An Electrostatic Interpretation of the Zeros of Paraorthogonal Polynomials on the Unit Circle

4 MB·English

by  Brian Simanek

AN ELECTROSTATIC INTERPRETATION OF THE ZEROS OF
PARAORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE
BRIAN SIMANEK
Abstract. We show that if µ is a probability measure with infinite support on the unit
6
circle having no singular component and a differentiable weight, then the corresponding
1
0 paraorthogonal polynomial Φn(z;β) solves an explicit second order linear differential equa-
2 tion. We also show that if τ (cid:54)= β, then the pair (Φ (z;β),Φ (z;τ)) solves an explicit first
n n

r order linear system of differential equations. One can use these differential equations to
p
deduce that the zeros of every paraorthogonal polynomial mark the locations of a set of
A
particles that are in electrostatic equilibrium with respect to a particular external field.

8
2
Keywords: Paraorthogonal polynomials, Generalized Lam´e differential equation, Electrostatic equilibrium

]
A Mathematics Subject Classification: Primary 42C05; Secondary 34M05, 78A30
C
.
h
t 1. Introduction
a
m
Given a positive probability measure µ with compact and infinite support in the com-
[
plex plane C, one can constuct the corresponding sequence of orthonormal polynomials

2
{ϕ (z;µ)} where ϕ (z;µ) is a polynomial of degree exactly n having positive leading co-
v n n≥0 n
2 efficient, which we denote by κ (µ). If we divide ϕ (z;µ) by κ (µ), then we obtain a monic
7 n n n
6 degree n polynomial, which we denote by Φn(z;µ). We will often suppress the µ dependence
5
inournotationifthereisnopossibilityforconfusion. Ourstudyoftheseobjectsismotivated
0
. by the fact that orthogonal polynomials have proven to be valuable tools in the study of
1
0 physical models.
5
The most well-studied collections of orthogonal polynomials come from measures of or-
1
: thogonality supported on the real line and are known as orthogonal polynomials on the real
v
Xi line (OPRL). Such polynomials have a variety of applications in spectral theory (see [31]),
r potential theory (see [24]), and the theory of special functions (see [15]). One of the key
a
features of OPRL is the three-term recurrence formula satisfied by the orthonormal polyno-
mials. Many well-known families of OPRL (including the Hermite polynomials, the Laguerre
polynomials, and the Jacobi polynomials) are polynomial solutions to particular families of
linear second order differential equations with polynomial coefficients. This property makes
these particular families of polynomials especially applicable as we will discuss later in more
detail.
A second collection of well-studied orthogonal polynomials comes from measures of or-
thogonality supported on the unit circle and are known as orthogonal polynomials on the
unit circle (OPUC). Just as OPRL has a close connection with self-adjoint operators, OPUC
1
2
has a close connection with unitary operators (see [31]). In analogy with the three-term
recurrence satisfied by the OPRL, the monic OPUC satisfy the Szego˝ recursion:
Φ (z;µ) = zΦ (z;µ)−α¯ Φ∗(z;µ),
n+1 n n n
where α ∈ D := {z : |z| < 1} and Φ∗(z) = znΦ (1/z¯). Therefore, to each infinitely sup-
n n n
ported probability measure µ on the unit circle, we can associate the sequence {α } of
n n≥0
Verblunsky coefficients. A theorem of Verblunsky states that such a sequence also deter-
mines an infinitely supported probability measure (see [28, Chapter 1]). The utility and
applicability of the Hermite, Laguerre, and Jacobi polynomials inspired interest in families
of OPUC that are solutions to linear second order differential equations. To this end, Ismail
and Witte proved the following result in [16]:
Theorem 1.1 (Ismail & Witte, 2001). Let w(z) = e−v(z) be differentiable in a neighborhood
of the unit circle, have moments of all integral orders, and assume that the integrals
(cid:90) v(cid:48)(z)−v(cid:48)(ζ) dζ
ζnw(ζ)
z −ζ iζ
|ζ|=1
exist for all integers n. Then the corresponding orthonormal polynomials satisfy the differ-
ential relation
κ (cid:90) v(cid:48)(z)−v(cid:48)(ζ)
ϕ(cid:48) (z) = n n−1ϕ (z)−iϕ∗(z) ϕ (ζ)ϕ∗(ζ)w(ζ)dζ
n κ n−1 n z −ζ n n
n |ζ|=1
(cid:90) v(cid:48)(z)−v(cid:48)(ζ)
+iϕ (z) ϕ (ζ)ϕ (ζ)w(ζ)dζ.
n n n
z −ζ
|ζ|=1
From this result, one can derive a second order differential equation to which ϕ (z) is a
n
solution; we refer the reader to [16] for details. The main purpose of our investigation is to
deriveacomparableresultforarelatedclassofpolynomialscalledparaorthogonal polynomials
on the unit circle (POPUC).
Given an infinitely supported probability measure µ on the unit circle and a complex
number β of modulus 1, we define the paraorthogonal polynomial Φ (z;β;µ) as the monic
n
degree n polynomial given by
Φ (z;β;µ) := zΦ (z;µ)−β¯Φ∗ (z;µ). (1)
n n−1 n−1
Paraorthogonal polynomials and their zeros have received considerable recent attention from
the research community (see for example [2, 8, 17, 18, 19, 27, 29, 35, 37]). It is well-
known and easy to show that all of the zeros of Φ (z;β;µ) are distinct and lie on the unit
n
circle. Furthermore, if τ (cid:54)= β are distinct complex numbers of modulus 1, then the zeros of
Φ (z;β;µ) and Φ (z;τ;µ) strictly interlace on the unit circle in that if x and y are two zeros
n n
of Φ (z;β;µ) and [x,y] is the arc of the unit circle that runs from x to y in the counter-
n
clockwisedirection,then[x,y]\{x,y}containsazeroofΦ (z;τ;µ). Althoughparaorthogonal
n
polynomials are not orthogonal polynomials, they often serve as an appropriate analog of
OPRL in settings where the real line is replaced by the unit circle (see for example [18, 35]).
One of our main results is the next theorem, which can be thought of as an analog of the
3
Ismail and Witte result that applies to paraorthogonal polynomials. In fact, the theorem
appliestoanydegreenmonicpolynomialsatisfyingtherelation(1)forsomecomplexnumber
β. As in [28], we write a measure on the unit circle as a measure in the variable θ ∈
[0,2π). For brevity and because there is no possibility for confusion, we will suppress the
µ-dependence of various quantities in our notation.
Theorem 1.2. Suppose dµ(θ) = w(θ)dθ is a probability measure on the unit circle, where
2π
w is continuous on [0,2π] (mod 2π) and differentiable on (0,2π) and let {α }∞ be the
n n=0
corresponding sequence of Verblunsky coefficients. If β ∈ C, then the polynomial y(z) =
Φ (z;β) defined by (1) solves the following differential equation on any domain including
n
infinity or zero on which the coefficients are meromorphic:
(cid:20)1−n h(cid:48) (z;β;β)(cid:21)
0 = y(cid:48)(cid:48)(z)+ − n y(cid:48)(z) (2)
z h (z;β;β)
n
(cid:20) (cid:21)
W[h (z;β;β),h (z;−β;β)] 1
n n
+ − ((n+zG (z))G (z)+J (z)(D (z)−nα )) y(z),
2β¯zh (z;β;β) z n n n n n−1
n
where
(cid:90) 2π |ϕ∗ (eiθ)|2w(cid:48)(θ)dθ
G (z) : = i n−1
n (z −eiθ) 2π
0
(cid:90) 2π ϕ∗ (eiθ)2w(cid:48)(θ)dθ
D (z) : = −iz n−1
n (z −eiθ)einθ 2π
0
(cid:90) 2π ϕ (eiθ)2w(cid:48)(θ) dθ
n−1
J (z) : = i
n (z −eiθ)ei(n−2)θ 2π
0
h (z;x;y) : = x¯(n(1−y¯α )+zG (z)+y¯D (z))−z(J (z)−y¯G (z)),
n n−1 n n n n
and W[f,g] denotes the Wronskian of f and g (that is, W[f,g] = fg(cid:48) −gf(cid:48)).
By allowing the parameter β to be arbitrarily chosen in C, we can apply our results to
a wide variety of polynomials including paraorthogonal polynomials and perturbations of
orthogonal polynomials without a precise understanding of the induced perturbation to the
measure of orthogonality.
InourapplicationsofTheorem1.2, wewillfocusonthecase|β| = 1sothatthepolynomial
Φ (z;β;µ) is a paraorthogonal polynomial. Notice that if we set β = α , then Φ (z;β;µ)
n n−1 n
is just the monic orthogonal polynomial Φ (z;µ). In this case, Theorem 1.2 yields a second
n
order differential equation solved by Φ (z;µ). If we set β = 0, then Theorem 1.2 yields
n
a second order differential equation solved by zΦ (z;µ), which can be rewritten as a
n−1
differential equation solved by Φ (z;µ). There is no reason to think that the second order
n−1
differential equations solved by Φ (z;µ) derived by these two methods will be the same as
n
each other or the same as the one produced by the results in [16] if those results apply. In
fact, we will see by example in Section 4.4 that these methods may yield different differential
equations. It is easy to see that the polynomial Φ (z;µ) satisfies many differential equations
n
4
if one allows the complexity of the coefficients to grow with n. Indeed, if
Φ(cid:48)(cid:48)(z;µ)+P(z)Φ(cid:48) (z;µ)+Q(z)Φ (z;µ) = 0,
n n n
and R(z) is any entire meromorphic function, then
Φ(cid:48)(cid:48)(z;µ)+Φ(cid:48) (z;µ)(P(z)+R(z)Φ (z;µ))+Φ (z;µ)(Q(z)−R(z)Φ(cid:48) (z;µ)) = 0.
n n n n n
In the next section we will discuss applications of Theorem 1.2 to electrostatics. In that
context, the coefficients in the second order differential equation determine the location of
chargesthatgenerateanelectricfieldandthezerosofΦ (z;β;µ)markthelocationofcharges
n
in equilibrium with respect to that field. It is not surprising that a particular configuration
of charges can be in equilibrium with respect to different electric fields, so from a physical
perspective, the lack of uniqueness of the differential equation is also expected.
An intermediate step in the proof of Theorem 1.2 is of interest in its own right and can
be stated as follows:
Theorem 1.3. Suppose µ is as in Theorem 1.2 and τ,β are distinct complex numbers.
The polynomials u(z) := Φ (z;β,µ) and v(z) := Φ (z;τ,µ) solve the following system of
n n
differential equations on any domain containing infinity or zero on which the coefficients are
meromorphic:
(cid:18) (cid:19) (cid:18) (cid:19)
h (z;β;β) h (z;τ;β)
u(cid:48)(z) = v(z) n −u(z) n (3)
¯ ¯
z(β −τ¯) z(β −τ¯)
(cid:18) (cid:19) (cid:18) (cid:19)
h (z;β;τ) h (z;τ;τ)
v(cid:48)(z) = v(z) n −u(z) n . (4)
¯ ¯
z(β −τ¯) z(β −τ¯)
The main restriction in the applicability of Theorems 1.2 and 1.3 is the requirement that
the measure be given by a continuous and differentiable weight function. However, for any
fixed n ∈ N and β ∈ ∂D, the map from measures to degree n paraorthogonal polynomials
¯
that map 0 to β is far from injective. In fact, in the pre-image of any such paraorthogonal
polynomial is a measure that satisfies the hypotheses of the above theorems. Indeed, we
have the following result, which is a consequence of the Bernstein-Szego˝ Theorem (see [28,
Theorem 1.7.8])
Theorem 1.4. Let µ be a probability measure with infinite support on the unit circle and
suppose β ∈ ∂D. Then
(cid:18) ei(n−1)θ dθ(cid:19)
Φ (z;β;µ) = Φ z;β;
n n ϕ (eiθ)ϕ∗ (eiθ) 2π
n−1 n−1
where ϕ (z) = ϕ (z;µ).
n−1 n−1
The following result will be relevant for our applications and is especially useful when
combined with Theorem 1.4.
Proposition 1.5. Suppose w is as in Theorem 1.2.
5
i) If w(cid:48)(θ) = f(eiθ) where f is analytic in a neighborhood of the unit circle, then the
functions G , D , and J (defined in Theorem 1.2) can be analytically continued from
n n n
{z : |z| > 1} to {z : |z| > r} for some r < 1.
ii) If w(cid:48)(θ) = f(eiθ) where f is a rational function without poles on the unit circle,
then the functions G , D , and J (defined in Theorem 1.2) can be meromorphically
n n n
continued from {z : |z| > 1} to all of C as rational functions with no poles except
possibly at zero and the poles of f inside D.
Proof. i) We present the proof for G ; the proof in the other two cases is even easier. We
n
will make use of the fact that
|ϕ∗(eiθ)|2 = ϕ∗(eiθ)ϕ (eiθ)e−inθ.
n n n
Therefore, we can write
(cid:90) 2π ϕ∗ (eiθ)ϕ (eiθ)f(eiθ)dθ (cid:90) ϕ∗ (ζ)ϕ (ζ)f(ζ)dζ
G (z) = i n−1 n−1 = n−1 n−1 . (5)
n (z −eiθ)einθ 2π (z −ζ)ζn+1 2π
0 |ζ|=1
If |z| > 1, then our hypotheses allow us to move the contour of integration to the circle
{ζ : |ζ| = r} for some r < 1 and hence we obtain an analytic continuation of G (z) to the
n
exterior of this circle.
ii) Given the form of G , D , and J , it suffices to show that if R(t) is a rational function
n n n
of t without poles on the unit circle, then when |z| > 1
(cid:90) 2π R(eiθ) dθ
z −eiθ 2π
0
is a rational function of z with no poles except possibly at 0 and the poles of R inside D.
By using the partial fraction decomposition of R, this follows from the fact that if P is a
polynomial and m is a non-negative integer, then
(cid:90) 2π P(eiθ) dθ P(0)
= , |z| > 1,
z −eiθ 2π z
0
(cid:40)
(cid:90) 2π 1 dθ 1 , |z| > 1, |x| > 1,
= z(−x)m
(z −eiθ)(eiθ −x)m 2π 1 , |z| > 1, |x| < 1.
0
z(z−x)m
(cid:3)
By combining the previous two results, we conclude that every paraorthogonal polynomial
is a solution on all of C to a differential equation of the form (2) that has rational coefficients.
This will be especially relevant for our applications.
In the next section, we will discuss an application of polynomial solutions to linear second
order differential equations. In Section 3 we will prove Theorems 1.2 and 1.3. The key idea
will be to use the Szego˝ recursion in several places to simplify various formulas. Finally, in
Section 4 we present some detailed examples that highlight the applications discussed in the
next section.
6
Acknowledgments. We would like to thank Mihai Stoiciu for much useful conversation
and also Andrei Martinez-Finkelshtein for useful feedback on this work and for bringing the
paper [1] to our attention. We would also like to thank the anonymous referee for directing
us to the observation that Theorem 1.2 can be applied even if |β| =(cid:54) 1.
2. Applications
In this section, we will highlight some important consequences and applications of the
main results of the previous section.
2.1. Electrostatics. Orthogonal polynomials have proven to be useful tools when studying
the equilibrium positions of electrons that repel each other via the two-dimensional Coulomb
interaction. ItwasStieltjesworkthatleadtothediscoverythatifn ≥ 3particlesareconfined
to the interval [−1,1] interacting by means of a logarithmic (i.e. Coulomb) potential, then
the unique energy minimizing configuration consists of one particle located at 1, one located
at −1, and the other n − 2 particles located at the (distinct) zeros of a particular Jacobi
polynomial (see [32, 33, 34] and see [36] for a proof). The key step in the proof of this
fact is rewriting the equilibrium condition on a collection of points as a second order linear
differential equation that is solved by the polynomial with zeros at precisely those points
and then recognizing that a Jacobi polynomial with the appropriate choice of parameters
is a solution to this differential equation (see [36] for details). Further developments in the
electrostatic interpretation of zeros of orthogonal polynomials on the real line can be found
in [3, 4, 12, 13, 14, 20].
Our main application of the results in Section 1 will be to demonstrate that the zeros of
certain families of POPUC are the locations of points that are in electrostatic equilibrium.
We will use the convention (as in [11]) that if a particle of charge q is located at a point
a ∈ C and a particle of charge p is located at a point b ∈ C, then the force on the particle
¯
at b due to the particle at a is 2pq/(b−a¯).
We will consider the problem of creating an electric field that will keep identical charges at
fixed points on the unit circle stationary. More precisely, suppose we are given a collection
{x ,...,x } ⊆ ∂D. We will demonstrate a way to find a number m, a collection of points
1 n
{a }m ⊆ C \ {x ,...,x }, and a set of real charges {q }m so that if a particle of charge
i i=1 1 n i i=1
+1 is placed placed at each x (j = 1,...,n) and a particle of charge q is placed at a
j i i
(i = 1,...,m), then the total force on the particle at each x is zero (j = 1,...,n). In
j
other words, if we have a collection of identically charged particles all lying on a concentric
circle, we will demonstrate a way to construct an electric field that will keep these particles
stationary. The points {a }m and charges {q }m comprise what we call a set of electric
i i=1 i i=1
field generators (in the language of [7], the charged particles at {x }n would be called
j j=1
mobile charges and the charged particles at {a }m would be called impurity charges). It
i i=1
is important to keep in mind that the charged particles at the points {x }n interact with
j j=1
each other as well as with the electric field generators.
With our motivation now clearly stated, we provide the following definitions:
7
Definition. i) Given a set of electric field generators {a }m and {q }m , we will say that
i i=1 i i=1
a collection of points {x ,...,x } located on a smooth curve Γ is in Γ-normal electrostatic
1 n
equilibrium if for each j = 1,...,n, the force at x is normal to Γ at x .
j j
ii) Given a set of electric fields generators {a }m and {q }m , we will say that a collection
i i=1 i i=1
of points {x ,...,x } ⊆ ∂D is in total electrostatic equilibrium if for each j = 1,...,n,
1 n
n m
(cid:88) 1 (cid:88) q
i
+ = 0. (6)
x −x x −a
j k j i
k=1 i=1
k(cid:54)=j
Notice that if Γ is the unit circle, then the Γ-normal electrostatic equilibrium condition
can be rewritten as
  
n m
(cid:88) 1 (cid:88) q
Imx  + i  = 0, j = 1,2,...,n, (7)
 j x −x x −a 
j k j i
k=1 i=1
k(cid:54)=j
so it is clear that a collection of points on the unit circle that is in total electrostatic equi-
librium is also in ∂D-normal electrostatic equilibrium, but the converse is false. Indeed, it is
well-known and easy to show that n particles of identical non-zero charge placed at the nth
roots of unity and subject to no external force are in ∂D-normal electrostatic equilibrium,
but are not in total electrostatic equilibrium.
The formula (7) is a restatement of the fact that if we place particles of charge +1 at each
x (j = 1,...,n) and a particle of charge q at each a (i = 1,...,m), then the condition (7)
j i i
is satisfied if and only if the force exerted on the particle at x is normal to the unit circle
j
at x for each j = 1,...,n (see [11, 22]). Similarly, the condition (6) is satisfied if and only
j
if the force exerted on the particle at x is equal to zero for each j = 1,...,n (see [23]).
j
As in the case of the interval, orthogonal polynomials are a useful tool when studying
electrostatic equilibria on the circle. In [6], Forrester and Rogers use Jacobi polynomials to
find a collection of 2n points on the unit circle that is in ∂D-normal electrostatic equilibrium
when the electric field generators consist of a particle with charge p at 1 and a particle with
charge q at −1. Their result assumes the added symmetry condition that each arc of the
unit circle connecting 1 to −1 contains an equal number of points.
Our main application is the following result:
Theorem 2.1. Given any collection of n ≥ 2 distinct points {x ,...,x } ⊆ ∂D, there exists
1 n
a set of electric field generators so that the collection {x ,...,x } is in total electrostatic
1 n
equilibrium.
In fact, given any n distinct points {x ,...,x } ⊆ ∂D, we can write down an explicit
1 n
algorithm for finding the electric field generators. We proceed as follows:
• Step 1: Define the measure µ on ∂D by
n
n
1 (cid:88)
µ = δ ,
n n xj
j=1
8
and define β := (−1)n+1(cid:81)n x¯ .
j=1 j
• Step 2: Perform Gram-Schmidt orthogonalization on the linearly independent set
{1,z,...,zn−1} in L2(∂D,µ ) to arrive at the sequence of orthonormal polynomials
n
{1,ϕ (z;µ ),...,ϕ (z;µ )}.
1 n n−1 n
• Step 3: Define the probability measure
1 dθ
dν := .
n |ϕ (eiθ;µ )|22π
n−1 n
• Step 4: Calculate the quantity h (z;β;β) for the measure ν in the domain {z : |z| >
n n
1}. ByProposition1.5, itwillbearationalfunctionS (z)/S (z)forsomepolynomials
1 2
S and S .
1 2
• Step 5: Place a particle of charge −1/2 at each zero of S , a particle of charge +1/2
1
at each zero of S , and a particle of charge 1(1−n) at zero.
2 2
The validity of the above algorithm will follow from the proof of Theorem 2.1. In order to
prove Theorem 2.1, we need to translate the equilibrium problem into a differential equation
so that we may apply the results of Section 1. The appropriate differential equation in this
setting is called a Lam´e equation, which we now discuss.
2.2. The Lam´e Equation. A generalized Lam´e differential equation is a differential equa-
tion of the form
(cid:32) (cid:33)
m
(cid:88) t S(z)
y(cid:48)(cid:48)(z)+ i y(cid:48)(z)+ y(z) = 0, (8)
z −w (cid:81)m (z −w )
i=1 i j=1 j
where S(z) is a polynomial of degree at most m−2. Suppose we can find a polynomial P(z)
of degree N with distinct zeros that solves (8) and also satisfies P(w ) (cid:54)= 0 for i = 1,...,m.
i
Let {p }N be the zeros of P. It is easy to verify that
j j=1
P(cid:48)(cid:48)(p ) (cid:88)N 2
j
= ,
P(cid:48)(p ) p −p
j j k
k=1
k(cid:54)=j
so setting y = P and z = p in (8) shows
j
N m
(cid:88) 1 (cid:88) t /2
i
+ = 0 j = 1,...,N.
p −p p −w
j k j i
k=1 i=1
k(cid:54)=j
We recognize this as the condition for total electrostatic equilibrium with external field
generated by a collection of charged particles located at {w }m , where the particle at w
i i=1 i
carriescharget /2. WewillseethattheproofofTheorem2.1followsfromapplyingtheabove
i
reasoning to the differential equation in Theorem 1.2 under the appropriate hypotheses. To
this end, Proposition 1.5 will be essential.
There is an extensive literature on the topic of generalized Lam´e differential equations and
theirrelevancetoelectrostatics. Wereferthereadertothereferences[4,20,21,22,23,25,26]
9
for further information. We also refer the reader to [3, 5, 9, 10] for further results concerning
applications of polynomial solutions to second order differential equations.
3. Proofs and Calculations
Proof of Theorems 1.2 and 1.3. Our proof begins very much in the same spirit as the proof
of [16, Theorem 2.1]. We begin by writing
(cid:88)n−1 (cid:90) 2π dθ
Φ(cid:48) (z;β) = ϕ (z) Φ(cid:48) (eiθ;β)ϕ (eiθ)w(θ)
n k n k 2π
0
k=0
(cid:88)n−1 (cid:90) 2π dθ
= ϕ (z) ieiθΦ(cid:48) (eiθ;β)eiθϕ (eiθ)w(θ)
k n k 2πi
0
k=0
(cid:88)n−1 (cid:90) 2π d dθ
= ϕ (z) [Φ (eiθ;β)]eiθϕ (eiθ)w(θ) .
k n k
dθ 2πi
0
k=0
We then integrate by parts to rewrite this as
(cid:88)n−1 (cid:90) 2π d dθ
Φ(cid:48) (z;β) = − ϕ (z) Φ (eiθ;β) [eiθϕ (eiθ)w(θ)]
n k n dθ k 2πi
0
k=0
(cid:88)n−1 (cid:90) 2π (cid:104) (cid:105) dθ
= i ϕ (z) Φ (eiθ;β) eiθϕ (eiθ)w(cid:48)(θ)−iw(θ)e2iθϕ(cid:48)(eiθ)+eiθϕ (eiθ) .
k n k k k 2π
0
k=0
where we used the continuity properties of w. For each m ∈ N, let us define K (z,t) :=
m
(cid:80)m ϕ (z)ϕ (t) to be the reproducing kernel for the measure µ and polynomials of degree
j=0 j j
at most m. By [37, Section 2.2], we can rewrite the above expression as
(cid:90) 2π dθ
Φ(cid:48) (z;β) = i Φ (eiθ;β)K (z,eiθ)e−iθw(cid:48)(θ)
n n n−1 2π
0
(cid:90) 2π(cid:16) (cid:17) dθ
+ϕ (z) eiθϕ (eiθ)+e2iθϕ(cid:48) (eiθ) Φ (eiθ;β)w(θ)
n−1 n−1 n−1 n 2π
0
(cid:90) 2π dθ
= i Φ (eiθ;β)K (z,eiθ)e−iθw(cid:48)(θ) +nΦ (z)(1−β¯α ), (9)
n n−1 n−1 n−1
2π
0
It follows from [37, Section 2.3] that (with ζ = eiθ)
κ ϕ (ζ)Φ (z;β ) ζΦ (ζ)
n−1 n−1 n ζ ¯ n−1
K (z,ζ) = , where β = .
n−1 z −ζ ζ Φ∗ (ζ)
n−1
10
If we plug this into (9) and simplify (using also the fact that ϕ (ζ) = ζ1−nϕ∗ (ζ) when
n−1 n−1
|ζ| = 1), we get
(cid:18) (cid:90) 2π Φ (eiθ;β)w(cid:48)(θ)ϕ∗ (eiθ)dθ(cid:19)
Φ(cid:48) (z;β) = Φ (z) n(1−β¯α )+izκ n n−1
n n−1 n−1 n−1 (z −eiθ)einθ 2π
0
(cid:90) 2π Φ (eiθ;β)w(cid:48)(θ)ϕ (eiθ)dθ
−iκ Φ∗ (z) n n−1
n−1 n−1 (z −eiθ)ei(n−1)θ 2π
0
= Φ (z)(n(1−β¯α )+zG (z)+β¯D (z))−Φ∗ (z)(J (z)−β¯G (z)), (10)
n−1 n−1 n n n−1 n n
One can also derive (10) from (9) by using the Christoffel-Darboux formula (see [30, Section
3]) to replace K (z;eiθ).
n−1
The relation (10) holds for all values of β ∈ C. Let τ ∈ C be distinct from β. Equation
(1) easily implies
¯
βΦ (z;τ)−τ¯Φ (z;β) Φ (z;β)−Φ (z;τ)
Φ (z) = n n , Φ∗ (z) = n n .
n−1 z(β¯−τ¯) n−1 τ¯−β¯
If we plug this into (10), we get the following system of ODE’s:
(cid:18)¯ ¯ ¯ ¯ (cid:19)
β(n(1−βα )+zG (z)+βD (z)) (J (z)−βG (z))
Φ(cid:48) (z;β) = Φ (z;τ) n−1 n n + n n (11)
n n z(β¯−τ¯) τ¯−β¯
(cid:18) ¯ ¯ ¯ (cid:19)
τ¯(n(1−βα )+zG (z)+βD (z)) (J (z)−βG (z))
n−1 n n n n
−Φ (z;β) +
n ¯ ¯
z(β −τ¯) τ¯−β
(cid:18)¯ (cid:19)
β(n(1−τ¯α )+zG (z)+τ¯D (z)) (J (z)−τ¯G (z))
Φ(cid:48) (z;τ) = Φ (z;τ) n−1 n n + n n (12)
n n z(β¯−τ¯) τ¯−β¯
(cid:18) (cid:19)
τ¯(n(1−τ¯α )+zG (z)+τ¯D (z)) (J (z)−τ¯G (z))
n−1 n n n n
−Φ (z;β) + .
n ¯ ¯
z(β −τ¯) τ¯−β
This completes the proof of Theorem 1.3. We continue with the proof of Theorem 1.2.
Equation (11) yields the following formula for Φ (z;τ):
n
z(β¯−τ¯)Φ(cid:48) (z;β)+Φ (z;β)(cid:0)τ¯(n(1−β¯α )+zG (z)+β¯D (z))−z(J (z)−β¯G (z))(cid:1)
n n n−1 n n n n .
¯ ¯ ¯ ¯
β(n(1−βα )+zG (z)+βD (z))−z(J (z)−βG (z))
n−1 n n n n
(13)
We can use (13) to write
(cid:18)
Φ(cid:48) (z;τ) = Φ(cid:48)(cid:48)(z;β)z(β¯−τ¯)h (z;β;β)+
n n n
Φ(cid:48) (z;β)(cid:0)(β¯−τ¯)W[h (z;β;β),z]+h (z;β;β)h (z;τ;β)(cid:1) (14)
n n n n
(cid:19)
+Φ (z;β)W[h (z;β;β),h (z;τ;β)] h (z;β;β)−2
n n n n