field generators so that the collection {x1,...,xn} is in total electrostatic equilibrium. In fact, given any n distinct points {x1,...,xn} ⊆ ∂D, we can deduce an explicit algorithm for finding the electric field generators. We proceed as follows: Step 1: Define the measure µn on ∂D by µn = 1 n

n

∑

j=1

δxj , and define β := (−1)n+1 ∏n

j=1 ¯

xj. Step 2: Consider the Gram-Schmidt orthogonalization process for the linearly independent set {1,z,...,zn−1} in L2(∂D,µn) to get the sequence of orthonormal polynomials {1,ϕ1(z;µn),...,ϕn−1(z;µn)}.

(OPCOP2017) OP , zeros and electrostatics UC 2017 14 / 32

Main result: Theorem 3 (Simanek, 2016)

Step 3: Define the probability measure dνn := 1 |ϕn−1(eiθ;µn)|2 dθ 2π . Step 4: Calculate the quantity hn(z;β;β) for the measure νn in the domain {z : |z| > 1}. Indeed, it will be a rational function S1(z)/S2(z) for some polynomials S1 and S2. Step 5: Place a particle of charge −1/2 at each zero of S1, a particle of charge +1/2 at each zero of S2, and a particle of charge 1

2(1−n) at zero.

Basic idea: To translate the equilibrium problem into a Lam´ e differential equation.

(OPCOP2017) OP , zeros and electrostatics UC 2017 15 / 32

Example: Lebesgue Polynomials

Let µ be Lebesgue measure on the circle. In this case αn−1 = 0 and w is constant so w′ = 0. Let us also assume β = 1, so that Φn(z;β) = zn −1 and hn(z;β;β) = n. With this choice, from Theorem 2 we see that particles of charge +1 located at the nth roots of unity are in total electrostatic equilibrium when the external field is generated by a charge of 1

2(1−n) located at the origin.

(OPCOP2017) OP , zeros and electrostatics UC 2017 16 / 32

Example: Chebyshev Polynomials

Consider the measure dµ(θ) = (1−cos(θ))dθ 2π = |1−eiθ|2 2 dθ 2π . For this measure, the Verblunsky coefficients satisfy αn = −(n +2)−1. The corresponding monic POPUC are Φn(z;−1) =(zn+1 −1)(z −1), with zeros located at the (n +1)st roots of unity, up to z = 1. hn(z;−1,−1) = P3,n(z)

P4,n(z),

where P3,n(z) = −n(nzn+2 −(n +2)zn+1 +z +1 = −n2(z −1)∏n+1

j=1 (z −pn,j)

P4,n(z) = (n +1)zn+1(z −1)

(OPCOP2017) OP , zeros and electrostatics UC 2017 17 / 32

Example: Chebyshev Polynomials

This means that particles of charge +1 located at the zeros of Φn(z;−1) are in ∂D-normal electrostatic equilibrium when the external field is generated by a single particle of charge +1 located at 1. On the other hand, the particles of charge +1 located at the zeros of Φn(z;−1) are in total electrostatics equilibrium when the external field is generated by particles of charge −1/2 at each pn,j and a particle of charge +1 at the origin.

(OPCOP2017) OP , zeros and electrostatics UC 2017 18 / 32

Outline

1

From classical to semiclassical orthogonal polynomials Electrostatic model for semiclassical OP

2

Electrostatic model for zeros of POPUC Orthogonal and Paraorthogonal polynomials on the unit circle Differential properties for POPUC Electrostatic interpretation Finding the electric field generators: an algorithmic approach Examples: Lebesgue and Chebyshev polynomials

3

Zeros of exceptional orthogonal polynomials Exceptional orthogonal polynomials Xm-Laguerre-(I) polynomials Xm-Laguerre-(II) polynomials

4

Zeros of Freud-Sobolev type orthogonal polynomials Freud-Sobolev type orthogonal polynomials Electrostatic models for the “even” Q2n and “odd” Q2n+1 subsequences

(OPCOP2017) OP , zeros and electrostatics UC 2017 19 / 32

Exceptional orthogonal polynomials

Exceptional orthogonal polynomials were introduced in the seminal work

• D. G´
• mez-Ullate, N. Kamran , R. Milson, An extended class of orthogonal

polynomials defined by a Sturm-Liouville problem, J. Math Anal. Appl. 359 (1) (2009) 352–367. Let {yn}n∈σ, σ = N−{i1,...,im} be a sequence of monic polynomials with degyn = n, orthogonal with respect to a weight and, in addition, they are also eigenfunctions of a second order differential operator, i.e. p(x)y′′

n +q(x)y′ n +r(x)yn = λnyn ,

where p, q, r are rational functions. σ is a numerable set of N called the degree sequence and m is a number of missing integers in the degree sequence, known as the codimension. Concerning the location of zeros of exceptional orthogonal polynomials, the seminal reference is

• D. G´
• mez-Ullate, F. Marcell´

an, R. Milson. Asymptotic and interlacing properties of zeros of exceptional Jacobi and Laguerre polynomials. J. Math. Anal. Appl. 399 (2013) 480–495.

(OPCOP2017) OP , zeros and electrostatics UC 2017 20 / 32

Definition: weighted Fej´ er constants

Next we analyze the electrostatic properties of zeros of Exceptional orthogonal

• polynomials. The basic reference in this section will be:

´ A.P . Horv´ ath, The electrostatic properties of zeros of exceptional Laguerre and Jacobi polynomials and stable interpolation, J. Approx. Theory 194 (2015), 87–107. Let Un := {u1,...,un} be any system of nodes on an interval I and 0 < w ∈ C2(I) be a weight function on I. Let ωUn(x) := ∏n

k=1(x −uk)

The weighted Fej´ er constants on I with respect to Un and w are: Ck := Ck,Un ,w = ω′′

Un

ω′

Un

(uk)+ w′ w (uk). We will investigate the local extrema of the energy function Tw(u1,...,un) =

n

∏

j=1

w(uj)

n

∏

1≤i<j≤n

(ui −uj)2. Basic idea: The differential equation of orthogonal polynomials, and its transformed version to a Schr¨

• dinger equation.

(OPCOP2017) OP , zeros and electrostatics UC 2017 21 / 32

Lemma (Horv´ ath, 2015)

Let pn(x) = ∏n

i=1(x −ζi), be a polynomial of degree n such that

p′′

n(x)+M(x;n)p′ n(x)+N(x;n)pn(x) = 0.

Let us assume that M(x;n) is the logarithmic derivative of a function wn(x) which is smooth enough, that is (logwn(x))′ = M(x;n). Let Twn(u1, ..., un) be the energy function with respect to wn. Then for i, j = 1,2,...

∂ logTwn (u1,...,un) ∂ui

(ζ1,...,ζn) = Ci,wn,Zn = 0,

∂ 2 logTwn (u1,...,un) ∂ui∂uj

(ζ1,...,ζn) =

2 (ζi−ζj)2 , ∂ 2 logTwn (u1,...,un) ∂u2

i

(ζ1,...,ζn) = − 2

3Φ(ζi),

where Φ(x) := Φwn(x) is the coefficient of the transformed differential equation: z′′

n(x)+Φ(x)zn(x) = 0,

which is satisfied by zn(x) = pn(x)

• wn(x).

(OPCOP2017) OP , zeros and electrostatics UC 2017 22 / 32

Xm-Laguerre-(I) polynomials

Let L(α−1)

k

(x) be the k-th degree classical Laguerre polynomial of parameter α −1. Let {LI,(α)

m,m+n}∞ n=0 denotes the sequence of exceptional Laguerre polynomials of

the first kind with codimension m ≥ 1. They are the orthogonal polynomials on (0,∞) with respect to the weight ˆ w(α)

m

:= xαe−x S2(x) , S(x) := s(α−1)

m

(x) := L(α−1)

m

(−x) LI,(α)

m,m+n satisfies the following differential equation on R{0,−y1,...,−ym}

y′′(x)+ α +1−x x − 2S′(x) S(x)

• y′(x)+

m +n x − α x 2S′(x) S(x)

• y(x) = 0.

(OPCOP2017) OP , zeros and electrostatics UC 2017 23 / 32

Xm-Laguerre-(I) polynomials

Lemma (G´

• mez Ullate et al, 2013): LI,(α)

m,m+n has n +m simple zeros: n regular

zeros x(α)

m,n,1,...,x(α) m,n,n ∈ (0, ∞) and m exceptional zeros

z(α)

m,n,1,...,z(α) m,n,m ∈ (−∞,0). Furthermore

lim

n→∞nx(α) m,n,i =

• j(α)

i

2 4 and as n → ∞ the exceptional zeros of LI,(α)

m,m+n converge to the zeros of

S(x) = L(α−1)

m

(−x). Here {j(α)

i

}i≥1 is the increasing sequence of the positive zeros of the Bessel function of the first kind. Theorem (Horv´ ath, 2015): Let α ≥ 1. Let us denote by Xm,n = {xm,n,1,...,xm,n,n} the positive zeros of LI,(α)

m,m+n. Xm,n is the Fekete set that is the unique set where

the logarithmic energy function (−logTυ) takes its infimum under the external field represented by

• υ(α+1)

m,n

• 1

2(n−1) ,

where υ(α+1)

m,n

= xα+1e−xP2(x) S2(x) , P(x) :=

m

∏

i=1

(x −zm,n,i).

(OPCOP2017) OP , zeros and electrostatics UC 2017 24 / 32

Xm-Laguerre-(II) polynomials

The type II exceptional Laguerre polynomials of codimension m ≥ 1 are denoted by {LII,(α)

m,m+n}∞ n=0.

Let us denote S(x) := L(−α−1)

m

(x) and let us assume that α > m −1. It is known that S has no zeros in [0, ∞) and LII,(α)

m,m+n-s are orthogonal on (0,∞)

with respect to the weight ˆ w(α) := xαe−x S2(x) LII,(α)

m,m+n satisfies

y′′(x)+ α +1−x x − 2S′(x) S(x)

• y′(x)+

n −m x − α x 2S′(x) S(x)

• y(x) = 0.

Lemma (G´

• mez Ullate et al, 2013): LII,(α)

m,m+n has n +m simple zeros, n regular

zeros x(α)

m,n,1,...,x(α) m,n,n ∈ (0, ∞) and 0 or 1 negative zero according to whether m is

even or odd. Furthermore, as n → ∞ the exceptional zeros of LII,(α)

m,m+n converge to

the zeros of S(x) = L(−α−1)

m

(x) and limn→∞ nx(α)

m,n,i =

• j(α)

i

2 4

.

(OPCOP2017) OP , zeros and electrostatics UC 2017 25 / 32

Xm-Laguerre-(II) polynomials

We can write LII,(α)

m,m+n = P(x)q(x), where P(x) is a polynomial of degree m with 0

• r 1 real zero.

We can examine the properties of the regular zeros of LII,(α)

m,m+n, i.e. the zeros of q.

One can write q′′(x)+

• M(x;n)+2P′

P (x)

• q′(x)+
• N(x;n)+ P′′

P (x)+M(x;n)P′ P (x)

• q(x) = 0,

where M(x;n) and N(x;n) are the coefficients in the former differential equation for LII,(α)

m,m+n

M(x;n) = α+1−x

x

− 2S′(x)

S(x) ,

N(x;n) = n−m

x

− α

x 2S′(x) S(x) .

Theorem (Horv´ ath, 2015): Let υ(x) = υ(α+1)

m,n

(x) = xαe−xP2(x)

S2(x)

. If α > m −1, and if n is large enough, then the set of regular zeros of LII,(α)

m,m+n is the unique set on

which the logarithmic energy function takes its minimum under the external field

• υ(α+1)

m,n

• 1

2(n−1) . (OPCOP2017) OP , zeros and electrostatics UC 2017 26 / 32

Outline

1

From classical to semiclassical orthogonal polynomials Electrostatic model for semiclassical OP

2

Electrostatic model for zeros of POPUC Orthogonal and Paraorthogonal polynomials on the unit circle Differential properties for POPUC Electrostatic interpretation Finding the electric field generators: an algorithmic approach Examples: Lebesgue and Chebyshev polynomials

3

Zeros of exceptional orthogonal polynomials Exceptional orthogonal polynomials Xm-Laguerre-(I) polynomials Xm-Laguerre-(II) polynomials

4

Zeros of Freud-Sobolev type orthogonal polynomials Freud-Sobolev type orthogonal polynomials Electrostatic models for the “even” Q2n and “odd” Q2n+1 subsequences

(OPCOP2017) OP , zeros and electrostatics UC 2017 27 / 32

Freud and Freud-Sobolev type orthogonal polynomials

Let {Fn(x)}n0 be Freud polynomials, associated to p,q =

• R p(x)q(x)e−x4dx.

TTRR xFn(x) = Fn+1(x)+a2

nFn−1(x),

n ≥ 1, where the recurrence coefficients an satisfy the so called string equation 4a2

n

• a2

n+1 +a2 n +a2 n−1

• = n.

Next, we consider the electrostatic behavior of zeros of the SMOP {Qn(x)}n0 corresponding to p,q1 = p,q+M0p(0)q(0)+M1p′(0)q′(0). They are known in the literature as Freud-Sobolev type orthogonal polynomials

(OPCOP2017) OP , zeros and electrostatics UC 2017 28 / 32

Electrostatic model for zeros of Q2n+1 (odd case)

Here M0 and M1 affects independently to the subsequences {Q2n(x)}n0 and {Q2n+1(x)}n0 respectively. M1 changes exclusively the equilibrium position of the zeros of {Q2n+1(x)}n0 under the action of the following external potential (Garza, Huertas,Marcell´ an,(2017)) Vext(x,2n +1) = 1 2ln u(x,2n +1)− 1 2ln x2e−x4 The first term above represents a short range potential corresponding to four unit charges located at the zeros of the quartic polynomial u(x,n). The coefficients of u(x,n) depend on n, an, M0, M1 and the confluent Kernels Kn(0,0) and K (1,1)

n

(0,0). When u(x,n) → u(x,2n +1) the dependence with M0 and Kn(0,0) vanishes The second term represents a long range potential associated with the weight function e−x4

(OPCOP2017) OP , zeros and electrostatics UC 2017 29 / 32

Electrostatic model for zeros of Q2n+1 (odd case)

This picture illustrates the variation of the zeros of the odd degree Freud-Sobolev type polynomials Q7(x) when M1 varies. Here, the value of M0 is not relevant.

• 1.4
• 1.2
• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2 1.4

• 0.4
• 0.2

0.2 0.4

(OPCOP2017) OP , zeros and electrostatics UC 2017 30 / 32

Electrostatic model for zeros of Q2n (even case)

The even case was analyzed in (Arceo, Huertas, Marcell´ an (2016)). In a similar way, M0 changes exclusively the equilibrium position of the zeros of the “even” subsequence {Q2n(x)}n0 under the action of the external potential Vext(x,2n) = 1 2ln u(x,2n)− 1 2ln x2e−x4.

1.5 1 0.5 0.5 1 1.5 0.5 0.5 1.0

(OPCOP2017) OP , zeros and electrostatics UC 2017 31 / 32

Thank you!

(OPCOP2017) OP , zeros and electrostatics UC 2017 32 / 32