Multiple orthogonal polynomials for a Nikishin system on a star-like
set
A. López-García 1
1
University of South Alabama
2
A. López-García (U. South Alabama)
E. Miña-Díaz 2
University of Mississippi
1 / 23
Banded Hessenberg operators
• Let H be a banded Hessenberg operator with only two non-zero diagonals of
the form


0
1


0
1




0
1


.

.
.
.
.
..
..
..
. .
 ..



.. 


.
0
H=


a0 0





a1 0




a2
0




a3


..
.
where an > 0 for all n ≥ 0. Number of zero diagonals in between is p ≥ 1.
A. López-García (U. South Alabama)
2 / 23
Banded Hessenberg operators
• Let H be a banded Hessenberg operator with only two non-zero diagonals of
the form


0
1


0
1




0
1


.

.
.
.
.
..
..
..
. .
 ..



.. 


.
0
H=


a0 0





a1 0




a2
0




a3


..
.
where an > 0 for all n ≥ 0. Number of zero diagonals in between is p ≥ 1.
• For Hn the principal n × n truncation of H, the monic polynomials
Qn (z) = det(z In − Hn ) satisfy
zQn (z) = Qn+1 (z) + an−p Qn−p (z),
n ≥ p,
with initial conditions Qj (z) = z j , j = 0, . . . , p.
A. López-García (U. South Alabama)
2 / 23
• If n ≡ ` mod (p + 1), 0 ≤ ` ≤ p, then
Qn (z) = z ` Ln (z p+1 ),
deg(Ln ) =
n−`
,
p+1
where the zeros of Ln are simple and lie on (0, ∞).
A. López-García (U. South Alabama)
3 / 23
• If n ≡ ` mod (p + 1), 0 ≤ ` ≤ p, then
Qn (z) = z ` Ln (z p+1 ),
deg(Ln ) =
n−`
,
p+1
where the zeros of Ln are simple and lie on (0, ∞).
• Define the star S+ := {z ∈ C : z p+1 ∈ [0, ∞)}.
p=2
Theorem (Eiermann–Varga 1993, He–Saff 1994, Ben Romdhane 2008)
Zeros of Qn lie on S+ .
A. López-García (U. South Alabama)
3 / 23
• If n ≡ ` mod (p + 1), 0 ≤ ` ≤ p, then
Qn (z) = z ` Ln (z p+1 ),
deg(Ln ) =
n−`
,
p+1
where the zeros of Ln are simple and lie on (0, ∞).
• Define the star S+ := {z ∈ C : z p+1 ∈ [0, ∞)}.
p=2
Theorem (Eiermann–Varga 1993, He–Saff 1994, Ben Romdhane 2008)
Zeros of Qn lie on S+ .
The zeros of Ln and Ln−1 interlace on (0, ∞).
A. López-García (U. South Alabama)
3 / 23
2
2
• Assume (an )∞
n=0 is bounded, so H : ` → ` is a bounded operator.
• Resolvent functions:
φj (z) = h(zI − H)−1 ~ej , ~e0 i,
j = 0, . . . , p − 1,
where ~ej is the jth element in the standard basis of `2 .
Then
Z
dνj (t)
φj (z) =
,
S+ z − t
for some compactly supported complex measure νj on S+ .
• (ν0 , . . . , νp−1 ) spectral measures for H.
A. López-García (U. South Alabama)
4 / 23
2
2
• Assume (an )∞
n=0 is bounded, so H : ` → ` is a bounded operator.
• Resolvent functions:
φj (z) = h(zI − H)−1 ~ej , ~e0 i,
j = 0, . . . , p − 1,
where ~ej is the jth element in the standard basis of `2 .
Then
Z
dνj (t)
φj (z) =
,
S+ z − t
for some compactly supported complex measure νj on S+ .
• (ν0 , . . . , νp−1 ) spectral measures for H.
Favard type theorem for Qn (Aptekarev–Kalyagin–Van Iseghem, 2000)
Assume that an > 0 for all n and the numbers an are uniformly bounded. Then the
polynomials Qn are multiple orthogonal with respect to the system (ν0 , . . . , νp−1 ).
This means that for each j = 0, . . . , p − 1 we have
Z
jn − j − 1k
Qn (z) z l dνj (z) = 0,
l = 0, . . . ,
.
p
S+
A. López-García (U. South Alabama)
4 / 23
Theorem (Delvaux–L., 2013)
Suppose that an > 0 for all n and the sequence (an ) is periodic with period r that is a
multiple of p. If the numbers a0 , a1 , . . . , ar −1 satisfy
r /p−1
Y
n=0
r /p−1
apn >
Y
n=0
r /p−1
apn+1 > · · · >
Y
apn+p−1 ,
n=0
then the polynomials Qn are multi-orthogonal with respect to a Nikishin system
(ν0 , . . . , νp−1 ) on S+ .
A. López-García (U. South Alabama)
5 / 23
Nikishin systems on star-like sets and an inverse spectral problem
• We consider an inverse spectral problem:
Starting from
(ν0 , . . . , νp−1 ) Nikishin system on S+
we want to analyze
asymptotic behavior of an and Qn .
A. López-García (U. South Alabama)
6 / 23
Nikishin systems on star-like sets and an inverse spectral problem
• We consider an inverse spectral problem:
Starting from
(ν0 , . . . , νp−1 ) Nikishin system on S+
we want to analyze
asymptotic behavior of an and Qn .
• Let p ≥ 2. To construct a Nikishin system on S+ , take (Γ0 , Γ1 , . . . , Γp−1 ) a system
of star-like sets given by
Γj = T −1 (∆j ),
j = 0, . . . , p − 1,
where T (z) = z p+1 and (∆j )p−1
j=0 are compact intervals such that

[0, +∞) if j even,
∆j ⊂
(−∞, 0] if j odd.
We also assume that ∆j ∩ ∆j+1 = ∅ for all j = 0, . . . , p − 2.
A. López-García (U. South Alabama)
6 / 23
Case p = 2
Γ1
Γ0
Nikishin system on Γ0 for p = 2:
Let (σ0 , σ1 ) be a system of positive, rotationally invariant measures supported on
(Γ0 , Γ1 ), respectively.
Then (ν0 , ν1 ) is the Nikishin system generated by (σ0 , σ1 ) if
dν0 (z) = dσ0 (z),
Z
dσ1 (t)
dν1 (z) =
dσ0 (z) = dhσ0 , σ1 i(z).
Γ1 z − t
A. López-García (U. South Alabama)
7 / 23
Case p = 2
Γ1
Γ0
Nikishin system on Γ0 for p = 2:
Let (σ0 , σ1 ) be a system of positive, rotationally invariant measures supported on
(Γ0 , Γ1 ), respectively.
Then (ν0 , ν1 ) is the Nikishin system generated by (σ0 , σ1 ) if
dν0 (z) = dσ0 (z),
Z
dσ1 (t)
dν1 (z) =
dσ0 (z) = dhσ0 , σ1 i(z).
Γ1 z − t
The measures (ν0 , ν1 ) are supported on Γ0 .
A. López-García (U. South Alabama)
7 / 23
Nikishin system on Γ0 for general p:
Start with positive, rotationally invariant measures (σ0 , σ1 , . . . , σp−1 ) on
(Γ0 , Γ1 , . . . , Γp−1 ), respectively.
A. López-García (U. South Alabama)
8 / 23
Nikishin system on Γ0 for general p:
Start with positive, rotationally invariant measures (σ0 , σ1 , . . . , σp−1 ) on
(Γ0 , Γ1 , . . . , Γp−1 ), respectively.
Then (ν0 , ν1 , . . . , νp−1 ) is the Nikishin system generated by (σ0 , σ1 , . . . , σp−1 ) if
dν0 (z) = dσ0 (z),
Z
dσ1 (t)
dν1 (z) =
dσ0 (z) = dhσ0 , σ1 i(z),
Γ1 z − t
Z
dhσ1 , σ2 i(t)
dσ0 (z) = dhσ0 , σ1 , σ2 i(z).
dν2 (z) =
z −t
Γ1
..
.
Z
dνp−1 (z) =
Γ1
dhσ1 , . . . , σp−1 i(t)
dσ0 (z) = dhσ0 , σ1 , . . . , σp−1 i(z)
z −t
The measures (ν0 , . . . , νp−1 ) are all supported on Γ0 .
A. López-García (U. South Alabama)
8 / 23
Hierarchy of measures on (Γ0 , . . . , Γp−1 )
• There is a hierarchy of measures sk ,j , 0 ≤ k ≤ j, 0 ≤ j ≤ p − 1 generated by the
measures (σ0 , σ1 , . . . , σp−1 ).
s0,0
s0,1
s1,1
s0,2
s1,2
s2,2
···
···
···
..
.
s0,p−1
s1,p−1
s2,p−1
..
.
sp−1,p−1
• The measures sk ,j , k ≤ j ≤ p − 1 are supported on Γk and form the k th layer of
the hierarchy.
A. López-García (U. South Alabama)
9 / 23
Hierarchy of measures on (Γ0 , . . . , Γp−1 )
• There is a hierarchy of measures sk ,j , 0 ≤ k ≤ j, 0 ≤ j ≤ p − 1 generated by the
measures (σ0 , σ1 , . . . , σp−1 ).
s0,0
s0,1
s1,1
s0,2
s1,2
s2,2
···
···
···
..
.
s0,p−1
s1,p−1
s2,p−1
..
.
sp−1,p−1
• The measures sk ,j , k ≤ j ≤ p − 1 are supported on Γk and form the k th layer of
the hierarchy.
• The measures sk ,j are defined inductively as follows. For k = j we set
sk ,k = σk ,
and for 0 ≤ k < j, we set
Z
dsk ,j (z) =
Γk +1
dsk +1,j (t)
dσk (z),
z −t
z ∈ Γk .
• The Nikishin system (ν0 , . . . , νp−1 ) is the first layer of the hierarchy:
νj = s0,j ,
A. López-García (U. South Alabama)
j = 0, . . . , p − 1.
9 / 23
Hierarchy of measures induced on (∆0 , . . . , ∆p−1 )
• The measures σj induce a hierarchy of measures µk ,j , 0 ≤ k ≤ j, 0 ≤ j ≤ p − 1 on
the real line that plays a key role in the analysis.
• The measures µk ,j , k ≤ j ≤ p − 1 are supported on ∆k and form the k th layer of
the hierarchy.
• The measures µk ,j are defined inductively as follows: For k = j we set
µk ,k = σk∗ := T ∗ σk
and for 0 ≤ k < j we set
Z
dµk ,j (t) =
t
∆k +1
• b
sk ,j (z) = z p+k −j µ
bk ,j (z p+1 ),
A. López-García (U. South Alabama)
dµk +1,j (s)
t −s
!
dσk∗ (t),
t ∈ ∆k .
µ
b ≡ Cauchy transform of µ.
10 / 23
Hierarchy of measures induced on (∆0 , . . . , ∆p−1 )
• The measures σj induce a hierarchy of measures µk ,j , 0 ≤ k ≤ j, 0 ≤ j ≤ p − 1 on
the real line that plays a key role in the analysis.
• The measures µk ,j , k ≤ j ≤ p − 1 are supported on ∆k and form the k th layer of
the hierarchy.
• The measures µk ,j are defined inductively as follows: For k = j we set
µk ,k = σk∗ := T ∗ σk
and for 0 ≤ k < j we set
Z
dµk ,j (t) =
t
∆k +1
• b
sk ,j (z) = z p+k −j µ
bk ,j (z p+1 ),
dµk +1,j (s)
t −s
!
dσk∗ (t),
t ∈ ∆k .
µ
b ≡ Cauchy transform of µ.
• For each k , the functions
(1, µ
bk ,k , µ
bk ,k +1 , . . . , µ
bk ,p−1 )
form an AT system on any interval disjoint with ∆k with respect to any multi-index
with non-increasing components.
A. López-García (U. South Alabama)
10 / 23
Definition of multiple orthogonal polynomials
For a Nikishin system (ν0 , . . . , νp−1 ) on Γ0 , let (Qn )∞
n=0 be the sequence of monic
polynomials of lowest degree that satisfy the multiple orthogonality conditions
Z
jn − j − 1k
Qn (x) x l dνj (x) = 0,
l = 0, . . . ,
,
j = 0, . . . , p − 1.
p
Γ0
A. López-García (U. South Alabama)
11 / 23
Definition of multiple orthogonal polynomials
For a Nikishin system (ν0 , . . . , νp−1 ) on Γ0 , let (Qn )∞
n=0 be the sequence of monic
polynomials of lowest degree that satisfy the multiple orthogonality conditions
Z
jn − j − 1k
Qn (x) x l dνj (x) = 0,
l = 0, . . . ,
,
j = 0, . . . , p − 1.
p
Γ0
Theorem
We have
1) Qn has maximal degree n.
2) If n ≡ ` mod (p + 1), 0 ≤ ` ≤ p, then Qn (z) = z ` Pn (z p+1 ), and Pn has exactly
n−`
simple zeros in the interior of ∆0 .
p+1
3) The zeros of Pn interlace on ∆0 .
4) The polynomials Qn satisfy the three-term recurrence relation
zQn (z) = Qn+1 (z) + an Qn−p (z),
j
Qj (z) = z ,
n ≥ p,
j = 0, . . . , p,
and an > 0 for all n ≥ p.
A. López-García (U. South Alabama)
11 / 23
The polynomials Pn in
Qn (z) = z ` Pn (z p+1 ),
n≡`
mod (p + 1)
satisfy multi-orthogonality conditions with respect to the measures µ0,j : For each
j = 0, . . . , p − 1,
Z
n + p` − 1 − j(p + 1)
`−j
Pn (t) t s dµ0,j (t) = 0,
≤s≤
.
p+1
p(p + 1)
∆0
A. López-García (U. South Alabama)
12 / 23
Second kind functions and associated polunomials
• Define functions Ψn,k , k = 0, . . . , p − 1 inductively (on k ) by
Ψn,0 := Qn ,
Z
Ψn,k −1 (t)
Ψn,k (z) =
dσk −1 (t),
z −t
Γk −1
• Ψn,k ∈ H(C \ Γk −1 ),
k = 1, . . . , p − 1.
k = 1, . . . , p − 1.
• Ψn,k satisfies orthogonality conditions with respect to the measures sk ,j ,
k ≤ j ≤ p − 1 in the k th layer of the Nikishin hierarchy.
For each j = k , . . . , p − 1,
Z
n−j −1
.
Ψn,k (z) z l dsk ,j (z) = 0,
0≤l≤
p
Γk
A. López-García (U. South Alabama)
13 / 23
• Assume n ≡ ` mod (p + 1), 0 ≤ ` ≤ p. For every k = 0, . . . , p − 1,
z k −` Ψn,k (z) = φn,k (z p+1 ),
where φn,k ∈ H(C \ ∆k −1 ).
A. López-García (U. South Alabama)
14 / 23
• Assume n ≡ ` mod (p + 1), 0 ≤ ` ≤ p. For every k = 0, . . . , p − 1,
z k −` Ψn,k (z) = φn,k (z p+1 ),
where φn,k ∈ H(C \ ∆k −1 ).
• The functions φn,k are "almost" second-kind functions:
φn,0 (z) = Pn (z),
R

 ∆k −1
φn,k (z) =
R

z
∆
t φn,k −1 (t)
z−t
k −1
A. López-García (U. South Alabama)
dσk∗−1 (t),
φn,k −1 (t)
z−t
if 1 ≤ k ≤ `,
dσk∗−1 (t), if ` < k ≤ p − 1.
14 / 23
• Assume n ≡ ` mod (p + 1), 0 ≤ ` ≤ p. For every k = 0, . . . , p − 1,
z k −` Ψn,k (z) = φn,k (z p+1 ),
where φn,k ∈ H(C \ ∆k −1 ).
• The functions φn,k are "almost" second-kind functions:
φn,0 (z) = Pn (z),
R

 ∆k −1
φn,k (z) =
R

z
∆
t φn,k −1 (t)
z−t
k −1
dσk∗−1 (t),
φn,k −1 (t)
z−t
if 1 ≤ k ≤ `,
dσk∗−1 (t), if ` < k ≤ p − 1.
• For each k , the function φn,k satisfy multi-orthogonality conditions with respect to
the measures µk ,j , k ≤ j ≤ p − 1 in the k th layer of the hierarchy on the real line.
A. López-García (U. South Alabama)
14 / 23
Counting the number of zeros of the functions φn,k
Proposition
Let 1 ≤ k ≤ p − 1, and suppose that
n = λp(p + 1) + r ,
Then
φn,k (z) = O(z
A. López-García (U. South Alabama)
−N(n,k )
0 ≤ r ≤ p(p + 1) − 1.
),
r −k +1
r
N(n, k ) = λ +
−
.
p
p+1
15 / 23
Counting the number of zeros of the functions φn,k
Proposition
Let 1 ≤ k ≤ p − 1, and suppose that
n = λp(p + 1) + r ,
Then
φn,k (z) = O(z
−N(n,k )
0 ≤ r ≤ p(p + 1) − 1.
),
r −k +1
r
N(n, k ) = λ +
−
.
p
p+1
Proposition
Let 1 ≤ k ≤ p − 1. Then all the zeros of φn,k in C \ (∆k −1 ∪ {0}) are simple and lie on
the interior of ∆k . The zeros of φn+1,k and φn,k interlace.
If Z (n, k ) is the number of zeros of φn,k in the interior of ∆k , then Z (n, k ) can be
computed inductively as follows: Set
Z (n, 0) =
n−`
,
p+1
n≡`
mod (p + 1),
and for k ≥ 1,
Z (n, k ) =

Z (n, k − 1) − N(n, k ),
if ` < k ,
Z (n, k − 1) + 1 − N(n, k ), if k ≤ `.
A. López-García (U. South Alabama)
15 / 23
Polynomials Pn,k associated with the functions φn,k
For 0 ≤ k ≤ p − 1, let Pn,k denote the monic polynomial whose zeros are the zeros of
φn,k in the interior of ∆k .
Proposition
The function φn,k satisfy the following orthogonality conditions with respect to varying
measures.
If k ≥ `, then
Z
φn,k (t) t l
dσk∗ (t)
= 0,
Pn,k +1 (t)
l = 0, . . . , Z (n, k ) − 1,
φn,k (t) t l
t dσk∗ (t)
= 0,
Pn,k +1 (t)
l = 0, . . . , Z (n, k ) − 1.
∆k
and if k < `, then
Z
∆k
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16 / 23
Ratio asymptotics
• We consider the p families
∞
Pn+1,k
,
Pn,k n=0
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k = 0, . . . , p − 1.
17 / 23
Ratio asymptotics
• We consider the p families
∞
Pn+1,k
,
Pn,k n=0
k = 0, . . . , p − 1.
• For every k = 0, . . . , p − 1 fixed,
deg(Pn+1,k ) − deg(Pn,k ) = Z (n + 1, k ) − Z (n, k )
is a periodic function of n with period p(p + 1). Moreover,
Z (n + 1, k ) − Z (n, k ) ∈ {−1, 0, 1},
A. López-García (U. South Alabama)
for all n.
17 / 23
Using a method devised by Aptekarev–López Lagomasino–Rocha for Nikishin systems
on the real line, we can prove:
Theorem
Let (ν0 , ν1 , . . . , νp−1 ) be a Nikishin system generated by the positive, rotationally
symmetric measures (σ0 , σ1 , . . . , σp−1 ) on the compact star-like sets (Γ0 , Γ1 , . . . , Γp−1 ).
Suppose that for all j = 0, . . . , p − 1, the measure σj satisfies
A. López-García (U. South Alabama)
dσj (x)
|dx|
> 0 for a.e. x ∈ Γj .
18 / 23
Using a method devised by Aptekarev–López Lagomasino–Rocha for Nikishin systems
on the real line, we can prove:
Theorem
Let (ν0 , ν1 , . . . , νp−1 ) be a Nikishin system generated by the positive, rotationally
symmetric measures (σ0 , σ1 , . . . , σp−1 ) on the compact star-like sets (Γ0 , Γ1 , . . . , Γp−1 ).
Suppose that for all j = 0, . . . , p − 1, the measure σj satisfies
dσj (x)
|dx|
> 0 for a.e. x ∈ Γj .
Then for any fixed r = 0, . . . , p(p + 1) − 1 and any fixed k = 0, . . . , p − 1, the following
limit holds
Pmp(p+1)+r +1,k (z)
(r )
lim
= Fk (z),
z ∈ C \ ∆k .
m→∞ Pmp(p+1)+r ,k (z)
(r )
The functions Fk are independent of the measures (σ0 , . . . , σp−1 ). They only depend
on the intervals (∆0 , . . . , ∆p−1 ).
A. López-García (U. South Alabama)
18 / 23
Using a method devised by Aptekarev–López Lagomasino–Rocha for Nikishin systems
on the real line, we can prove:
Theorem
Let (ν0 , ν1 , . . . , νp−1 ) be a Nikishin system generated by the positive, rotationally
symmetric measures (σ0 , σ1 , . . . , σp−1 ) on the compact star-like sets (Γ0 , Γ1 , . . . , Γp−1 ).
Suppose that for all j = 0, . . . , p − 1, the measure σj satisfies
dσj (x)
|dx|
> 0 for a.e. x ∈ Γj .
Then for any fixed r = 0, . . . , p(p + 1) − 1 and any fixed k = 0, . . . , p − 1, the following
limit holds
Pmp(p+1)+r +1,k (z)
(r )
lim
= Fk (z),
z ∈ C \ ∆k .
m→∞ Pmp(p+1)+r ,k (z)
(r )
The functions Fk are independent of the measures (σ0 , . . . , σp−1 ). They only depend
on the intervals (∆0 , . . . , ∆p−1 ).
This result complements results by
Rakhmanov (1977)
Aptekarev–López Lagomasino–Rocha (2005)
L. (2010)
A. López-García (U. South Alabama)
18 / 23
Consequently, we obtain:
1) Ratio asymptotics for the original polynomials Qn : For any fixed
r = 0, . . . , p(p + 1) − 1,
lim
Qmp(p+1)+r +1 (z)
(r )
= z F0 (z p+1 ),
Qmp(p+1)+r (z)
lim
Qmp(p+1)+r +1 (z)
F (z p+1 )
,
= 0 p
Qmp(p+1)+r (z)
z
m→∞
z ∈ C \ Γ0 ,
if r 6≡ p mod (p + 1),
(r )
m→∞
A. López-García (U. South Alabama)
z ∈ C \ (Γ0 ∪ {0}),
if r ≡ p mod (p + 1).
19 / 23
Consequently, we obtain:
1) Ratio asymptotics for the original polynomials Qn : For any fixed
r = 0, . . . , p(p + 1) − 1,
lim
Qmp(p+1)+r +1 (z)
(r )
= z F0 (z p+1 ),
Qmp(p+1)+r (z)
lim
Qmp(p+1)+r +1 (z)
F (z p+1 )
,
= 0 p
Qmp(p+1)+r (z)
z
m→∞
z ∈ C \ Γ0 ,
if r 6≡ p mod (p + 1),
(r )
m→∞
z ∈ C \ (Γ0 ∪ {0}),
if r ≡ p mod (p + 1).
2) Asymptotics for the recurrence coefficients an : For any fixed
r = 0, . . . , p(p + 1) − 1,
lim amp(p+1)+r = a(r ) > 0.
m→∞
A. López-García (U. South Alabama)
19 / 23
Consequently, we obtain:
1) Ratio asymptotics for the original polynomials Qn : For any fixed
r = 0, . . . , p(p + 1) − 1,
lim
Qmp(p+1)+r +1 (z)
(r )
= z F0 (z p+1 ),
Qmp(p+1)+r (z)
lim
Qmp(p+1)+r +1 (z)
F (z p+1 )
,
= 0 p
Qmp(p+1)+r (z)
z
m→∞
z ∈ C \ Γ0 ,
if r 6≡ p mod (p + 1),
(r )
m→∞
z ∈ C \ (Γ0 ∪ {0}),
if r ≡ p mod (p + 1).
2) Asymptotics for the recurrence coefficients an : For any fixed
r = 0, . . . , p(p + 1) − 1,
lim amp(p+1)+r = a(r ) > 0.
m→∞
3) As z → ∞,
(r )
F0 (z) = 1 −
(r )
a(r )
+O
z
F0 (z) = z − a(r ) + O
A. López-García (U. South Alabama)
1
z2
1
,
z
,
if r 6≡ p mod (p + 1),
if r ≡ p mod (p + 1).
19 / 23
(r )
Relations between the functions Fk
and the values a(r )
• For every λ = 0, . . . , p − 1,
(λ(p+1)+p)
F0
A. López-García (U. South Alabama)
(z) = z F (λ(p+1)) (z)
20 / 23
(r )
Relations between the functions Fk
and the values a(r )
• For every λ = 0, . . . , p − 1,
(λ(p+1)+p)
F0
(z) = z F (λ(p+1)) (z)
• So for every λ = 0, . . . , p − 1,
a(λ(p+1)+p) = a(λ(p+1)) .
A. López-García (U. South Alabama)
20 / 23
(r )
Relations between the functions Fk
and the values a(r )
• For every λ = 0, . . . , p − 1,
(λ(p+1)+p)
F0
(z) = z F (λ(p+1)) (z)
• So for every λ = 0, . . . , p − 1,
a(λ(p+1)+p) = a(λ(p+1)) .
• From the three-term recurrence relation zQn = Qn+1 + an Qn−p , we get
r −1
Y
(r )
(i)
F0 (z),
a(r ) = 1 − F0 (z)
r 6≡ p mod (p + 1),
i=r −p
r −1
Y
(r )
(i)
F0 (z),
a(r ) = z − F0 (z)
r ≡ p mod (p + 1).
i=r −p
A. López-García (U. South Alabama)
20 / 23
• The previous relations imply that
m0 (p+1)+p−1
m(p+1)+p−1
Y
(i)
F0 (z)
i=m(p+1)
A. López-García (U. South Alabama)
=
Y
(i)
F0 (z),
0 ≤ m, m0 ≤ p − 1,
m 6= m0 .
i=m0 (p+1)
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• The previous relations imply that
m0 (p+1)+p−1
m(p+1)+p−1
Y
(i)
F0 (z)
=
(i)
F0 (z),
0 ≤ m, m0 ≤ p − 1,
m 6= m0 .
i=m0 (p+1)
i=m(p+1)
m0 (p+1)+p−1
m(p+1)+p−1
X
Y
a(r ) =
r =m(p+1)
A. López-García (U. South Alabama)
X
a(r ) ,
0 ≤ m, m0 ≤ p − 1,
m 6= m0 .
r =m0 (p+1)
21 / 23
• The previous relations imply that
m0 (p+1)+p−1
m(p+1)+p−1
Y
(i)
F0 (z)
=
(i)
F0 (z),
0 ≤ m, m0 ≤ p − 1,
m 6= m0 .
i=m0 (p+1)
i=m(p+1)
m0 (p+1)+p−1
m(p+1)+p−1
X
Y
a(r ) =
X
a(r ) ,
0 ≤ m, m0 ≤ p − 1,
m 6= m0 .
r =m0 (p+1)
r =m(p+1)
• We also have
0
(m0 (p+1))
1 − F0
(z)
a(m (p+1))
=
,
(m(p+1))
a(m(p+1))
1 − F0
(z)
z ∈ C \ ∆0 ,
with m, m0 as before.
A. López-García (U. South Alabama)
21 / 23
Main ideas in the proof of ratio asymptotics
1) For every k = 0, . . . , p − 1,
deg(Pn+1,k ) − deg(Pn,k ) = Z (n + 1, k ) − Z (n, k )
is periodic in n with period p(p + 1).
A. López-García (U. South Alabama)
22 / 23
Main ideas in the proof of ratio asymptotics
1) For every k = 0, . . . , p − 1,
deg(Pn+1,k ) − deg(Pn,k ) = Z (n + 1, k ) − Z (n, k )
is periodic in n with period p(p + 1).
2) The zeros of Pn+1,k and Pn,k interlace, hence for every fixed
r = 0, . . . , p(p + 1) − 1, the families
Pmp(p+1)+r +1,k ∞
,
k = 0, . . . , p − 1
Pmp(p+1)+r ,k m=0
are normal in C \ ∆k , respectively.
A. López-García (U. South Alabama)
22 / 23
Main ideas in the proof of ratio asymptotics
1) For every k = 0, . . . , p − 1,
deg(Pn+1,k ) − deg(Pn,k ) = Z (n + 1, k ) − Z (n, k )
is periodic in n with period p(p + 1).
2) The zeros of Pn+1,k and Pn,k interlace, hence for every fixed
r = 0, . . . , p(p + 1) − 1, the families
Pmp(p+1)+r +1,k ∞
,
k = 0, . . . , p − 1
Pmp(p+1)+r ,k m=0
are normal in C \ ∆k , respectively.
3) There exists Λ ⊂ N such that
lim
m∈Λ
Pmp(p+1)+r +1,k (z)
(r )
= Fk (z),
Pmp(p+1)+r ,k (z)
C \ ∆k ,
for every k = 0, . . . , p − 1.
(r )
The goal is to show that the limits Fk are independent of the subsequence Λ.
A. López-García (U. South Alabama)
22 / 23
4) Using the orthogonalities of φn,k with respect to varying measures and results by
de la Calle – López Lagomasino on relative and ratio asymptotics of OPRL with
respect to varying measures, we can show
(r )
Fk
(r )
(r )
is a Szegő function with weights involving Fk −1 and Fk +1 ,
for every k = 0, . . . , p − 1.
A. López-García (U. South Alabama)
23 / 23
4) Using the orthogonalities of φn,k with respect to varying measures and results by
de la Calle – López Lagomasino on relative and ratio asymptotics of OPRL with
respect to varying measures, we can show
(r )
Fk
(r )
(r )
is a Szegő function with weights involving Fk −1 and Fk +1 ,
for every k = 0, . . . , p − 1.
5) Using the boundary value behavior of Szegő functions, we obtain a system of
(r )
boundary value equations for the functions Fk , k = 0, . . . , p − 1. Using a result of
Aptekarev on systems of boundary value equations for harmonic functions, we
(r )
show that functions Fk are unique.
A. López-García (U. South Alabama)
23 / 23