Ismail M., Koelink E. (editors) Theory and Applications of Special Functions
Подождите немного. Документ загружается.
Hilbert space asymptotics for Orthogonal Polynomials
373
and asymptotics function
DiL)
(x)
=
[2i sin(x/2)1-~
[2
COS(X/~)]-~C~~)(X)
(2.31)
x
exp(ix[n+
(M
+
N)/2])
+
(x
-+
-x).
The following lemma shows that the remainder function
sLL)
(x)
=
Dn (x)
-
DLL)
(x) (2.32)
has exponential decay in
L~
sense
as
L
--+
oo.
Lemma
2.2.
Assuming
(2.20)
and d
E
(0,
d+), we have
with the
bound uniform for
n
E
N.
Proof.
By (2.17) and (2.30)-(2.32) we have
By virtue of boundedness of w,(x) and the estimates (2.6), the lemma
follows.
0
From (2.32)-(2.33) we deduce
To obtain exponential decay of
IIPn
-
Drill,
therefore, we need only es-
timate
Pn
-
Dn
I I
(2n-1)11.
The significance of the choice
L
=
2n
-
1
is that it ensures that the only exponentials present in the expansion
(L)
of
c,
(x) exp(inx) are those occurring in the expansion of Pn(x). For
M
=
N
=
0, it is therefore immediate that we have equalities
(x)
=
Bn
(x)
+
cnmBm (x)
374
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
for certain
G,,
dn,
E
R.
Now the functions on the rhs are manifestly
bounded on [0,
TI,
whereas the functions on the lhs have a divergence
as
x
+
0 and/or
as
x
+
T
whenever
M
>
1
and/or
N
>
1,
cf. (2.31).
Thus the restriction to We in the next lemma is essential.
Lemma
2.3.
Assume w(x) belongs to We (2.18).
Then the equalities
(2.36) hold true.
Proof. For w
E
Wo.o we have already established (2.36). Consider next
w
E
W~J.
Then (2.'30)-(2.31) yield
-
sin(n
+
l)x
L
0AL)
(x)
=
+
xat
sin(n
+
1
-
1)x
sin x sin x
1=1
Since we have
sin(k
+
l)x
sin x
=
Bk(4
+
C
tkl& (x),
l<k
we deduce (2.36).
Now let
w
E
Wllo. Then we obtain
sin(n
+
1/2)x
L
sin(n
+
112
-
1)s
0LL)
(x)
=
sin(x/2)
+
Cai:
sin(x/2)
'
1=1
and since
sin(k
+
1/2)x
sin(x/2)
=
&(x)
+
x
sklB1
(I)
9
l<lc
we obtain again (2.36). Finally, for
w
E
Wo,l we get
so (2.36) follows from
Hence the lemma is proved.
We are now prepared to obtain the principal result of this paper.
Theorem
2.4.
Let
w
E
We and d
E
(0,
d+). Then we have the estimates
llPn
-
Drill
=
O(exp(-2nd)))
n
+
00,
(2.43)
Hilbert space asymptotics for Orthogonal Polynomials
375
~n
-
1
=
O(exp(-2nd)), n
-t
oo.
(2.44)
Proof.
By virtue of Lemma 2.3 we may invoke the expansions (2.36),
which we rewrite as
cf. (2.32). Taking the inner product with pnPn and using (2.22), we
obtain
-
I
+
Pn
(pn,
SL~~-'))
.
Pn
-
(2.46)
By the Schwarz inequality and Lemma 2.2, we have
Now pn is positive, so in view of the upper bound (2.28) we may deduce
Hence the estimate (2.44) follows.
As a consequence of (2.44) and (2.26), we now obtain an O(exp(-nd))-
bound on
11
Pn
-
Dn
11.
To arive at the sharper bound (2.43), we introduce
d'
E
(d, d+) .
(2.49)
Now we take the inner product of (2.45) with
Pi
for
I
<
n, which yields
due to (2.22)
0
=
dnr
+
(PI,
sL~~-')
(2
.50)
From the Schwarz inequality and (2.33) with d
+
d' we obtain
Hence we have
Telescoping and using (2.44) with d
+
d',
we now get
Finally, recalling (2.49), we deduce (2.43).
0
376
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
3.
F'urt her developments
In this section we derive some corollaries of the above results and
consider the weight function class
WM,N
with
M
and/or
N
greater than
1. First, we obtain the asymptotics of the recurrence coefficients an,
bn
in (1.5).
Proposition
3.1.
Let
w
E
We
and d
E
(0, d+). Then we have
bn
=
O(exp(-2nd))) n
-+
oo.
(3.2)
Proof. From the relations (1.3)-(1.5) it is easily seen that
Therefore, (3.1) is clear from (2.44). To prove (3.2)) we begin by noting
that Dn(x) (1.8) satisfies the recurrence
Now the recurrence (1.5) entails
where
v
denotes multiplication by cos x. Telescoping, we obtain
Using (3.4) and then (2.22)) we get
Finally, using the Schwarz inequality and the estimates (2.43)) (2.27))
we deduce (3.2).
0
Next, we consider the special case where the function
f
(2)
defining
~(x) (recall (2.3)) is a polynomial.
Proposition
3.2.
Let
w
E
We
and suppose that in the c,-expansion
(2.4) we have
a:
=
O,
k
>
L, L
E
N.
(3.8)
Then we have
Pn(x)
=
D,(x), 2n
2
L
+
I.
(3.9)
Proof. From (2.30)-(2.32) we deduce
Hilbert space asymptotics for Orthogonal Polynomials
377
Thus we have
s~~~-')(x) =0, 2n2 L+1, (3.11)
so that (2.45) yields
Taking the inner product with Pl,l
5
n, we deduce from (2.22) that
dnl
=
0 for
1
<
n, and that pn
=
1
for
I
=
n. Hence (3.9) follows.
For w in our class Wolo, the result contained in this proposition can
already be found in (Szegd, 1975)) by looking rather hard in Section 12.4
('Bernstein-Szegd polynomials'). In words, it says that, for w
E
We cor-
responding to a polynomial
f
(z)
of degree L, the dominant asymptotics
function Dn(x) coincides with Pn(x) for 2n
2
L+
1.
As we have pointed
out before, no such result can hold for w
E
W
\
We, since Pn(x) is
bounded on (0, T), whereas in that case D,(x) is unbounded. Even so,
in view of the results (2.21)-(2.29) (which are valid for all w
E
W), one
might be inclined to guess that the sequence of numbers
converges exponentially to
0
as n
+
oo.
As we show next, this is not
the case.
Proposition
3.3.
Let w
E
W
\
We. Then we have
Proof.
To ease the exposition, we detail the proof for the case w
E
WO,J
and then indicate how to proceed in general. We assume {c,)
E
l2 so as
to arrive at a contradiction.
Consider the identification map
I
:
3-10,~
=
L~([o,
TI,
w(x)dx)
-+
3-10,~
=
L~
([0,
TI,
w,(x)dx)
,
(3.15)
F(x)
H
2 cos(x/2) F(x).
Recalling (2.17)) we see that
I
is an isometric isomorphism. Using ob-
vious notation, we therefore have
Next, recalling (2.19)) we see that we may view (ID,) (x) as the dom-
inant asyrnptotics function
D~+~
(x) corresponding to the polynomial
378
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
pn+1(x) in
',Yo,o
that arises from w,(x). Doing so, it follows from the
exponential decay of
f!n+l
-
Dn+1
(which we proved in Section 2) and
II
-
II
the assumption
{k)
E
l2 that the sequence
is in 12.
We are now in the position to invoke a known completeness result,
cf. Theorem
A
on p. 72 of Higgins' monograph (Higgins, 1977). Since
I
is unitary, the sequence {IPn)z=o yields an orthonormal base in ',Yo,o.
Since the sequence {An)~=o is in Z2(IV), the pertinent completeness result
w
says that the sequence
{a+;}
is complete in This yields the
n=O
desired contradiction, since
Po
is orthogonal to
pl,
p2,
.
.
.
Once this special case is understood, it will be clear how to proceed
for any
WM,~
with M and/or N greater than 1: one can identify
WM,~
with one of the four spaces in
We
(2.18) (by reducing
M
and N modulo
2), and use (2.19)
as
in the special case (M, N)
=
(0,2) to arrive at a
contradiction.
0
Next, we note that thanks to (2.26) and (2.11)) we have the equiva-
lence
lim cn
=
0
H
lim pn
=
1.
n+w
n+w
(3.18)
Consider now the weight function integral
Evidently, we have
Po(z)
=
Po
=
I(W)-~/~,
so using (3.3) we obtain
From this we readily infer a second equivalence
lirn
/in
=
1
o
lim an
=
1,
lirn
n
ay2
=
I(u).
n-+w
n+m
nhw
(3.22)
j=O
Hilbert space asymptotics for Orthogonal Polynomials
379
At this point we insert a parenthetical remark. For the class We of
weight functions, we have already shown that pn goes to 1. The class
contains in particular the weight function w~w (1.11)) for the choice
of parameters (1.12) yielding the Askey-Wilson integral I(wAw) given
by Eq. (6.1.1) in the monograph by Gasper and Rahman (Gasper and
Rahman, 1990). Since the self-adjoint recurrence coefficients an,AW are
explicitly known, our results yield the explicit formula for I(wAW) as a
corollary. Quite likely, this relation between I(w) and the aj-product has
been noticed before for the Askey-Wilson case, but we have not found
this in the literature.
Of course, the normalization (2.1) of our weight functions is critical
for the relations just pointed out. Indeed, when we switch from a given
w(x)
E
W to Xw(x),
X
>
0, then we should multiply
Pn
by to
retain unit norm. Thus the coefficients pn change to ~-'/~p,, whereas
the recurrence coefficients a,,
bn
are invariant.
To elaborate on the normalization issue, let us
fix
c,
E
C,
and consider
Clearly, we have
Now the polynomials P,,R(x) obviously converge to the polynomials
Pnll(x) for
R
1
1,
so the natural expectation is that the sequence p,,l
still converges to
1
as
n
-+
oo.
Unfortunately, there seems to be no short
and simple way to control the pertinent interchange of limits.
In any case, the sequence p,,~ does have limit
1
for
n
--+
oo.
Equiva-
lently, we have
lim pn=
1,
QWE
W,
n+m
(3.25)
so that we also have
lim c,=O, lim an=l, lirn na~~=I(w), QwEW. (3.26)
n--too
n+m
n+oo
To substantiate this assertion, we invoke a limit theorem that can be
found in Szeg6's monograph, cf. Theorem 12.7.1 in (Szeg6, 1975). The
crux is that our normalization requirement (2.1) implies
380
THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS
(To verify this for w
E
Wo,0, use the substitution z
=
e-" and note
that
f
(z)
E
A
has a one-valued analytic logarithm for lzl
5
1
such that
ln(f (0))
=
0. To handle w
E
WM,~, the above approximation (3.24) can
be used.)
With the convergence question answered, it is natural to ask about
the rate of convergence for w
E
WM,~ with M or N greater than
1.
In
our last proposition we answer this for the simplest w(x) in each class.
Proposition
3.4.
Assume w
E
W
\
We corresponds to c,(x)
=
1.
Then
we have
an=1+O(n-2), n-+a, (3.28)
pn=l+O(n-l), n-a, (3.29)
Proof. Inspecting (2.17), we see that the functions Pn(x) are propor-
(M-1/2,N-112)
tional to the Jacobi polynomials
Pn
(cos x). Their self-adjoint
recurrence coefficients are given by
cf., e.g., (Koekoek and Swarttouw, 1994). Now a straightforward calcu-
lation yields
a;
=
1
+
C(M, ~)n-~
+
0
(n-3)
,
n
-r
oo,
(3.32)
C(M, N) -[M(M
-
1)
+
N(N
-
1)]/2. (3.33)
Hence (3.28) follows.
Since we have already seen that pn converges to
1,
we may invoke
(3.20)-(3.22), yielding
00
=
n
a:.
(3.34)
j=n
From this representation and (3.32) we readily deduce (3.29). Finally,
(2.26) and (3.29) entail (3.30).
0
It is natural to expect that the asymptotics (3.28)-(3.30) holds true
for all w
E
WM,~. After submission of this paper we learned that this is
indeed the case, as follows from work by Kuijlaars, McLaughlin, van Ass-
che and Vanlessen (Kuijlaars et al., 2003); moreover, we were informed
that Kuijlaars has obtained results that imply in particular uniform ex-
ponential decay of Pn(x)
-
Dn(x) on [0,
T]
for
w
E
We (Kuijlaars, 2003).
Both of these references make essential use of Riemann-Hilbert problem
techniques (Deift, 1999).
Hilbert space asymptotics for Orthogonal Polynomials
381
Acknowledgments
We are indebted to
M.
Ismail for several illuminating discussions and
comments. We would also like to thank
W.
van Assche for welcome infor-
mation on pertinent literature. Finally, we acknowledge useful comments
and suggestions from two referees, one of whom made us aware of the
references (Kuijlaars et al.,
2003)
and (Kuijlaars,
2003).
References
Askey, R. A. and Wilson, J. A. (1985). Some basic hypergeometric orthogonal poly-
nomials that generalize Jacobi polynomials. Mem. Amer. Math. Soc., 54(319).
Deift, P. (1999). Orthogonal polynomials and random matrices: A Riemann-Hilbert
approach, volume 3 of Courant Lecture Notes in Mathematics. New York University
Courant Institute of Mathematical Sciences, New York, NY.
Gasper, G. and Rahman, M. (1990). Basic Hypergeometric Series, volume 35 of Ency-
clopedia of Mathematics and its Applications. Cambridge Univ. Press, Cambridge.
Higgins, J. R. (1977). Completeness and basis properties of sets of special functions,
volume 72 of Cambridge Tracts in Mathematics. Cambridge Univ. Press, Cam-
bridge.
Ismail, M.
E.
H.
(1986). Asymptotics of the Askey-Wilson and q-Jacobi polynomials.
SIA M
J.
Math. Anal., 17: 1475-1482.
Ismail, M.
E.
H. and Wilson, J. A. (1982). Asymptotic and generating relations for
the q-Jacobi and
4(~3
polynomials.
J.
Approx. Theory, 36:43-54.
Koekoek, R. and Swarttouw, R. F. (1994). The Askey-scheme of hypergeometric or-
thogonal polynomials and its q-analogue. report 94-05, Delft University of Tech-
nology.
Kuijlaars, A.
B.
J.
(2003). Riemann-Hilbert analysis for orthogonal polynomials. In
Koelink,
E.
and van Assche, W., editors, Summer School on Orthogonal Polynomi-
als and Special Functions, Leuven 2002, Lecture Notes in Mathematics. Springer-
Verlag. To appear.
Kuijlaars, A.
B.
J., McLaughlin, K. T.-R., van Assche, W., and Vanlessen, M. (2003).
The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials
on [-I,
11.
math.CA/0111252. Preprint.
Ruijsenaars, S.
N.
M. (2002). Factorized weight functions vs. factorized scattering.
Comm. Math. Phys., 228:467-494.
Ruijsenaars, S.
N.
M.
(2003). Relativistic Lam6 functions: Completeness vs. polyno-
mial asymptotics. Indag. Math. N.S., ll(3). Special issue dedicated to Tom Koorn-
winder.
SzegB,
G.
(1975). Orthogonal polynomials, volume XXIII of Colloquium Publications.
American Mathematical Society, Providence, RI, fourth edition.
van Diejen,
J.
F.
(2003). Asymptotic analysis of (partially) orthogonal polynomials
associated with root systems. Intern. Math. Res. Notices, (7):387-410.
ABEL-ROTHE TYPE GENERALIZATIONS
OF JACOBI'S TRIPLE PRODUCT
IDENTITY
Michael Schlosser*
Institut fur Mathematik der Universitat Wien
Strudlhofgasse
4
A-1090
Wien
Austria
Abstract
Using a simple classical method we derive bilateral series identities from
terminating ones. In particular, we show how to deduce Ramanujan's
l$l
summation from the q-Pfaff-Saalschutz summation. Further, we
apply the same method to our previous q-Abel-Rothe summation to
obtain, for the first time, Abel-Rothe type generalizations of Jacobi's
triple product identity. We also give some results for multiple series.
Keywords: q-series, bilateral series, Jacobi's triple product identity, Ramanujan's
lGl
summation, q-Rothe summation, q-Abel summation, Macdonald
identities,
A,
series,
U(n)
series.
1.
Introduction
Jacobi's (Jacobi, 1829) triple product identity,
is one of the most famous and useful identities connecting number the-
ory and analysis. Many grand moments in number theory rely on this
result, such as the theorems on sums of squares (cf. (Gasper and Rah-
man, 1990, Sec. 8.11)), the Rogers-Ramanujan identities (cf. (Gasper
and Rahman, 1990, Sec.
2.7)),
or Euler's pentagonal number theorem
(cf. (Bressoud, 1999,
p.
51)). In addition to this identity, different exten-
sions of it, including Ramanujan's (Hardy, 1940) summation formula
*The author
was
supported by an APART grant of the Austrian Academy of Sciences
O
2005
Springer Science+Business Media, Inc.