Hua C., Wong R. Differential Equations and Asymptotic Theory in Mathematical Physics
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22
Case
111.
x
E
(-1,1),
so
la1
=
IpI,
cr
#
p.
In this case
a
comparison
function is
(1
-
t/p)-’/2
(1
-
t/a)-1/2
(1
-
Ct!/P)1/2
.
+
dt)
=
(1
-
P/a)1/2
With
,B
=
e-i6,
cr
=
eie,
so
x
=
cose, and we obtain
Therefore
einOei9/2
e--inOe--iO/2
pn(cose)
w
+
fidm
fid?%ETe’
or
equivalently
In fact (5.10) is uniform in
6
for
c
5
0
5
T
-
E
and
E
E
(0,~).
when we know the singular part of a generating function.
the orthogonal expansion
The above example is typical. Darboux’s asymptotic method is useful
Next we discuss Poisson kernels. Let
f
E
L2(E,p).
Then
f
will have
00
f(x)
N
c
fnpn(x),
fn
=
/
f(x)pn(z)dp(x),
(5.11)
and
{p,}
are orthonormal with respect to
p
and complete in
L2(E,
p).
The
expansion (5.11) is an expansion in
L2(E,
p).
We try, to write an integral
representation for the orthogonal series. Let us explore
a
formal approach,
and write
n=O
E
00
00”
It is not clear how to produce
f
from the above right-hand side unless the
integral is
s,
S(y
-
x)f(y)dy,
6
being
a
Dirac delta function.
To
remedy
23
this we introduce a damping factor into the series by replacing
pn(z)pn(y)
by
pn(z)pn(y)tn,
0
<
t
<
1
and hope for
(5.12)
with
Pt(z,y)
defined in (5.2). The result is that (5.12) holds under some
general conditions and the analysis will be greatly simplified if
P,(z,y)
is
nonnegative for
z,
y
E
E,
and
t
E
(0,l). For Hermite polynomials the
Poisson kernel is
Formula (5.13) is called the Mehler formula. Clearly
Pt(z,
y)
2
0
for
z,
y
E
R,
0
<
t
<
1. Indeed (5.12) is the real inversion formula
for
the Gauss-
Weierstrass transform
63.
Poisson kernels are bilinear generating functions.
The Poisson kernel for Laguerre polynomials is
a
multiple of
where
F
is
00
Z"
F(z)
=
n=O
c
n!
(a
+
(5.14)
(5.15)
and is related to the modified Bessel function
I,,
see
58,
".It is clear that
the closed form (5.14), known in the literature
as
the Hille-Hardy formula,
exhibits the positivity of the Poisson kernel for Laguerre polynomials when
t
E
[0,
1). Motivated by the forms (5.13) Sarmanov and Bratoeva
53
con-
sidered bilinear series such that
(5.16)
for
c,
E
R.
Theorem
5.2.
(Sarmanov and Bratoeva)
If
C,"=,
ci
converges then the
condition
(5.16)
holds for all
(2,
y)
E
JR
x
R
if
and only
if
there is a positive
measure
p
such that
1
c,
=
1,
t"dp(t),
n
=
0,1,.
. . .
(5.17)
24
It is clear that the sequences
{cn}
characterized by Theorem
5.2
form a
convex set whose extreme points are the sequences
{t"},
for
t
E
(-1,l);
which correspond to measures having a single atom at
t.
Sarmanov
52
proved the corresponding theorem for the Laguerre series by requiring
(5.18)
for all sequences
{cn}
such that
C~=oc~
<
00,
and
a
>
-1.
Sarmanov's
theorem states that
(5.18)
holds for all
2
2
0,y
2
0
if and only if
c,
=
Jt
tn
dp(t)
with
p
a positive measure. Askey gave an informal argument
which helps explain these results. Here again the extreme points of the set
of sequences
{cn}
is
{rn
:
0
5
T
<
1)
and establishe the positivity of the
series side in the Hille-Hardy formula
(5.14).
The series for the Poisson kernel for Jacobi polynomial is stated on page
102
of Bailey as
where
(5.20)
The representation
(5.19)
shows the positivity of the Poisson kernel for
z,y
E
[-I,
11,
t
E
[O,l)
in the full range of the parameters
a
>
-1,P
>
-1.
Bailey also states the kernel
which is positive for
z,
y
E
[-1,1],
t
E
[O,l)
with
a
>
-1,P
>
-1
but with
the additional constraint
a
+
/3
>
-1.
The question of proving the positivity of
Pt(z,
y)
when the polynomials
depend on parameters have been the subject of many research papers, see
the monograph by Askey
4,
the treatise by Gasper and Rahman
22
and the
references in them.
25
Observe that the derivation of (5.9) and (5.10) did not use the orthogo-
nality of
P,(x).
In fact we started with the recursive definition (5.3)-(5.4)
and applied Darboux's method. Indeed (5.9) and
(5.10)
show that
P,(x)
is oscillatory when
x
E
(-1,l)
and has exponential growth in
C
\
[-1,1].
The oscillatory nature of
P,
indicates that the zeros are dense in [-1,1].
Several mathematicians worked on the problem of determining the
asymptotic behavior of
p,(x)
for
n
large from the qualitative of the re-
currence coefficients
{a,}
and
{b,}
in (2.2). The model polynomials for
this analysis are the Chebyshev polynomials
of
the first kind
{T,(x)},
T,(COS
8)
=
cos
no,
(5.22)
xT,(x)
=
-Tn+1(Z)
1
+
--ZL-l(X).
1
2 2
(5.23)
The normalized weight function
is
(1
-
x')-'/'/n.
A
sample of results in
this direction is the following theorem of Nevai from
46.
Theorem
5.3.
Assume that
b,
4
0,
and
a,
4
0
in
such a way that
n(la,
-
1/21
+
Ibnl)
converges. Then {p,(x)} are orthonormal with
respect to a probability measure
p,
and
p'
is supported on [-1,1]. The
discrete part
of
p
is finite (may be empty) and lies outside
(-1,l).
Moreover
ca
holds
for
x
=
cos
8
for
0
<
8
<
n
and
'p
depend on
8
but
not on
n.
Note that the normalized weight function for
{T,}
is
(l-x')-'/'/n,
and
for
n
>
0,
{a
T,(x)}
are the orthonormal polynomial. Therefore (5.24)
reduces the asymptotics of general
{p,(x)}
to the asymptotics of T,(cos
8)
with
8
shifted by
'p(8)ln
+
o(l/n).
In other words
p,(x)
behaves
as
if
p,
is
fi
T,.
Another theorem of Ne~ai~~ is the following.
Theorem
5.4.
Assume that
Cf"(lu,
-
1/21
+
Ibnl)
<
00.
Then
p'(x)
is supported on [-1,1] and
p
may have a discrete part outside
(-1,l).
Moreover with
x
=
cos
8,0
<
8
<
7r,
the
limiting
relation
holds.
26
We next derive an integral representation for
a
general Jacobi polyno-
mial. This representation has the form
s,
enf(z)g(z)dz,
so
one can apply
the method
of
steepest descent to this integral and determine the large
n
asymptotics of the polynomials. The details
of
the steepest descent method
are in Wong’s article in these proceedings,
65.
The Jacobi polynomials
{
P?”’(x)}
have the series representation
(-n)k(a+P+n+l)k
(I)n.
1
--Ic
(5.26)
+
l)n
k!(a
+
1)k
pp’P’(,)
=
n!
The Pfaff-Kummer transformation is
provided that the series on both sides are convergent.
Now
apply
(5.27)
to
(5.26)
to get
we establish
The use of the binomial theorem and Cauchy’s theorem leads to
dz
(5.29)
where
C
is
a
closed contour
so
that
-2(z
f
1)-’
be in the exterior
of
C.
The integral representation
(5.29)
has
a
form suitable for the application
of the steepest descent method. In fact the integral representation
(5.29)
can be used to derive the generating function
P(a’p’(x)
n
=
&
s,(1+
(x
+
1)2/2)”+”(1+
(x
-
1)Z/2)”+PF,
00
2a-kpR-1
(5.30)
(1
-
t
+
R)
a
(1
+
t
+
R)
P
’
c
PpP)
(z)t”
=
n=O
where
R
=
dl
-
2xt
+
t2,
58.
A
different proof is in
50.
One can apply
Darboux’s method to the generating function
(5.30)
and determine the
asymptotic behavior of
Pp”)(x)
for large
n
and fixed
x
in different parts
27
of the complex x-plane.
The interested reader is strongly encouraged to
apply Darboux's method to
(5.30),
or apply the steepest descent method
to
(5.29)
and establish the following theorem.
Theorem
5.5.
Let
a
and
,8
be real. Then
a+P
FpP'(x)
M
(x
-
1)-"/2(x
+
1)-P/2
[(x
+
1)1/2
+
(x
-
1)'/2]
71+1/2
(5.31)
[x
+
(2
-
1)'/2]
,
1 1
X-
(x2
-
1y4
for
z
E
C
\
[-1,1]. On the other hand
if0
<
6
<
T,
then
PpP)(coSo)
=
-
k(Q)
cos(N6
+
y)
+
0
(n-312)
,
d=
(5.32)
where
k(6)
=
(~in(e/2))-~-'/~(cos(e/2)>-P-~/~,
7r
(5.33)
N
=
n
+
(a+P+
1)/2,
y
=
-(a
+
1/2)-.
2
When
a
=
,8,
the generating function
(5.30)
reduces to a generating func-
tion for the ultraspherical polynomials since
(5.34)
We made no attempt to cover the application of Riemann-Hilbert tech-
niques to deriving large degree asymptotics of orthogonal polynomials. This
is a powerful technique and is covered in Deift's monograph
17.
The state
of the art of this approach is covered in the recent survey article
of
Arno
Kuijlaars
39.
We conclude this section by describing a technique to recover the or-
thogonality measure(s)
p
when the recurrence relation
(2.2)
is given.
Theorem
5.6.
Let
po
=
1, pl
=
(x
-
bo)/al, and
bn
E
R,n
2
0,an
>
0,n
>
0,
and assume that {pn(x)}
is
generated
by
xpn(x)
=
an+lPn+l(x)
+
bnpn(x)
+
anpn-l(x),
n
>
0.
Then there exists a probability measure
p
such that ~,pm(x)pn(x)
dp(x)
=
hm,n
*
Theorem
5.6
gives a converse to the fact that orthogonal polynomials
satisfy three term recurrence relations. We shall refer to Theorem
5.6
as the
spectral theorem for orthogonal polynomials. Some authors call it Favard's
28
theorem but it was in the literature before Favard’s paper was written. The
probability measure
p
whose existence is guaranteed by Theorem
5.6
may
not be unique.
We now introduce
a
second solution
{p:(z)}
to
(2.2)
by the initial con-
ditions
and requiring
p:
satisfies
(2.2).
Theorem
5.7.
(Markov).
Assume that the recursion coefficients
{a,}
and {b,} are bounded. Then the measure
in
Theorem
5.4
is unique, sup-
ported on a compact set
E
c
R
and satisfies
(5.36)
Moreover, the convergence
in
(5.36)
is uniform on compact subsets
of
@\E.
Note that if
p
is unique then
(5.36)
continues to holds but
E
=
supp(p)
may not be bounded. The convergence in
(5.36)
is still uniform on compact
subsets of
C
\
E.
The Perron-Stieltjes inversion formula is
F(z)
=
!k@
z-t
(5.37)
if and only if
(5.38)
J1”
F(u
-
if)
-
F(u
+
if)
p(t)
-
p(s)
=
lim
du.
€’Of
2lri
Theorem
5.7
is an instance where the asymptotics of polynomials give
crucial information about their orthogonality measure.
6.
Applications
This section is devoted to some problems which naturally lead
to
three
term recurrence relations and determining the orthogonality measure
p
or
its support sheds some light on the original problem.
The first example is birth and death processes. Consider
a
stationary
Markov chain whose state space is the nonnegative integers. Let
p,,,(t)
be the transition probability to go from state
m
to state
n
in time
t.
Here
29
pm,n(t)
does not depend on the initial time. We assume that
Xmt
+
o(t)
,
n=m+l,
pm,n(t)
=
pnt
+
o(t),
n=m-1,
(6.1)
{
1
-
(Am
+
pm)t
+
o(t)
7~
=
m,
and
pm,n(t)
=
o(t),
if
Im
-
n1
>
1.
The birth rates
Am
and death rates
pm
satisfy
po
2
0,
pm
>
0,
m
>
0,
and
Am
>
0
for
m
2
0.
The Chapman-Kolmogorov equations which describe this process are
Ijm,n(t)
=
L-lPrn,n-l(t)
+
Pn+lPm,n+l(t)
-
(An
+
pn)pm,n(t),
(6.2)
Ijm,n(t)
=
Xmpm+l,n(t)
+
pmPm-l,n(t)
-
(Am
+
prn)pm,n(t),
(6.3)
where
jl
means
%.
Let us attempt to solve (6.2)-(6.3) by the separation of
variables
prn,n(t)
=
f(t)QmFn.
Therefore
f'(t)/f(t)
is independent oft,
so
set it equal to
-x.
Now (6.2)
indicates that
Fn
depends on
z,
so
we denote it by
Fn(x).
jFrom (6.2) we
get
-xFn(x)
=
Xn-lFn-l(x)
-
(An
+
pn)Fn(x)
+
pn+lFn+l(x),
(6.4)
with
Fo(x)
arbitrary,
so
we take
F-l(x)
:=
0
and
Fo(z)
=
1.
(6.5)
(6.6)
Similarly
Q-l(x)
=
0
and
QO(z)
=
1,
and
-xQn(x)
=
XnQn+l(x)
-
(An
+
~n)Qn(x)
+
~nQn-l(x).
Indeed (6.4), (6.6) and the initial conditions imply
Thus we have determined a solution and we multiply by separation con-
stants and sum or integrate over all choices
of
x.
Therefore
30
As
t
4
0'1
~1~2..
.
pnPm,n(t)
+
dm,n.
Thus
1
Fm(x)Fn(x)dp(x)
=
tmbm,n,
(6.9)
so
it seems that
{F,}
are orthogonal with respect to
p.
At
t
4
co
our
pm,n(t)
cannot have exponential growth, hence
x
E
(0,
co),
and we obtain
pm,n(t)
=
-
LW
e-"tFm(z)Fn(x)dp(x).
(6.10)
Observe that (6.10) gives the solution in a factored form and it is not
difficult to analyze the large
t
behavior of
p,,,(t)
from (6.10).
It turns out the
p
must be a positive, hence
{F,}
are orthogonal with
respect to
p.
Furthermore the application
of
the chain sequence techniques
of
97
to (6.5) and (6.6) show that all zeros of
F,
and
Qn
lie in
(0,
co).
The integral representation (6.10) has been established by Karlin and
McGregor in
35,
36.
Later Karlin and McGregor
38
studied random walks
on the state space of the nonnegative integers and defined another sequence
of orthogonal polynomials. The random walk polynomials are generated by
Ro(x)
=
1,
Ri(x)
=
(1
+
po/A0)2,
(6.11)
5'm
Since the recurrence coefficients in (6.12) are bounded. Theorem
5.7
shows
that
{R,(x)}
are orthogonal with respect to
a
measure supported on a
compact set. Theorem 7.5 proves that all the zeros of
R,
belong to
(-1,
l),
for all
n.
From this fact, one can prove that
{Rn(x)}
are orthogonal with
respect to a measure supported on a subset of
[-I,
11.
The Laguerre polynomials are
{F,}
polynomials when
p,
=
n,A,
=
n
+
Q
+
1.
The ultraspherical polynomials
are
multiples of random walk
polynomials with
p,
=
n,
A,
=
n
+
2v. In fact the Jacobi, Laguerre, Hahn,
Meixner, the Charlier polynomials, or there various special cases, are birth
and death process polynomials or random walk polynomials.
The random walk polynomials corresponding to
An=cn+p, p,=n, p>O,c>O
(6.13)
are very interesting. The case
c
=
1
is the ultraspherical polynomials. In
the case
c
=
0,
the polynomials
{R,(x)}
are orthogonal with respect to
a
discrete measure. This measure together with some explicit represen-
tations were found in 1958, independently and using completely different
31
techniques, by Carlitz
l4
and, Karlin and McGregor
37.
In 1984 Askey
and Ismail analyzed the full model (6.13). They proved that the orthog-
onality measure is absolutely continuous on [-2&/(c
+
l),
2&/(c
+
l)].
When
c
#
1
it has an infinite discrete part supported in
[-1,
-2&/(c
+
l)],
[2&/(c
+
l),
11.
The points f2&/(c
+
l),
do not support positive
masses but are the only limit points of the points supporting positive
point masses. Moreover contains explicit and asymptotic formulas for
the polynomials and their generating functions. The techniques used
axe
Markov’s theorem, Darboux’s method,
as
well as standard special function
techniques.
Another problem which leads to orthogonal polynomials is the
so
called
J-Matrix Method. The idea is
to
start with a Schrodinger operator on
R,
d2
dx2
T
:=
--
f
V(X).
(6.14)
The operator
T
is densely defined on
L2(R)
and is symmetric. The idea is
to find a complete orthonormal basis in the domain of
T
such that
T
the
matrix representation of
T
in
{pn(x)}
is tridiagonal, that is
p,TPndx
=
0,
if
Im
-
n1
>
1.
J,
-
We now diagonalize
T,
that is let
T$E
=
E$E
and assume
n=O
(6.15)
Observe that
E$n(E)
=
(E$E,
pn)
=
(T$E,
pn)
=
(TPn-l$n-l+ Tpn+n
+
Tpn+l$n+lpn, pn).
Therefore
E$n(E) =$n+l(E)(TPn+l, pn)
+
$n(E)(Tpni
~n)
+
+n-l(E)(Tpn-l,
~n).
(6.16)
If
(Tpn,pn*l)
#
0,
then (6.16) is a recurrence relation for a sequence of
polynomials
{&(E)}.
It
is
easy to see that (6.16) can be reduced to
(2.2)
if and only if
(TP,
Pn-l)(Tpn-l, pn)
>
0.
This is indeed the case since
(Tv,,pn-l)
=
(pn,Tpn-l)
=
(Tpn-l,pn).
Heller, Reinhardt and Yamani
24
introduced this method and applied it
to