The Riemann-Hilbe p oblem o
ma ix- alued o hogonal polynomials1
Manuel Domínguez de la Iglesia
Depa men o Ma hema ics, K. U. Leu en
Semina classical analysis
Leu en, No embe 5, 2008
1join wo k wi h And ei Ma ínez Finkelsh ein
P elimina ies
The RH p oblem o OMP
An example
Ou line
1P elimina ies
2The RH p oblem o OMP
3An example
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
Ou line
1P elimina ies
2The RH p oblem o OMP
3An example
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
Le Wbe a N×Naweigh ma ix such ha dW (x) = W(x)dx.
We can cons uc a amily o OMP such ha
ZR
Pn(x)W(x)P∗
m(x)dx =δn,mI,n,m⩾0
Pn(x) = γn(xn+an,n−1xn−1+···) = γnb
Pn(x)
The ma ix- alued polynomials o he second kind, de ined by
Qn(x) = ZR
Pn( )W( )
−xd ,n⩾0
(Pn)nand (Qn)nsa is y a h ee e m ecu ence ela ion
Pn( ) = An+1Pn+1( ) + BnPn( ) + A∗
nPn−1( ),n⩾0
de (An+1)6=0,Bn=B∗
n
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
Le Wbe a N×Naweigh ma ix such ha dW (x) = W(x)dx.
We can cons uc a amily o OMP such ha
ZR
Pn(x)W(x)P∗
m(x)dx =δn,mI,n,m⩾0
Pn(x) = γn(xn+an,n−1xn−1+···) = γnb
Pn(x)
The ma ix- alued polynomials o he second kind, de ined by
Qn(x) = ZR
Pn( )W( )
−xd ,n⩾0
(Pn)nand (Qn)nsa is y a h ee e m ecu ence ela ion
Pn( ) = An+1Pn+1( ) + BnPn( ) + A∗
nPn−1( ),n⩾0
de (An+1)6=0,Bn=B∗
n
Manuel Domínguez de la Iglesia The RH p oblem o OMP

P elimina ies
The RH p oblem o OMP
An example
Le Wbe a N×Naweigh ma ix such ha dW (x) = W(x)dx.
We can cons uc a amily o OMP such ha
ZR
Pn(x)W(x)P∗
m(x)dx =δn,mI,n,m⩾0
Pn(x) = γn(xn+an,n−1xn−1+···) = γnb
Pn(x)
The ma ix- alued polynomials o he second kind, de ined by
Qn(x) = ZR
Pn( )W( )
−xd ,n⩾0
(Pn)nand (Qn)nsa is y a h ee e m ecu ence ela ion
Pn( ) = An+1Pn+1( ) + BnPn( ) + A∗
nPn−1( ),n⩾0
de (An+1)6=0,Bn=B∗
n
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
Le Wbe a N×Naweigh ma ix such ha dW (x) = W(x)dx.
We can cons uc a amily o OMP such ha
ZR
Pn(x)W(x)P∗
m(x)dx =δn,mI,n,m⩾0
Pn(x) = γn(xn+an,n−1xn−1+···) = γnb
Pn(x)
The ma ix- alued polynomials o he second kind, de ined by
Qn(x) = ZR
Pn( )W( )
−xd ,n⩾0
(Pn)nand (Qn)nsa is y a h ee e m ecu ence ela ion
Pn( ) = An+1Pn+1( ) + BnPn( ) + A∗
nPn−1( ),n⩾0
de (An+1)6=0,Bn=B∗
n
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
The coe icien s o he TTRR sa is y
An=γn−1γ−1
n,Bn=γn(an,n−1−an+1,n)γ−1
n
The TTRR o monic OMP
xb
Pn(x) = b
Pn+1(x) + αnb
Pn(x) + βnb
Pn−1(x),n⩾0
αn=an,n−1−an+1,n,βn= (γ∗
nγn)−1(γ∗
n−1γn−1)
Second-o de di e en ial equa ions o hype geome ic ype
P00
n(x)F2(x) + P0
n(x)F1(x) + Pn(x)F0(x) = ΛnPn(x),n⩾0
deg Fi⩽i,ΛnHe mi ian
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
The coe icien s o he TTRR sa is y
An=γn−1γ−1
n,Bn=γn(an,n−1−an+1,n)γ−1
n
The TTRR o monic OMP
xb
Pn(x) = b
Pn+1(x) + αnb
Pn(x) + βnb
Pn−1(x),n⩾0
αn=an,n−1−an+1,n,βn= (γ∗
nγn)−1(γ∗
n−1γn−1)
Second-o de di e en ial equa ions o hype geome ic ype
P00
n(x)F2(x) + P0
n(x)F1(x) + Pn(x)F0(x) = ΛnPn(x),n⩾0
deg Fi⩽i,ΛnHe mi ian
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
Also we ind a solu ion o he in e se
(Yn)−1=

−ZR
W( )b
P∗
n−1( )
−zd γ∗
n−1γn−1−1
2πiZR
W( )b
P∗
n( )
−zd
2πib
P∗
n−1(z)γ∗
n−1γn−1b
P∗
n(z)


The Liou ille-Os og adski o mula
Qn(z)P∗
n−1(z) − Pn(z)Q∗
n−1(z) = A−1
n
The He mi ian p ope y
Qn(z)P∗
n(z) = Pn(z)Q∗
n(z)
Manuel Domínguez de la Iglesia The RH p oblem o OMP

P elimina ies
The RH p oblem o OMP
An example
The h ee- e m ecu ence ela ion
I we call R=Yn+1(Yn)−1and deno ing
Yn(z) = I+1
zYn
1+On(1/z2)znI0
0z−nI,z→∞
Yn+1(z) = zI +Yn+1
111 −Yn
111 −Yn
112
Yn+1
121 0Yn(z)
γ∗
n−1γn−1= − 1
2πiYn
121 = − 1
2πiYn−1
1−1
12
αn=Yn
111 −Yn+1
111,βn=Yn
112Yn
121
Bn=γnYn
111 −Yn+1
111γ−1
n,A∗
n=γnYn
112Yn
121γ−1
n−1
Yn+1
121Yn
112 =Yn
112Yn+1
121 =I,Yn
111 +Yn
1∗
22 =0
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
The h ee- e m ecu ence ela ion
I we call R=Yn+1(Yn)−1and deno ing
Yn(z) = I+1
zYn
1+On(1/z2)znI0
0z−nI,z→∞
Yn+1(z) = zI +Yn+1
111 −Yn
111 −Yn
112
Yn+1
121 0Yn(z)
γ∗
n−1γn−1= − 1
2πiYn
121 = − 1
2πiYn−1
1−1
12
αn=Yn
111 −Yn+1
111,βn=Yn
112Yn
121
Bn=γnYn
111 −Yn+1
111γ−1
n,A∗
n=γnYn
112Yn
121γ−1
n−1
Yn+1
121Yn
112 =Yn
112Yn+1
121 =I,Yn
111 +Yn
1∗
22 =0
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
The h ee- e m ecu ence ela ion
I we call R=Yn+1(Yn)−1and deno ing
Yn(z) = I+1
zYn
1+On(1/z2)znI0
0z−nI,z→∞
Yn+1(z) = zI +Yn+1
111 −Yn
111 −Yn
112
Yn+1
121 0Yn(z)
γ∗
n−1γn−1= − 1
2πiYn
121 = − 1
2πiYn−1
1−1
12
αn=Yn
111 −Yn+1
111,βn=Yn
112Yn
121
Bn=γnYn
111 −Yn+1
111γ−1
n,A∗
n=γnYn
112Yn
121γ−1
n−1
Yn+1
121Yn
112 =Yn
112Yn+1
121 =I,Yn
111 +Yn
1∗
22 =0
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
The h ee- e m ecu ence ela ion
I we call R=Yn+1(Yn)−1and deno ing
Yn(z) = I+1
zYn
1+On(1/z2)znI0
0z−nI,z→∞
Yn+1(z) = zI +Yn+1
111 −Yn
111 −Yn
112
Yn+1
121 0Yn(z)
γ∗
n−1γn−1= − 1
2πiYn
121 = − 1
2πiYn−1
1−1
12
αn=Yn
111 −Yn+1
111,βn=Yn
112Yn
121
Bn=γnYn
111 −Yn+1
111γ−1
n,A∗
n=γnYn
112Yn
121γ−1
n−1
Yn+1
121Yn
112 =Yn
112Yn+1
121 =I,Yn
111 +Yn
1∗
22 =0
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
The h ee- e m ecu ence ela ion
I we call R=Yn+1(Yn)−1and deno ing
Yn(z) = I+1
zYn
1+On(1/z2)znI0
0z−nI,z→∞
Yn+1(z) = zI +Yn+1
111 −Yn
111 −Yn
112
Yn+1
121 0Yn(z)
γ∗
n−1γn−1= − 1
2πiYn
121 = − 1
2πiYn−1
1−1
12
αn=Yn
111 −Yn+1
111,βn=Yn
112Yn
121
Bn=γnYn
111 −Yn+1
111γ−1
n,A∗
n=γnYn
112Yn
121γ−1
n−1
Yn+1
121Yn
112 =Yn
112Yn+1
121 =I,Yn
111 +Yn
1∗
22 =0
Manuel Domínguez de la Iglesia The RH p oblem o OMP

P elimina ies
The RH p oblem o OMP
An example
The ke nel
Le (Pn)nan o hono mal amily. Then we ha e
Kn(x,y) =
n−1
X
j=0
P∗
j(y)Pj(x) = P∗
n−1(y)AnPn(x) − P∗
n(y)A∗
nPn−1(x)
x−y
This ke nel has he ollowing p ope ies
1Kn(x,y) = K∗
n(y,x)
2Kn(x,y) = RRKn(s,y)W(s)Kn(x,s)ds
We also ha e ha
Kn(x,y) = 1
2πi(x−y)0I(Yn)−1
+(y)(Yn)+(x)I
0
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
The ke nel
Le (Pn)nan o hono mal amily. Then we ha e
Kn(x,y) =
n−1
X
j=0
P∗
j(y)Pj(x) = P∗
n−1(y)AnPn(x) − P∗
n(y)A∗
nPn−1(x)
x−y
This ke nel has he ollowing p ope ies
1Kn(x,y) = K∗
n(y,x)
2Kn(x,y) = RRKn(s,y)W(s)Kn(x,s)ds
We also ha e ha
Kn(x,y) = 1
2πi(x−y)0I(Yn)−1
+(y)(Yn)+(x)I
0
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
The ke nel
Le (Pn)nan o hono mal amily. Then we ha e
Kn(x,y) =
n−1
X
j=0
P∗
j(y)Pj(x) = P∗
n−1(y)AnPn(x) − P∗
n(y)A∗
nPn−1(x)
x−y
This ke nel has he ollowing p ope ies
1Kn(x,y) = K∗
n(y,x)
2Kn(x,y) = RRKn(s,y)W(s)Kn(x,s)ds
We also ha e ha
Kn(x,y) = 1
2πi(x−y)0I(Yn)−1
+(y)(Yn)+(x)I
0
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
A di e en ial equa ion
We conside weigh ma ices o he o m
W(x) = ρ(x)T(x)T∗(x),
whe e Tsa is ies T0(x) = G(x)T(x).
Conside
Xn(z) = Yn(z)J(z) = Yn(z)ρ(z)1/2T(z)0
0ρ(z)−1/2(T(z))−∗
Then d
dz Xn(z)Xn(z)−1is en i e and nea in ini y i beha es like
I+Y1
z+O(z−2) 1
2
ρ0(z)
ρ(z)+G(z)0
0−1
2
ρ0(z)
ρ(z)−G(z)∗!I−Y1
z+O(z−2)
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
The Lax pai
Le Ynbe he solu ion o he RH o Wand conside
Xn(z) = Yn(z)e−z2/2eAz 0
0ez2/2e−A∗z
Xn+1(z) = zI +Yn+1
111 −Yn
111 −Yn
112
Yn+1
121 0Xn(z)
d
dz Xn(z) = −zI +A2Yn
112
−2Yn
121 zI −A∗Xn(z)
Compa ibili y condi ions
2(βn+1−βn) = Aαn−αnA+I
αn=1
2(A+ (γ∗
nγn)−1A∗(γ∗
nγn))
Manuel Domínguez de la Iglesia The RH p oblem o OMP

P elimina ies
The RH p oblem o OMP
An example
The Lax pai
Le Ynbe he solu ion o he RH o Wand conside
Xn(z) = Yn(z)e−z2/2eAz 0
0ez2/2e−A∗z
Xn+1(z) = zI +Yn+1
111 −Yn
111 −Yn
112
Yn+1
121 0Xn(z)
d
dz Xn(z) = −zI +A2Yn
112
−2Yn
121 zI −A∗Xn(z)
Compa ibili y condi ions
2(βn+1−βn) = Aαn−αnA+I
αn=1
2(A+ (γ∗
nγn)−1A∗(γ∗
nγn))
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
Ladde ope a o s
Lowe ing ope a o
b
P0
n(z) = Ab
Pn(z) − b
Pn(z)A+2βnb
Pn−1(z)
The e o e
βn=1
2(nI +an,n−1A−Aan,n−1)
Raising ope a o
b
P0
n(z)=−2b
Pn+1(z) + 2zb
Pn(z) + Ab
Pn(z) − b
Pn(z)A−2αnb
Pn(z)
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
Ladde ope a o s
Lowe ing ope a o
b
P0
n(z) = Ab
Pn(z) − b
Pn(z)A+2βnb
Pn−1(z)
The e o e
βn=1
2(nI +an,n−1A−Aan,n−1)
Raising ope a o
b
P0
n(z)=−2b
Pn+1(z) + 2zb
Pn(z) + Ab
Pn(z) − b
Pn(z)A−2αnb
Pn(z)
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
Second-o de di e en ial equa ions
In oduce he ollowing di e en ial/di e ence ope a o s
R1=∂1+∂0A,L1=2βnE−1+AE0
R2=∂1+∂0(A−2zI),L2= −2E1+ (A−2αn)E0
∂k=dk
dzk,Ek (n) = (n+k)
The ladde ope a o s a e equi alen o
b
Pn(z)R1=L1b
Pn(z),and b
Pn(z)R2=L2b
Pn(z)
The e o e, he OMP b
Pnsa is y wo second-o de di e en ial equa ions
b
Pn(z)R1R2=L1b
Pn(z)R2=L1L2b
Pn(z),
b
Pn(z)R2R1=L2b
Pn(z)R1=L2L1b
Pn(z)
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
Second-o de di e en ial equa ions
In oduce he ollowing di e en ial/di e ence ope a o s
R1=∂1+∂0A,L1=2βnE−1+AE0
R2=∂1+∂0(A−2zI),L2= −2E1+ (A−2αn)E0
∂k=dk
dzk,Ek (n) = (n+k)
The ladde ope a o s a e equi alen o
b
Pn(z)R1=L1b
Pn(z),and b
Pn(z)R2=L2b
Pn(z)
The e o e, he OMP b
Pnsa is y wo second-o de di e en ial equa ions
b
Pn(z)R1R2=L1b
Pn(z)R2=L1L2b
Pn(z),
b
Pn(z)R2R1=L2b
Pn(z)R1=L2L1b
Pn(z)
Manuel Domínguez de la Iglesia The RH p oblem o OMP

P elimina ies
The RH p oblem o OMP
An example
Second-o de di e en ial equa ions
In oduce he ollowing di e en ial/di e ence ope a o s
R1=∂1+∂0A,L1=2βnE−1+AE0
R2=∂1+∂0(A−2zI),L2= −2E1+ (A−2αn)E0
∂k=dk
dzk,Ek (n) = (n+k)
The ladde ope a o s a e equi alen o
b
Pn(z)R1=L1b
Pn(z),and b
Pn(z)R2=L2b
Pn(z)
The e o e, he OMP b
Pnsa is y wo second-o de di e en ial equa ions
b
Pn(z)R1R2=L1b
Pn(z)R2=L1L2b
Pn(z),
b
Pn(z)R2R1=L2b
Pn(z)R1=L2L1b
Pn(z)
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
We will p oo he ollowing:
Bo h equa ions a e equi alen
They a e also equi alen o he second-o de di e en ial equa ion o
hype geome ic ype
b
P00
n(z) + b
P0
n(z)(2A−2zI) + b
Pn(z)(A2−2J) = (−2nI +A2−2J)b
Pn(z)
The i s one is
b
P00
n(z) + 2b
P0
n(z)(A−zI) + b
Pn(z)A(A−2zI) =
−4βnb
Pn(z) + 2βn(A−2αn−1)b
Pn−1(z) − 2Ab
Pn+1(z) + A(A−2αn)b
Pn(z)
And he second
b
P00
n(z) + 2b
P0
n(z)(A−zI) + b
Pn(z)(A−2zI)A−2b
Pn(z) =
−4βn+1b
Pn(z) + 2(A−2αn)βnb
Pn−1(z) − 2Ab
Pn+1(z) + (A−2αn)Ab
Pn(z)
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
We will p oo he ollowing:
Bo h equa ions a e equi alen
They a e also equi alen o he second-o de di e en ial equa ion o
hype geome ic ype
b
P00
n(z) + b
P0
n(z)(2A−2zI) + b
Pn(z)(A2−2J) = (−2nI +A2−2J)b
Pn(z)
The i s one is
b
P00
n(z) + 2b
P0
n(z)(A−zI) + b
Pn(z)A(A−2zI) =
−4βnb
Pn(z) + 2βn(A−2αn−1)b
Pn−1(z) − 2Ab
Pn+1(z) + A(A−2αn)b
Pn(z)
And he second
b
P00
n(z) + 2b
P0
n(z)(A−zI) + b
Pn(z)(A−2zI)A−2b
Pn(z) =
−4βn+1b
Pn(z) + 2(A−2αn)βnb
Pn−1(z) − 2Ab
Pn+1(z) + (A−2αn)Ab
Pn(z)
Manuel Domínguez de la Iglesia The RH p oblem o OMP
P elimina ies
The RH p oblem o OMP
An example
We will p oo he ollowing:
Bo h equa ions a e equi alen
They a e also equi alen o he second-o de di e en ial equa ion o
hype geome ic ype
b
P00
n(z) + b
P0
n(z)(2A−2zI) + b
Pn(z)(A2−2J) = (−2nI +A2−2J)b
Pn(z)
The i s one is
b
P00
n(z) + 2b
P0
n(z)(A−zI) + b
Pn(z)A(A−2zI) =
−4βnb
Pn(z) + 2βn(A−2αn−1)b
Pn−1(z) − 2Ab
Pn+1(z) + A(A−2αn)b
Pn(z)
And he second
b
P00
n(z) + 2b
P0
n(z)(A−zI) + b
Pn(z)(A−2zI)A−2b
Pn(z) =
−4βn+1b
Pn(z) + 2(A−2αn)βnb
Pn−1(z) − 2Ab
Pn+1(z) + (A−2αn)Ab
Pn(z)
Manuel Domínguez de la Iglesia The RH p oblem o OMP