The reproducing kernel structure arising from a combination of continuous and discrete orthogonal polynomials into Fourier systems

0.22 MB·English

by  Luis Daniel Abreu

7
0
0
2 The reproducing kernel structure arising from a combination
of continuous and discrete orthogonal polynomials into
n
a Fourier systems
J

5
Lu´ıs Daniel Abreu

]
A
In memory of Joaquin Bustoz
C
.
h
Abstract. Westudymappingpropertiesofoperatorswithkernelsdefinedvia
t
a acombination ofcontinuous anddiscreteorthogonal polynomials,whichpro-
m videanabstractformulationofquantum (q-)Fouriertypesystems. Weprove
Ismail´s conjecture regarding the existence of a reproducing kernel structure
[
behindthesekernels,byestablishingalinkwithSaitoh´stheoryoflineartrans-
3 formations in Hilbert space. The results are illustrated with Fourier kernels
v withultrasphericalweights,theircontinuousq-extensionsandgeneralizations.
0 Asabyproductofthisapproach,anewclassofsamplingtheoremsisobtained,
9 aswellasNeumanntypeexpansions inBesselandq-Besselfunctions.
1
1
0
6
1. Introduction
0
/ The Gegenbauer expansion of the two variable complex exponential in terms
h
t of the ultraspherical polynomials
a
m t −ν ∞
(1.1) eixt =Γ(ν) ik(ν+k)J (t)Cν(x)
: 2 ν+k k
v (cid:18) (cid:19) k=0
X
i
X has the remarkable feature of being at the same time an expansion in a Neumann
series of Bessel functions. The usefulness of this expansion was made very clear in
r
a a paper authored by Ismail and Zhang, where it was used to solve the eigenvalue
problem for the left inverse of the differential operator, on L2 spaces with ultras-
pherical weights [20]. The consideration of the q-analogue of this diagonalization
problem led the authors to extend Gegenbauer´s formula to the q-case. This task
required the introduction of a new q-analogue of the exponential, a two variable
function denoted by (x;t) which became known in the literature as the curly q-
q
E
exponential function, bearing the name from its notational convention. Ismail and
2000Mathematics Subject Classification. Primary42C15, 44A20; Secondary 33C45, 33D45,
94A20
Keywords and phrases. Reproducing kernels, q-Fourier series, orthogonal polynomials, basic hy-
pergeometricfunctions,samplingtheorems
Partial financial assistance by FCG, FCT post-doctoral grant SFRH/BPD/26078/2005 and
CMUC..
1
2 LU´ISDANIELABREU
Zhang´s formula is
(1.2)
(x;it)= t−ν(q;q)∞ ∞ ikqk2/4(1−qk+ν)J(2) (2t;q)C (x;qν q).
Eq ( qt2;q2)∞(qν+1;q)∞ (1 qν) ν+k k |
− k=0 −
X
The functions involved in this formula will be defined in section 4. Since its intro-
duction, the function was welcomed as a proper q-analogue of the exponential
q
E
function, since it was suitable to provide a satisfactory q-analogue of the Fourier
theory of integral transformations and series developments. This suitability was
made concrete by Bustoz and Suslov in [5], where the authors introduced the sub-
ject of q-Fourier series. Some of the subsequent research activity has been already
collected in a book [29]. Among recent developments not yet included in this
book, we quote the orthogonality relations for sums of curly exponential functions
[23], obtained using spectral methods, and the construction of a q-analogue of the
Whittaker-Shannon-Kotel´nikov sampling theorem [19]. The designation ”Quan-
tum” has appear often in recent literature on q-analysis, as in the monographs
[22] and [21]. This designation is very convenient, since q-special functions are
intimately connected with representations of quantum groups [6].
An abstract formulation designed to capture the essential properties of q-
Fourier type systems was proposed in [15] and we proceed to describe it here. Let
µ be a measure on the real line and p (x) a complete orthonormal system in
n
{ }
L2(µ). Let (t ) be a sequence of points on the realline and assume that r (x) is
j n
{ }
a discrete orthonormalsystem with orthogonality relation
∞
ρ(t )r (t )r (t )=δ
j n j m j mn
j=0
X
and with dual orthogonality
∞
δ
mn
r (t )r (t )= .
k n k m
ρ(t )
n
k=0
X
Assume alsothat the the system r (x) is complete in l2( ρ(t )δ ). Now define
{ n } j tj
a sequence of functions F (x) by
n
{ } P
∞
(1.3) F (x)= r (t )p (x)u
n k n k k
k=0
X
where u isanarbitrarysequenceofcomplexnumbersintheunitcircle. Thefol-
k
{ }
lowingtheoremisduetoIsmailandcomprisesinanabstractformthefundamental
fact behind the theory of basic analogs of Fourier series on a q-quadratic grid [5].
Theorem A [15] The system F (x) is orthogonal and complete in L2(µ).
n
{ }
To give an idea of what is involved in this statement, we sketch Ismail´s argu-
ment. Since, by (1.3), r (t ) are the Fourier coefficients of F in the basis u p ,
k n n k k
{ }
the use of Parseval´sformula gives
∞
δ
mn
F (x)F (x)dµ(x)= r (t )r (t )=
n m k n k m
ρ(x )
Z k=0 n
X
and the orthogonality relation is proved. To show the completeness, choose f
∈
L2(µ)andassume F (x)f(x)dµ(x)=0foralln. AgainParseval´sformulaimplies
n
∞
f u r (t ) = 0 for all m, where f are the Fourier coefficients of f in the
k=0 k k k m R k
P
REPRODUCING KERNELS 3
basis p . Now the completeness of r implies f =0. Therefore f = 0 almost
k k k
{ } { }
everywhere in L2(µ).
In[16](see alsosection24.2ofthe monograph[21]), Ismailposedthe problem
of studying the mapping properties of operators with kernels defined as above and
conjectured that there was a reproducing kernel Hilbert space structure behind
these operators. We will show that Ismail´s conjecture is true. Our approach will
revealareproducingkernelstructurereminiscentofthewellknownstructureofthe
Paley-Wiener space of functions bandlimited to a real interval. However, even in
thecasewhenthesystem F (x) isthesetofthecomplexexponentials,weobtain
n
{ }
results that, as far as our knowledge goes, seem to be new. When the system
F (x) is the set of basis functions of the q-Fourier series constructed with the
n
{ }
function (x;it), we will obtain results that complement the investigations done
q
E
in [29] and [19]. In particular it will be shown that a sampling theorem related
to the one derived in [19] lives in a reproducing kernel Hilbert space and that the
correspondentq-analoguesoftheSincfunctionprovideanorthogonalbasisforthat
space.
Before outlining the paper we want to make clear that the techniques we are
using already exist in some antecedent form. In particular, section two contains
ideasfromIsmailtheoryofgeneralizedq-FourierseriesandfromSaitoh´stheoryof
lineartransformationsinHilbertspace. However,theirparticularcombinationhere
leads to new conclusions and sheds new light in the emerging theory of q-Fourier
series. Itrevealsanelementarystructureunderlyingmanysystemsinvolvingseveral
special functions simultaneously. The results in section 2.2 and section 3, 4 and 5
were never explicitly stated before.
The paper can be summarized as follows. The next section contains the de-
scriptionofthereproducingkernelstructurebehindtheabstractsettingoftheorem
A. An integraltransformationbetween two Hilbert spacesis defined in the context
ofSaitoh´stheoryoflineartransformations,basisforbothspacesareprovidedand
the formula for the reproducing kernel of the image Hilbert space is deduced. In
thiscontextasamplingtheoremappears,generalizingtheonein[19]. Theremain-
ing sections consider three applications of these results, using specific systems of
orthogonal polynomials as well as Bessel functions and their generalizations. The
firstapplicationisassociatedtoformula(1.1)andsystemsofcomplexexponentials.
The second application is linked to (1.2) and to systems of curly q-exponentials,
andwewritethe reproducingkernelasa φ basichypergeometricfunction. These
2 2
two examples explore the interplay between Lommel polynomials and Bessel func-
tionsandthecorrespondingrelationsbetweentheirq-analogues. Inthelastsection
we consider a constructionof a generalcharacter,designedoriginally in the papers
[20], [17] and [15]. It allows to extend the interplay between Bessel functions and
Lommel polynomials to a more general class of functions. Using this construc-
tion we will make a brief discussion about the application of our results to spaces
weighted by Jacobi weights and their q-analogues.
2. The reproducing kernel structure
Let H be a class of complex valued functions, defined in a set X C, such
⊂
that H is a Hilbert space with the norm of L2(X,µ). The function k(s,x) is a
reproducing kernel of H if
i) k(.,x) H for every x X;
∈ ∈
4 LU´ISDANIELABREU
ii) f(x)= f(.),k(.,x) for every f H, x X.
h i ∈ ∈
The space H is said to be a reproducing kernel Hilbert space if it contains a
reproducing kernel. From a structural point of view, the correct approach to the
study of our problem is via Saitoh´s theory of linear transforms of Hilbert space.
2.1. Preliminaries on Saitoh´s theory of linear transformations in
Hilbert space. This theory can be found in works by Saitoh [26], [27] and we
proceed to give a brief account of the results that we are going to use. An account
ofthe resultsquotedinthissectioncanalsobefoundinHigginsrecentsurvey[13].
For each t belonging to a domain D, let K belong to H (a separable Hilbert
t
space). Then,
k(t,s)= K ,K
h t siH
is defined on D D and is called the kernelfunction of the map K . Now consider
t
×
the set of images of H by the transformation
(Kg)(t)= g,K =f(t)
h tiH
and denote this set of images by R(K). The following theorem can be found in
[26]:
TheoremBThekernel k(t,s)determinesuniquelyareproducingkernelHilbert
space for which it is the reproducing kernel. This reproducing kernel Hilbert space
is precisely R(K) and it can have no other reproducing kernel.
Now,supposethat K (t D)iscompleteinH sothatK isonetoone. Then
t
{ } ∈
we have
Kg = g .
k kR(K) k kH
The following theorem, due to Higgins [12], will be critical on the remainder. We
willuse only the following ”orthogonalbasiscase”,a specialcase ofthe theoremin
[12]:
Theorem C With the notations established earlier, we have: If there exists
t (n I Z)such that K is an orthogonal basis, we then have the sampling
{ n} ∈ ⊂ { tn}
expansion
k(t,t )
n
f(t)= f(t )
n
k(t ,t )
n∈I n n
X
in R(K), pointwise over I, and uniformly over any compact subset of D for which
K is bounded.
t
k k
2.2. The reproducing kernel for q-Fouriertype systems. Inthissection
we will show the existence of a reproducing kernel structure behind the abstract
settingofTheoremA.Theresultswillfollowfromthestudyofthemappingproper-
tiesofanintegraltransformwhosekernelisobtainedfromthesequenceoffunctions
r and p .
k k
{ } { }
Theorem1. ThereexistsakernelK(x,t)satisfyingtherequirementsofSaitoh´s
theory of linear transformations, such that K(x,x )= λ F (x), where F (x) is
n n n n
{ }
the orthogonal sequence of functions in Theorem A and λ is a sequence of real
n
{ }
numbers.
Proof. Our first technical problem comes from the fact that, when r is
k
{ }
a discrete system of orthogonal polynomials with a determinate moment problem,
REPRODUCING KERNELS 5
then r (t) l2 if and only if x is a mass point for the measure of orthogonality.
k
{ }∈
For this reason the series
∞
r (t)p (x)u
k k k
k=0
X
would diverge if t is not such a point (this is pointed out in Section 5 of [15]).
Since we want our kernel to be defined for every t, we will assume the existence of
an auxiliary system of functions (x) l2 for every t real, and such that every
k
{J }∈
function interpolates r at the mass points x in the sense that
k k n
J { }
1
(2.1) (x )=λ r ( )
k n n k
J x
n
for every k = 0,1,... and n = 0,1,...and some constant λ independent of k. Now
n
we can use the functions (t) to define a kernel K(x,t) as
k
J
∞
(2.2) K(x,t)= (t)p (x)u .
k k k
J
k=0
X
SuchakerneliswelldefinedandbelongstoL2(µ),sinceitisasumofbasisfunctions
of L2(µ). From (1.3), (2.1) and (2.2) we have
∞
K(x,x ) = (x )p (x)u
n k n k k
J
k=0
X
∞
1
= λ r ( )p (x)u
n k k k
x
n
k=0
X
= λ F (x).
n n
(cid:3)
Remark 1. Observe that theorem 1 and theorem A with t = 1 show that
n xn
K(x,x ) is an orthogonal basis for the space L2(µ).
n
Remark 2. Intheabstract formulation itmaynotbeclearwhytheconstantλ
n
must be present. Actually the construction would work without it, but for technical
reasons that will become evident upon consideration of examples we prefer to use it.
Otherwise, careful bookkeeping of the normalization constants would be required in
the remaining sections.
Remark 3. Itwill beseeninthelastsectionthatageneralconstructivemethod
is available in order to find the function under very natural requirements on the
k
J
polynomials r .
k
Now define an integral transformation F by setting
(Ff)(t)= f(x)K(x,t)dµ(x).
Z
We will study this transform as a map whose domain is the Hilbert space L2(µ).
Endowing the range of F with the inner product
(2.3) Ff,Fg = f,g
h iF(L2(µ)) h iL2(µ)
then F(L2(µ)) becomes a Hilbert space isometrically isomorphic to L2(µ) under
the isomorphism F. Using Saitoh´s theory with D = R, K = K(.,t), H = L2(µ)
t
and R(K)=F(L2(µ)), we obtain at once:
6 LU´ISDANIELABREU
Theorem 2. The transform F is a Hilbert space isomorphism mapping the
space L2(µ) into F(L2(µ)). The space F(L2(µ)) is a Hilbert space with reproducing
kernel given by
(2.4) k(t,s)= K(x,t)K(x,s)dµ(x).
Z
AninterestingfeatureofthisparticularsettingisthateveryfunctioninF(L2(µ))
has two different expansions: One is the sampling expansion naturally associated
withthereproducingkernelstructure,theotheristheexpansioninthebasis (x).
n
J
It should be remarkedthat, in most of the previously known sampling expansions,
these two expansions were the same. These expansion results are summarized in
the next theorem.
Theorem 3. Every function f of the form
(2.5) f(x)= u(t)K(t,x)dµ(t)
Z
with u L2(µ), admits an expansion
∈
∞
(2.6) f(t)= a (t)
n n
J
n=0
X
where the coefficients a are given by
k
a =u u,p (.)
n nh n iL2(µ)
and a sampling expansion
k(x,t )
n
(2.7) f(x)= f(t ) .
n
k(t ,t )
n n
X
The sum in (2.7) converges absolutely. Furthermore, it converges uniformly in
every set such that K(.,t) is finite.
k kL2(µ)
Proof. We already know by default that p (x) is a basis for L2(µ). It
n
{ }
remains to prove that (t) is a basis for F(L2(µ)). Observe that
n
{J }
(Fp )(t) = p (x)K(x,t)dµ(x)
n n
Z
∞
= (t)u p (x)p (x)dµ(x)
k k n k
J
k=0 Z
X
= (t)u .
n n
J
Since p (x) is a basis for L2(µ) and F is an isomorphism between L2(µ) and
n
{ }
F(L2(µ)), then u (x) is a basis for F(L2(µ)). To prove the last assertion of
n n
{ J }
the theorem, observethat the function f defined by (2.5) belongs to F(L2(µ)) and
therefore can be expanded in the basis u (x) . The Fourier coefficients of this
n n
{ J }
expansion are
a = f,u (.) = Fu,F(p (.)) = u,p (.)
n h nJn iF(L2(µ)) h n iF(L2(µ)) h n iL2(µ)
wherewe haveused(2.3) inthe lastidentity. The sampling expansionfollowsfrom
applying theorem C to our setting and using remark 1. (cid:3)
REPRODUCING KERNELS 7
Remark 4. The construction of this section has never appeared before in the
literature, but it is reminiscent of the reproducing kernel structure of the Paley-
Wienerspace. Intheclassicalsituations(seeforexample[8]and[24]foranaccount
of these constructions with several examples) generalizing this structure, there is an
integral transform whose kernel is defined as
(2.8) K(x,t)= S (t)e (x)
k k
where e (x) is an orthogonal basis forXthe domain Hilbert space and S (t) is a
k k
sequence of functions such that there exists a sequence t satisfying the sampling
n
{ }
property
(2.9) S (t )=a δ
k n n n,k
As an instance, take S (t)= sinπ(t−k) and e (x)=eikx. Then (2.8) is
k π(t−k) k
∞
sinπ(t k)
eitx = − eikx
π(t k)
k=−∞ −
X
and K(x,t) is the kernel of the Fourier transform. The corresponding reproduc-
ing kernel Hilbert space is the Paley-Wiener space. The root of these ideas is in
Hardy´s groundbreaking paper [11]. For an application of this classical set up to
Jackson q-integral transforms and the third Jackson q-Bessel function, see [1]. In
our construction we made a modification of this classical setting: Instead of the se-
quence of functions S , with the sampling property (2.9), we considered a sequence
k
of functions , interpolating an orthogonal system r in the sense of (2.1).
k k
{J } { }
And we have seen that the essential properties of classical reproducing kernel set-
tings are kept. However, this modification allows to recognize a class of reproducing
kernel Hilbert spaces that were obscured until now. This will become clear in the
next section.
3. The Fourier system with ultraspherical weights
The nth ultraspherical (or Gegenbauer) polynomial of order ν is denoted by
Cν(x). These polynomials satisfy the orthogonality relation
n
1 (2ν) √πΓ(ν+ 1)
Cν(x)Cν(x)(1 x2)ν−1/2dx= n 2 δ
n m − n!(ν+n)Γ(ν) m,n
Z−1
and form a complete sequence in the Hilbert space L2[( 1,1),(1 x2)ν−1/2]. For
− −
typographicalconvenience we will introduce the following notation for this Hilbert
space:
Hν =L2[( 1,1),(1 x2)ν−1/2].
− −
The Bessel function of order ν, J (x), is defined by the power series expansion
ν
∞
( 1)n z ν+2n
(3.1) J (z)= − .
ν
n!Γ(ν+n+1) 2
nX=0 (cid:16) (cid:17)
ThenthLommelpolynomialoforderν,denotedbyh (x), isrelatedtotheBessel
n,ν
functions by the relation
1 1
(3.2) Jν+k(x)=hk,ν( )Jν(x) hk−1,ν−1( )Jν−1(x).
x − x
8 LU´ISDANIELABREU
The Lommel polynomials satisfy the discrete orthogonality relation
∞
1 1 1 1
h ( )h ( )= δ
(j )2 n,ν+1 ±j m,ν+1 ±j 2(ν+n+1) nm
ν,k ν,k ν,k
k=0
X
and the dual orthogonality
∞
1 1
2(ν+n+1)h ( )h ( )=(j )2δ .
k,ν+1 k,ν+1 ν,k nm
±j ±j
ν,n ν,m
k=0
X
They form a complete orthogonal system in the l2 space weighted by the discrete
measurewithrespecttowhichtheyareorthogonal. Wewillusethesetwocomplete
orthogonalsystems in our first illustration of the general results. Set
k!(ν+k)
p (x)= Cν(x)
k s (2ν)k k
and
rk(t)= 2(ν+n)hk,ν−1(t).
Consider also p
(t)= 2(ν+n)J (t).
k ν+k
J
Denoteby jν,k the kth zeroofthe Bepsselfunctionoforderν. Substituting x=jν,n
in (3.2), the following interpolating property is obtained
1 J (j )
ν+k ν,n
(3.3) hk,ν−1( )= .
jν,n −Jν−1(jν,n)
The interpolating property (3.3) will play the role of (2.1) with λ = 1 .
n −Jν−1(jν,n)
Consider also the sequence of complex numbers u defined as
n
{ }
u =ik
k
and set
∞
2k!
Kν(x,t)=√2 ik(ν+k) J (t)Cν(x).
s(2ν)k ν+k k
k=0
X
Now, theorem 1 tells us that Kν(x,j ) is an orthogonal basis of the space Hν.
ν,n
Moreover,formula (1.1) implies
πt
K21(x,t)=eixt
2
r
andtherefore we can think of Γ(ν) t νKν(x,t) as a one parametergeneralization
2
of the complex exponential kernel that may be worth of further study. In the
(cid:0) (cid:1)
special case ν = 1 we recover the well known orthogonality and completeness of
2
the complex exponentials eiπnx in L2( 1,1).
{ } −
The transformation F is defined, for every f Hν, as
∈
1
(Ff)(t)= f(x)Kν(x,t)(1 x2)ν−1/2dx
−
Z−1
and theorem 2 gives that the reproducing kernel of F(Hν) is
1
kν(t,s)= Kν(x,t)Kν(x,s)(1 x2)ν−1/2dx.
−
Z−1
REPRODUCING KERNELS 9
When ν =1/2 this becomes
π√tssin(t s)
k12(t,s)= − .
2 (t s)
−
Since k!(ν+k)Cν(t) formsabasisofthespaceHν,thenalso in 2(ν+n)J (x) =
{ (2ν)k n } { ν+n }
F k!q(ν+k)Cν(t) isabasisofthespaceF (Hν). Inthissituationptheorem3reads:
{ (2ν)k n }
q
Theorem 4. Let f be a function of the form
1
(3.4) f(t)= u(x)Kν(x,t)(1 x2)ν−1/2dx
−
Z−1
where u Hν. Then f can be written as
∈
∞
(3.5) f(t)= a J (t)
n ν+n
n=0
X
with the coefficients a given by
n
2n! 1
(3.6) a =in(ν+n) u(x)Cν(x)(1 x2)ν−1/2dx
n s(2ν)n Z−1 n −
and also as the sampling formula
∞
kν(t,j )
ν,n
f(t)= f(j ) .
ν,n kν(j ,j )
ν,n ν,n
n=0
X
Remark 5. Expansions of the type (3.5) are known as Neumann series of
Bessel functions (see chapter 16 of [25]).
Remark 6. When ν =1/2 the above sampling theorem states that every func-
tion of the form
t −21 1
f(t)= u(x)eixtdx,
2
(cid:18) (cid:19) Z−1
with u L2[( 1,1)], can be represented as
∈ −
∞
t sin(t 2πn)
f(t)= f(2πn) − .
2πn (t 2πn)
n=0 r −
X
4. The q-Fourier system with q-ultraspherical weights
We proceed to describe the q-analogue of the previous situation. Choose a
number q such that 0 < q < 1. The now classical notational conventions from [9]
and [21] for q-infinite products and basic hypergeometric series will be used often.
The q-exponential function that we talked about in the introduction is defined
in terms of basic hypergeometric series as
( t;q12)∞ q41eiθ,q41e−iθ 1
Eq(x;t)= (−qt2;q2)∞2φ1(cid:18) q12 (cid:12)q2,−t(cid:19)
(cid:12)
wherex=cosθ. Thecontinuousq-ultrasphericalpolynomi(cid:12)alsoforderνaredenoted
by Cν(x;qν q) and satisfy the orthogonality (cid:12)
n |
1 (1 qν)(q2ν;q)
Cν(x;qν q)Cν(x;qν q)w(x;qν q)dx= − nδ
n | m | | (1 qn+ν)(q;q) m,n
Z−1 − n
10 LU´ISDANIELABREU
where the weight function w(x;β q) is
|
(q,q2ν;q)∞(e2iθ,e−2iθ;q)∞
w(cosθ;β q)= ,(0<θ <π).
| sinθ(2πqν,qν+1;q)∞(βe2iθ,βe−2iθ;q))∞
The polynomials Cν(x;qν q) form a basis of the Hilbert space Hν defined as
{ n | } q
Hν =L2[( 1,1),w(x;qν q)].
q − |
The second Jackson q-Bessel function of order ν is defined by the power series
∞
J(2)(x;q)= (qν+1;q)∞ ( 1)n (x/2)ν+2n qn(ν+n).
ν (q;q)∞ − (q;q)n(qν+1;q)n
k=0
X
Since this is the only q-Bessel function to be used in the text, we will drop the
superscript for shortness of the notation and write J (x;q) = J(2)(x;q). The q-
ν ν
Lommel polynomials associated to the Jackson q-Bessel function of order ν are
denoted by hn,ν−1(x;q). These polynomials were defined in [14] by means of the
relation
1 1
(4.1) qnν+n(n−1)/2Jν+n(x;q)=hn,ν( ;q)Jν(x;q) hn−1,ν−1( ;q)Jν−1(x;q).
x − x
Theq-Lommelpolynomialssatisfytheorthogonalityrelation,foracertainconstant
A (ν+1) which is not explicitly known (see section 14.4 of [21]):
n
∞
A (ν+1) 1 1 qnν+n(n+1)/2
n
h ( ;q)h ( ;q)= δ
(j (q))2 n,ν+1 ±j (q) m,ν+1 ±j (q) 1 qn+ν+1 nm
ν,n ν,n ν,n
n=1 −
X
and the dual orthogonality
∞
(1 qn+ν+1) 1 1 (j (q))2
ν,n
− h ( ;q)h ( ;q)= δ .
qnν+n(n+1)/2 n,ν+1 ±j (q) m,ν+1 ±j (q) A (ν+1) nm
ν,n ν,n k
n=1
X
Consider
(1 qk+ν)(q;q)
p (x)= − kC (x;qν q),
k s(1 qν)(q2ν;q)k k |
−
(1 qk+ν)
r (t)= − q−kν/2−k(k−1)/4h (2t;q)
k s (1 qν) k,ν
−
and
Jk(t)=s(1(1−qkq+ν)ν)qkν/2+k(k4−1)Jν+k(2t;q).
−
The parameters u will be given by
k
u =ik.
k
Denote by j (q) the kth zero of J (x;q). Setting t= j (q) in (4.1) we have the
ν,k ν ν,k
interpolating property
1 J (j (q);q)
(4.2) hk,ν−1( ;q)= qkν+k(k−1)/2 ν+k ν,n .
jν,n(q) − Jν−1(jν,n(q);q)