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Multivariable BC type Askey-Wilson polynomials with partly discrete
orthogonality measure
Stokman, J.V.
Publication date
1997
Published in
The Ramanujan Journal
Link to publication
Citation for published version (APA):
Stokman, J. V. (1997). Multivariable BC type Askey-Wilson polynomials with partly discrete
orthogonality measure. The Ramanujan Journal, 1, 275-297.
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Multivariable BC Type Askey-Wilson Polynomials
With Partly Discrete Orthogonality Measure
JASPER V. STOKMAN jasper@wins.uva.nl
Faculty WINS, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
Received May 3, 1996; Accepted October 17, 1996
Abstract. The multivariable BC type Askey-Wilson polynomials are considered for a parameter domain such that
the orthogonality measure has partly discrete and partly continuous support.
Keywords: BC type Askey-Wilson polynomials, multivariable orthogonal polynomials, second order q-difference
operator
1991 Mathematics Subject Classification: Primary – 33D45; Secondary – 33D25
1. Introduction
One variable Askey-Wilson polynomials depend (apart from q) on four parameters a, b, c, d and are orthogonal with respect to a measure with partly discrete and partly continuous support (cf. [1]). The discrete part of the orthogonality measure vanishes when the modulus of each of the four parameters a, b, c, d is≤ 1.
In [3] it was shown that the Macdonald polynomials associated with root system BC (which depend on three parameters) can be extended to a five parameter family of multi-variable orthogonal polynomials (parameters a, b, c, d and t). This five parameter family of orthogonal polynomials is a multivariable analog of the four parameter family of Askey-Wilson polynomials, with the parameters a, b, c, d playing the same role as in the one variable case, and with t∈ (−1, 1) an extra deformation parameter. In [3], the modulus of each of the parameters a, b, c, d was assumed to be≤ 1. Consequently, the orthogonality measure for the corresponding multivariable Askey-Wilson polynomials has completely continuous support. In [8], the orthogonality measure for the multivariable Askey-Wilson polynomials was introduced without the restriction that the modulus of each of the parameters a, b, c, d is≤ 1, for deformation parameters t = qk (k∈ N). In general, this introduces discrete
parts to the orthogonality measure as in the one variable case. In this paper, we will give detailed proofs for the results in [8].
First, we will recapitulate in section 2 the definition of the one variable Askey-Wilson polynomials. We use here some new notations, which turn out to be more convenient for the generalization to the multivariable case. Furthermore, the method of proof which will be used in the multivariable case, is sketched in the one variable setting.
In section 3 we introduce the orthogonality measure for the multivariable Askey-Wilson polynomials. The proof of the orthogonality uses a crucial proposition which leads to
the selfadjointness of a second order q-difference operator for which the Askey-Wilson polynomials are eigenfunctions. The proof of this proposition will be given in section 4.
The results of the present paper can be extended to arbitrary deformation parameter t∈ (0, 1) by developing a residue calculus for the orthogonality measure of the multivari-able Askey-Wilson polynomials (see Remark 8). Details will be given in a forthcoming paper.
Notations: N = {1, 2, . . .} will be the natural numbers and N0the natural numbers
to-gether with 0. Empty sums are equal to 0, empty products are equal to 1. We will use the concept of selfadjoint operator in the formal sense: A Hermitian linear operator with respect to a scalar product on a vector space.
2. One variable Askey-Wilson polynomials
The results in this section were proved in [1]. We reformulate these results in a way which will be convenient for the generalization to the multivariable case. We first introduce some notations. Fix q∈ (0, 1). The q-shifted factorial is defined by
(a; q)i:= i−1 Y j=0 ¡ 1− qja¢ (i∈ N0), (a; q)∞:= ∞ Y j=0 ¡ 1− qja¢. We denote products of q-shifted factorials by
(a1, . . . , am; q)i := m Y j=1 (aj; q)i (i∈ N0), (a1, . . . , am; q)∞:= m Y j=1 (aj; q)∞.
The basic hypergeometric series are given by
r+1φr µ a1, . . . , ar+1 b1, . . . , br ; q, x ¶ := ∞ X k=0 (a1, . . . , ar+1; q)k (q, b1, . . . , br; q)k xk.
Let α, β ∈ C. For a function f : C → C and an integer N ∈ Z we define the Jackson q-integral from α to β of the function f , truncated at N , by
Z β α f (x)dq,Nx := Z β 0 f (x)dq,Nx− Z α 0 f (x)dq,Nx, Z β 0 f (x)dq,Nx := N X k=0 f (βqk)¡βqk− βqk+1¢ if N ≥ 0, Z β 0 f (x)dq,Nx := 0 if N < 0.
The Jackson q-integral from α to β for (continuous) functions f is obtained by taking the limit N → ∞, Z β α f (x)dqx := lim N→∞ Z β α f (x)dq,Nx.
An important property of the truncated Jackson q-integral is the partial q-integration rule; let N∈ N0, then Z β 0 ¡ D−qf¢(x)g(x)dq,Nx = − Z β 0 f (x)¡Dq+g¢(x)dq,Nx (2.1) + f (β)g(βq−1)− f(βqN +1)g(βqN)
with the backward q-derivative D−q given by
¡
Dq−f
¢
(x) := f (x)− f(qx)
(1− q)x (2.2)
and the forward q-derivative D+
q given by
¡
Dq+f¢(x) := f (q
−1x)− f(x)
(1− q)x . (2.3)
We will call f (β)g(βq−1) resp. −f(βqN +1)g(βqN) in (2.1) the upper resp. lower
stock-term. For continuous functions f and g, the partial q-integration rule for Jackson q-integrals (see [4, Proposition 2.1]) can be obtained from (2.1) by taking the limit N → ∞ at both sides of equation (2.1).
Definition 1 (Parameter domain for the Askey-Wilson polynomials.)
The set of parameters (a, b, c, d) which satisfy the conditions
(1) a, b, c, d are real, or if complex, then they appear in conjugate pairs, (2) ab, ac, ad, bc, bd, cd /∈ R≥1:={r ∈ R | r ≥ 1},
will be denoted by VAW.
If (a, b, c, d)∈ VAW, then the parameters a, b, c, d satisfy the following two properties.
1. Suppose that e∈ {a, b, c, d} with |e| ≥ 1, then e ∈ R.
2. At most two of the four parameters a, b, c, d have modulus≥ 1. If there are two, then one is positive and the other is negative.
Let (a, b, c, d) ∈ VAW. For e ∈ {a, b, c, d} with |e| > 1, let Ne be the largest positive
integer such that|eqNe| > 1. Take N
e := −1 for e ∈ {a, b, c, d} with |e| ≤ 1. Let C
be the unit circle in the complex plane. When integrating over the unit circle, we always traverse the unit circle in the counterclockwise direction. Define a Hermitian formh., .iAW
on C[x + x−1] by:
with hp1, p2iAW,0:= 1 2πi Z x∈C p1(x)p2(x)wAW,c(x; a, b, c, d; q) dx x, (2.5) respectively hp1, p2iAW,1:= 2 X e∈{a,b,c,d} Z e x=0 p1(x)p2(x)wAW,d(x; e; f, g, h; q) dq,Nex (1− q)x, (2.6)
and with the following weight functions:
wAW,c(x; a, b, c, d; q) :=
¡
x2, x−2; q¢ ∞
(ax, ax−1, bx, bx−1, cx, cx−1, dx, dx−1; q)∞ (2.7)
and for e∈ {a, b, c, d} with |e| > 1 and i ∈ {0, . . . , Ne}, wAW,d(eqi; e; f, g, h; q) :=
¡
e−2; q¢ ∞
(q, ef, f /e, eg, g/e, eh, h/e; q)∞ (2.8) ×
¡
e2, ef, eg, eh; q¢
i
(q, eq/f, eq/g, eq/h; q)i ¡ 1− e2q2i¢ (1− e2) µ q ef gh ¶i
with f, g, h∈ C such that e, f, g, h is a permutation of a, b, c, d. Note that the summation over e in (2.6) can be restricted to those e with|e| > 1, because the q-Jackson integral truncated at N is zero if N < 0. Since (a, b, c, d)∈ VAW, the summation is therefore over
at most two parameters, and the corresponding mass points eqiare real. Furthermore, the measure dq,Nex
(1−q)xfor|e| > 1 is just the counting measure for the set {e, . . . , eq
Ne}, i.e. Z e x=0 f (x) dq,Nex (1− q)x = Ne X k=0 f (eqk).
We prefer the notation in terms of the truncated Jackson integral because it gives a more transparent notation in the multivariable setting.
The denominator of the discrete weight wAW,d(eqi; e) is well defined and non zero for
(a, b, c, d)∈ VAWif e∈ {a, b, c, d} with |e| > 1 and i ∈ {0, . . . , Ne}. Indeed if egfh = 0,
then read the factors of the form (eq/f ; q)ifiin the denominator asQi−1 k=0
¡
f− eqk+1¢.
The denominator of the continuous weight function wAW,c(x) for x∈ C and (a, b, c, d) ∈ VAW can become zero when e∈ {±q−j}j∈N0 for some e∈ {a, b, c, d}. These possible
zeros in the denominator of wAW,c(x) are compensated by zeros in the numerator. Indeed,
since (a, b, c, d)∈ VAW, this follows from the fact that the numerator can be rewritten as
¡
x2, x−2; q¢∞=¡x,−x, x−1,−x−1; q¢∞¡qx2, qx−2; q2¢∞.
The discrete weights wAW,d(eqi; e) in (2.6) are strict positive, and the continuous weight
function wAW,c(x) in (2.5) is positive for x on the unit circle, henceh., .iAW is positive
pAWl (x; a, b, c, d; q) :=4φ3 µ q−l, ql−1abcd, ax, ax−1 ab, ac, ad ; q, q ¶ , l∈ N0. (2.9) Then pAW
n (x) is a polynomial of degree n in x + x−1, and
Theorem 1 ([1, Theorem 2.5])
hpAW n , p
AW
m iAW = 0 if n6= m. (2.10)
The proof of Theorem ?? is essentially in two steps. For (a, b, c, d) ∈ VAW with e /∈
{±1, ±q−1
2,±q−1,±q−32, . . .} for all e ∈ {a, b, c, d}, let C be a closed contour which
seperates the poles of wAW,cconverging to zero from the poles converging to infinity (see
[1, Theorem 2.1]). For contour C, one has (see [1, Theorem 2.3])
Z x∈C pAWn (x)pAW m (x)wAW,c(x; a, b, c, d; q) dx x = 0 if n6= m.
Now deforming the contour C back to the unit circle C, one picks up residues at x = f qj, f−1q−j for f ∈ {a, b, c, d} with |f| > 1 and j ∈ {0, . . . , N
f}. Since wAW,d ¡ f qj; f¢= RESx=f qj µ waw,c(x) x ¶ =− RESx=f−1q−j µ wAW,c(x) x ¶ (2.11)
for f ∈ {a, b, c, d} with |f| > 1 and j ∈ {0, . . . , Nf} (these poles of wAW,care simple
because of the conditions on the parameters), Theorem ?? follows for a dense subset of the parameter domain VAW. A continuity argument in the parameters a, b, c, d gives then the
result for all (a, b, c, d)∈ VAW.
{pAW
n (x; a, b, c, d; q)| n ∈ N0} is a basis of C[x + x−1] consisting of eigenfunctions for
the second order q-difference operator
DAWa,b,c,d,q:= vAW− (x; a, b, c, d; q)Dq−+ v + AW(x; a, b, c, d; q)D + q (2.12) with vAW− (x; a, b, c, d; q) := −(1 − q)x(1 − ax)(1 − bx)(1 − cx)(1 − dx) (1− x2)(1− qx2) , (2.13) vAW+ (x; a, b, c, d; q) := (1− q)x(a − x)(b − x)(c − x)(d − x) (1− x2)(q− x2) . (2.14)
The eigenvalue corresponding to the eigenfunction pAW
n (x; a, b, c, d; q) is given by EnAW(a, b, c, d; q) := q−1abcd(qn− 1) + (q−n− 1). (2.15) So DAWis selfadjoint with respect toh., .iAW for (a, b, c, d)∈ VAW, because the
basis with respect toh., .iAW. The selfadjointness of DAW with respect toh., .iAW can be
proved by direct calculations, without using the existence of an orthogonal basis consisting of eigenfunctions for DAW. We sketch here globally the idea of this proof. We will use
these ideas later on in the multivariable setting. Let (a, b, c, d)∈ VAW such that e /∈ {±1, ±q−
1
2,±q−1,±q−32, . . .} and such that f 6=
eqNe+1 for all e, f ∈ {a, b, c, d} with |e| > 1. Define w
AW,d(eq−1; e) := 0 for e ∈
{a, b, c, d}, and let wAW,d(eqNe+1; e) for e∈ {a, b, c, d} with |e| > 1 be given by formula
(2.8) with i = Ne+ 1. We have the following functional relation
vAW+ ¡gqm+1¢ wAW,d ¡ gqm+1; g¢ (1− q)gqm+1 =−v − AW(gq m)wAW,d(gqm; g) (1− q)gqm (2.16)
for g∈ {a, b, c, d} with |g| > 1 and for m ∈ {−1, . . . , Ng}.
Apply now the partial q-integeration rule 2#{e ∈ {a, b, c, d} | |e| > 1} times on the ex-pressionhDAWml, mniAW,1(where mn(x) = xn+ x−n(n∈ N0)) and use the functional
relation (2.16), then
hDAWml, mniAW,1− hml, DAWmniAW,1
is a sum of 2#{e ∈ {a, b, c, d} | |e| > 1} lower stockterms (the upper stockterms are all zero). On the other hand,
hml, DAWmniAW,0− hDAWml, mniAW,0
can be rewritten as an integral over a closed contourD (see Definition 3, section 4 for the definition ofD) with integrand
−1
(1− q)x2πiml(qx)mn(x)v −
AW(x)wAW,c(x).
The integrand has 2#{e ∈ {a, b, c, d} | |e| > 1} simple poles within the contour D, namely at the points
{eqNe,¡eqNe+1¢−1 | e ∈ {a, b, c, d}, |e| > 1}.
Then the sum of the 2#{e ∈ {a, b, c, d} | |e| > 1} residues is equal to the sum of the
2#{e ∈ {a, b, c, d} | |e| > 1} stockterms; both sums are equal to 2 X e:|e|>1 hl,n(eqNe) wAW,d(eqNe; e) (1− q)eqNe , where hl,n(x) = ml(x)mn(qx)v−AW(x)− ml(qx)mn(x)v−AW(x).
Hence selfadjointness follows for a dense subset of the parameter domain, and a continuity argument gives the result for all the parameter values.
3. The orthogonality measure for BC type Askey-Wilson polynomials
Let{²i}ni=1be the standard orthonormal basis in Rn. Let Snbe the symmetric group (group
of permutations of the set{1, . . . , n}). Snacts on the set{²i}ni=1by w.²i:= ²w(i). LetW
be the Weyl group of type BCn.W is isomorphic to a semidirect product of {±1}nand Sn,
and it acts by sign changes and permutations on the set{±²1, . . . ,±²n}. Let z1, . . . , znbe
independent variables, thenW acts in a natural way on the algebra A := C[z1±1, . . . , zn±1]
(by identifying zi±1with z±²iand using the action ofW on {±²
1, . . . ,±²n}). Denote AW
for the subalgebra ofW-invariant functions in A. A basis for AWis given by the monomials
{mλ| λ ∈ P+}, where P+is the set of partitions of length≤ n, P+:={µ ∈ Nn0| µ1≥ . . . ≥ µn}, and mλ(z) := X µ∈Wλ zµ, with zµ = zµ1 1 . . . z µn
n . TheW-orbit of λ ∈ P+⊂ Znis taken with respect to the natural
action ofW on Zn. The k-torus Tk is by definition the k-fold direct product of the unit
circleC in Ck,
Tk:=C×k={(w1, . . . , wk)∈ Ck | |wi| = 1 (i = 1, . . . , k)}.
The counterclockwise orientation on the unit circleC induces an orientation on Tk.
Define for (a, b, c, d)∈ VAW, for t = qkwith k∈ N and for m ∈ {0, . . . , n} a Hermitian
formh., .ia,b,c,dm,q,t : AW × AW → C by hf, gim:= 2m¡n m ¢ (2πi)n−m X e1,...,em Z e1 z1=0 .. Z em zm=0 Z .. Z (zm+1,...,zn)∈Tn−m (3.1) f (z)g(z)∆AW,m(z) dq,Ne1z1 (1− q)z1 . . . dq,Nemzm (1− q)zm dzm+1 zm+1 . . .dzn zn
for f, g∈ AW with weight function ∆
AW,m(z; a, b, c, d; q, qk) for z = (z1, . . . , zn) = (e1qi1, . . . , emqim, zm+1, . . . , zn) defined by ∆AW,m ¡ z; a, b, c, d; q, qk¢ := Ãm Y l=1 wAW,d(zl; el; fl, gl, hl; q) ! (3.2) × Ã n Y r=m+1 wAW,c(zr; a, b, c, d; q) ! δk(z), with
δk(z) := Y 1≤i<j≤n ¡ zizj, zizj−1, z−1i zj, zi−1z−1j ; q ¢ k, (3.3)
where wAW,c(x; a, b, c, d; q), resp. wAW,d(x; el; fl, gl, hl; q) is given by (2.7) resp. (2.8),
and el, fl, gl, hla permutation of a, b, c, d for l = 1, . . . , m. The sum is taken over ej ∈
{a, b, c, d} for all j = 1, . . . , m. Note that a sum with at least one |ej| ≤ 1 gives zero
contribution in (3.1). For m = 0 the summation sign in (3.1) should be skipped, and the integration is over (z1, . . . , zn)∈ Tn. Note that the dependance on k appears only in the
interaction factor δk(z) of the weight function. Finally define a Hermitian formh., .i a,b,c,d AW,q,qk on AW by hf, giAW := n X m=0 hf, gim. (3.4)
The integration domain WAW,m(a, b, c, d; q, qk) of the Hermitian formh., .imis given by WAW,0= Tn, respectively WAW,m= ½ (z1, . . . , zn)∈ Cn ¯¯ ¯¯zi∈ [ e:|e|>1 {e, . . . , eqNe} (i = 1, . . . , m) (3.5) and (zm+1, . . . , zn)∈ Tn−m ¾
for m > 0 if there exists an e∈ {a, b, c, d} with |e| > 1, and WAW,m=∅ otherwise.
The following lemma implies that the Hermitian formh., .iAW is positive definite.
Lemma 1 Let q∈ (0, 1), (a, b, c, d) ∈ VAW and t = qkwith k∈ N.
(a) ∆AW,m(z)≥ 0 for z ∈ WAW,mand m∈ {0, . . . , n}.
(b) Let 0 < m ≤ n. Let zi ∈ ∪e:|e|>1{e, . . . , eqNe} for i = 1, . . . , m. Then
∆AW,m(z1, . . . , zm, wm+1, . . . , wn) = 0 for all (wm+1, . . . , wn) ∈ Tn−m if and only
if zi= qlzjfor certain l∈ {0, . . . , k − 1} and certain i, j ∈ {1, . . . , m}, i 6= j.
Proof: Fix z ∈ WAW,m. The positivity of the factors wAW,d(zl) (l∈ {1, . . . , m}) and wAW,c(zl) (l∈ {m + 1, . . . , n}) in ∆AW,m(z) is known from the one variable case (see
section 2, or see [1]). Hence it is sufficient to check the lemma for δk(z). We check the
positivity of the factors
φki,j(z) :=¡zizj, zizj−1, zi−1zj, z−1i zj−1; q
¢
k
in δk(z) for all 1≤ i < j ≤ n.
For 1 ≤ i < j ≤ n with j ≥ m + 1 we have φki,j(z) = |¡zizj, zi−1zj; q
¢
k|
2 because |zi| = 1 (if i ≥ m + 1) resp. zi∈ R (if i ≤ m) and |zj| = 1.
For 1≤ i < j ≤ m we have to consider two cases.
(1) zi = eqpi, zj = eqpj with e ∈ {a, b, c, d}, |e| > 1 and pi, pj ∈ {0, . . . , Ne}. Since
the modulus of e is≥ 1, we have e ∈ R. If 0 ≤ pi− pj < k, then
¡
zjz−1i ; q
¢
if 0≤ pj− pi < k, then ¡ zizj−1; q ¢ k = 0. Hence φ k i,j(z) = 0 if 0≤ |pi− pj| < k. If pi− pj ≥ k, then φki,j(z) = kY−1 l=0 ¡ 1− ql+pi+pje2¢ ¡1− ql+pi−pj¢ ¡1− ql+pj−pi¢ ¡1− ql−pi−pje−2¢
is strict positive because we have
¡
1− ql+pi+pje2¢< 0 , ¡1− ql+pi−pj¢> 0,
¡
1− ql−pi−pje−2¢> 0 , ¡1− ql+pj−pi¢< 0
for all l∈ {0, . . . , k − 1}. A similar argument gives φki,j(z) > 0 for pj− pi≥ k.
(2) zi= eqpi, zj= f qpjwith e, f ∈ {a, b, c, d}, e 6= f, |e|, |f| > 1 and pi∈ {0, . . . , Ne}, pj ∈ {0, . . . , Nf}. We have e, f ∈ R and one of the parameters is positive and the other
is negative. Hence we have φk
i,j(z) > 0.
The lemma follows now directly from (1) and (2), since at most two of the four parameters a, b, c, d have modulus≥ 1.
The dominance order on P+is given by
µ≤ λ ⇔ j X l=1 µl≤ j X l=1 λl ∀j = 1, . . . , n (3.6) for λ, µ∈ P+.
Definition 2 Let q∈ (0, 1), (a, b, c, d) ∈ VAW and t = qk, k∈ N.
The Askey-Wilson polynomials {PλAW(z; a, b, c, d; q, t)| λ ∈ P+} are defined by the
following two conditions:
(1) PλAW = mλ+
P
µ<λ;µ∈P+cλ,µmµfor certain cλ,µ∈ C,
(2)hPAW
λ , mµiAW = 0 if µ < λ.
Since h., .iAW is positive definite by Lemma 1, we have that conditions (1) and (2)
uniquely determine PAW
λ . It is clear that{PλAW| λ ∈ P+} is a basis of AW and that
hPAW
λ , PµAWiAW = 0 if µ < λ, resp. µ > λ. Orthogonality of the Askey-Wilson
polynomials is not obvious when the degrees λ, µ∈ P+of the polynomials are incompatible
with respect to the partial order≤. Full orthogonality of the Askey-Wilson polynomials will be proved in Theorem 3.
In the one variable case, the Askey-Wilson polynomials are independent of t = qk
because the scalar producth., .iAW is independent of t when n = 1. In fact, the scalar
producth., .iAW reduces to (2.4) when n = 1, so the Askey-Wilson polynomials as defined
in Definition 2 are exactly the monic one variable Askey-Wilson polynomials as defined in section 2 for n = 1. So in the one variable case we have for l∈ N0,
PlAW(x; a, b, c, d; q) = (ab, ac, ad; q)l
al(ql−1abcd; q) l 4φ3 µ q−l, ql−1abcd, ax, ax−1 ab, ac, ad ; q, q ¶ .
If (a, b, c, d) ∈ VAW with|e| ≤ 1 for all e ∈ {a, b, c, d} then the scalar product h., .iAW
is given by integration over the n-torus Tnwith respect to the weight function ∆AW,0(z).
Since this is exactly the scalar product considered by Koornwinder [3], Definition 2 for
(a, b, c, d) ∈ VAW with|e| ≤ 1 for all e ∈ {a, b, c, d} gives exactly the five parameter
family of multivariable Askey-Wilson polynomials with discrete deformation parameter t = qk(k∈ N) which was introduced by Koornwinder for arbitrary t ∈ (−1, 1) in [3].
To prove full orthogonality of the Askey-Wilson polynomials, we will use a specific second order q-difference operator for which the Askey-Wilson polynomials are joint eigenfunc-tions. The second order q-difference operator Da,b,c,dAW,q,tis defined by (cf. [3])
DAW := n X j=1 ³ φ−AW,j(z)Dqj,−+ φ+AW,j(z)Dj,+q ´ , (3.7) with Dj,−
q and Dj,+q the backward resp. forward q-derivative acting on the jth coordinate,
so ¡ Dqj,−f ¢ (z) := f (z)− (Tq,jf ) (z) (1− q)zj , ¡Dqj,+f ¢ (z) := ¡ Tq−1,jf ¢ (z)− f(z) (1− q)zj ,
where Tq±1,jis the q±1-shift in the jth coordinate,
¡
Tq±1,jf
¢
(z) := f (z1, . . . , zj−1, q±1zj, zj+1, . . . , zn).
The functions φ−AW,j(z) and φ+AW,j(z) are given by
φ−AW,j(z; a, b, c, d; q, t) := vAW− (zj; a, b, c, d; q) Y l6=j (1− tzlzj) ¡ 1− tzl−1zj ¢ (1− zlzj) ¡ 1− zl−1zj ¢ , (3.8) φ+AW,j(z; a, b, c, d; q, t) := vAW+ (zj; a, b, c, d; q) Y l6=j (t− zlzj) ¡ t− z−1l zj ¢ (1− zlzj) ¡ 1− z−1l zj ¢. (3.9) See (2.13) resp. (2.14) for the definition of vAW− resp. vAW+ . Koornwinder proved the following proposition (see [3, Lemma 5.2] and the remark after [9, Proposition 4.1]).
Proposition 1 Let λ∈ P+and q∈ (0, 1). For arbitrary a, b, c, d, t∈ C we have
DAWmλ=
X
µ≤λ
Eλ,µAWmµ (3.10)
with EAW
λ,µ(a, b, c, d; q, t)∈ C depending polynomially on a, b, c, d and t. The leading term EAW
λ,λ(a, b, c, d; q, t) will be denoted by E AW
λ (a, b, c, d; q, t) and is given by
EλAW := n X j=1 ¡ q−1abcdt2n−j−1(qλj − 1) + tj−1(q−λj− 1)¢. (3.11)
In particular, DAW maps AW into itself. An operator satisfying (3.10) for all λ∈ P+is
called a triangular operator with respect to the basis of monomials and with respect to the partial order≤.
In order to prove that DAWis diagonalized by the Askey-Wilson polynomials for (a, b, c, d)
∈ VAW and t = qk, k ∈ N, it will be sufficient to prove that DAW is selfadjoint with
respect to the scalar producth., .iAW (cf. [5], or [9, Proposition 2.2]).
In order to prove that DAWis selfadjoint with respect toh., .iAW, we split DAWfor every r∈ {0, . . . , n} into two parts:
DAW = DrF+ DrL, (3.12)
with DFr the sum of the first r terms in (3.7),
DrF := r X j=1 ³ φ−AW,j(z)Dj,q−+ φ+AW,j(z)Dqj,+ ´ (r = 0, . . . , n) (3.13) and with DL
r the sum of the last n− r terms in (3.7), DrL:= n X j=r+1 ³ φ−AW,j(z)Dj,q−+ φ+AW,j(z)Dj,+q ´ (r = 0, . . . , n) (3.14) (in particular, DF
0 ≡ 0 and DnL ≡ 0). For (a, b, c, d) ∈ VAW and t = qk(k ∈ N), write L1(W
AW,0) for the L1functions on the torus WAW,0= Tnwith respect to the measure
∆AW,0(z) dz1 z1 . . .dzn zn , z∈ Tn.
If furthermore|e| > 1 for certain e ∈ {a, b, c, d}, then the L1-functions on W
AW,mwith
respect to the measure
∆AW,m(z) dq,Ne1z1 (1− q)z1 . . .dq,Nemzm (1− q)zm dzm+1 zm+1 . . .dzn zn ,
z = (z1, . . . , zn) = (e1ql1, . . . , emqlm, zm+1, . . . , zn) ∈ WAW,m, are denoted by L1(W
AW,m) for m = 1, . . . , n.
For two functions f, g : WAW,m → C with z 7→ f(z)g(z) in L1(WAW,m), define
hf, gimby formula (3.1). The following proposition is crucial for the proof that DAW is
selfadjoint with respect toh., .iAW.
Proposition 2 Let q∈ (0, 1), (a, b, c, d) ∈ VAW and t = qkwith k∈ N and suppose that e /∈ {±1, ±q−1/2,±q−1,±q−3/2, . . .} for all e ∈ {a, b, c, d}. Suppose furthermore, that f 6= eqNe+1for all e, f∈ {a, b, c, d} with |e| > 1.
For µ, λ∈ P+and r∈ {1, . . . , n}, we have that the map
z7→ (DFrmλ)(z)mµ(z) resp. z7→ (DLr−1mλ)(z)mµ(z)
is in L1(W
AW,r) resp. in L1(WAW,r−1), and
hDF
Remark 1 For parameters (a, b, c, d)∈ VAWsatisfying the extra condition that|e| ≤ 1 for
all e∈ {a, b, c, d}, Proposition 2 should be read as the statement that the proposition holds for r = 1, i.e. that the map z7→ (DAWmλ)(z)mµ(z) is in L1(WAW,0) for all λ, µ∈ P+
and that DAW is selfadjoint with respect toh., .i0=h., .iAW.
We will deal with the proof of Proposition 2 in section 4. For the one variable case, the idea of the proof is sketched in section 2 (note that for n = 1, formula (3.15) with r = 1 is equivalent to the selfadjointness of DAW with respect toh., .iAW).
As a corollary, we have
Theorem 2 Let q∈ (0, 1), (a, b, c, d) ∈ VAW and t = qk, k∈ N. DAW is selfadjoint with respect toh., .iAW, i.e. we have
hDAWf, giAW =hf, DAWgiAW ∀f, g ∈ AW. (3.16)
Proof: It is sufficient to prove (3.16) for f = mλ, g = mµwith arbitrary λ, µ ∈ P+.
Fix q ∈ (0, 1) and fix k ∈ N. Denote t = qk. Both sides of (3.16) depend continu-ously on (a, b, c, d) ∈ VAW because DAWmλ =
P
µ≤λE AW
λ,µmµ with Eλ,µAW depending
continuously on a, b, c, d by Proposition 1, and hmλ, mµiAW depends continuously on
(a, b, c, d) ∈ VAW. So it is sufficient to prove selfadjointness for (a, b, c, d) ∈ VAW
satisfying the extra conditions stated in Proposition 2.
Under these extra assumptions on the parameters (a, b, c, d) ∈ VAW we deduce
induc-tively from (3.15) that
j X l=0 hDAWmλ, mµil= j X l=0 hmλ, DAWmµil+hDjLmλ, mµij− hmλ, DjLmµij
for j∈ {0, . . . , n}. For j = n this yields
hDAWmλ, mµiAW =hmλ, DAWmµiAW
because DL n ≡ 0.
Corollary 1 Let q∈ (0, 1), (a, b, c, d) ∈ VAW and t = qk, with k∈ N, then
DAWPλAW = EλAWPλAW ∀λ ∈ P+. (3.17)
We can prove now full orthogonality of the Askey-Wilson polynomials.
Theorem 3 Let q∈ (0, 1), (a, b, c, d) ∈ VAW and t = qkwith k∈ N, then
hPAW
λ , PµAWiAW = 0 if λ6= µ. (3.18)
Proof: We first give the proof of the theorem assuming that the following statement is valid:
(S) For (a, b, c, d)∈ VAW, k ∈ N and λ, µ ∈ P+ with λ6= µ and abcd 6= −1, we have EAW
λ (a, b, c, d; q, q
k)6= EAW
µ (a, b, c, d; q, qk) as Laurent polynomials in q.
Fix k ∈ N and λ, µ ∈ P+ with λ 6= µ, and let V
λ,µ(k) be the set of parameters
(q, a, b, c, d) ∈ (0, 1) × VAW such that EλAW(a, b, c, d; q, q
k) 6= EAW
µ (a, b, c, d; q, qk).
Theorem 2 and Corollary 1 imply that
hPAW λ , P AW µ i a,b,c,d AW,q,qk = 0 (3.19)
for (q, a, b, c, d)∈ Vλ,µ(k). Since Vλ,µ(k)⊂ (0, 1) × VAW is dense by statement (S), it
remains to prove that the left hand side of (3.19) depends continuously on (q, a, b, c, d)∈
(0, 1)× VAW. This follows from the fact thathmν, mσi a,b,c,d
AW,q,qkdepends continuously on
(q, a, b, c, d)∈ (0, 1) × VAW for all ν, σ∈ P+and [9, Proposition 2.3].
We prove now statement (S). Fix (a, b, c, d)∈ VAW with abcd 6= −1 and fix k ∈ N.
Suppose that EλAW(a, b, c, d; q, q k ) = EµAW(a, b, c, d; q, q k ) (3.20)
as Laurent polynomials in q for certain λ, µ∈ P+. We have to prove that λ = µ. Formula (3.20) implies (abcd) n X i=1 ³ qλi+k(n−i)− qµi+k(n−i) ´ = n X i=1 ³ q−µi−k(n−i)+1− q−λi−k(n−i)+1 ´ (3.21)
as Laurent polynomials in q. If we expand the left hand side of (3.21) in powers of q,
(abcd) n X i=1 ³ qλi+k(n−i)− qµi+k(n−i) ´ =X l∈Z cLlql, (3.22) then cL
l = 0 except for finitely many l and cLl = abcd or cLl =−abcd if cLl 6= 0. We have abcd∈ R and abcd < 1, because (a, b, c, d) ∈ VAW. We assumed that abcd 6= −1, so
|cL
l| 6= 1 for all l ∈ Z. Expanding the right hand side of (3.21) in powers of q gives n X i=1 ³ q−µi−k(n−i)+1− q−λi−k(n−i)+1 ´ =X l∈Z cRlql (3.23) with cR
l = 0 except for finitely many l and|cRl | = 1 if cRl 6= 0. Since (3.22) is the same
Laurent polynomial in q as (3.23) by (3.21), we must have cR
l = cLl = 0 for all l∈ Z. In particular we have n X i=1 q−µi−k(n−i) = n X i=1 q−λi−k(n−i) (3.24)
as Laurent polynomials in q. Since λ, µ∈ P+, we get µi+ k(n− i) = λi+ k(n− i) for
Remark 2 The method of proving full orthogonality with the help of a triangular,
selfad-joint second order q-difference operator was introduced by Macdonald in [5]. In [5] this technique is used for proving full orthogonality of the Macdonald polynomials.
Koornwinder [3] introduced the orthogonality measure and the Askey-Wilson polynomi-als for (a, b, c, d)∈ VAW with|e| ≤ 1 for all e ∈ {a, b, c, d} and for continuous parameter t ∈ (−1, 1). Theorem 2, Corollary 1 and Theorem 3 were proved for these parameter values. For a survey of the techniques we used in this section, see [9, section 2].
Remark 3 In [6], the zonal spherical functions on a one parameter family of quantum
Grassmannians of fixed rank are recognized as multivariable Askey-Wilson polynomials involving parameters (a, b, c, d) in a two parameter subset of VAW and t = q (i.e. k = 1).
Parameters (a, b, c, d) ∈ VAW with|e| > 1 for some e ∈ {a, b, c, d} occur in this two
parameter subset of VAW.
Remark 4 The multivariable big and little q-Jacobi polynomials which were introduced
in [7], are limit cases of the Askey-Wilson polynomials (cf. [9]). The support of the orthogonality measure for the big resp. little q-Jacobi polynomials consists of infinitely many discrete mass points. In [8] it is shown that formally for t = qk (k∈ N), only the completely discrete part of the orthogonality measure for the Askey-Wilson polynomials survives under these limit transitions, and that the support of the completely discrete part tends to the support of the orthogonality measure for the big resp. little q-Jacobi polynomials.
4. Proof of Proposition 2
Fix parameters q, a, b, c, d and k which satisfy the conditions given in Proposition 2. If|e| ≤
1 for all e∈ {a, b, c, d}, then the functions z 7→ (DAWmλ)(z)mµ(z) are in L1(WAW,0)
for µ, λ∈ P+because D
AW maps AW into itself (Proposition 1) and the weight function
∆AW,0is continuous on Tn. So for the first assertion of Proposition 2, i.e. that the maps
z7→ (DFrmλ)(z)mµ(z) resp. z7→ (DrL−1mλ)(z)mµ(z) (λ, µ∈ P+) (4.1)
are in L1(WAW,r) resp. L1(WAW,r−1) for parameters satisfying the conditions of
Propo-sition 2, we may assume that|e| > 1 for some e ∈ {a, b, c, d} in view of Remark 1. It is sufficient to check that the maps z 7→ φ±AW,j(z) (given by (3.8) resp. (3.9)) are in
L1(W
AW,r) for j≤ r resp. in L1(WAW,r−1) for j≥ r, so we need to check that factors in
the denominator of φ±j with zeros on the (q-)integration domain WAW,rresp. WAW,r−1,
cancel out with factors in the numerator of the weight function ∆r(z) (if j ≤ r) resp.
∆r−1(z) (if j≥ r).
Fix parameters q, a, b, c, d and k which satisfy the conditions given in Proposition 2, and assume that|e| > 1 for some e ∈ {a, b, c, d}. Let j ≤ r. The factors (1−zl−1zj) with l≤ r, l6= j in the denominator of φ−AW,j(z) resp. φ+AW,j(z) have zeros on WAW,r. These factors
cancel out with factors in the numerator of δk(z). No other factors in the denominator of φ±AW,j(z) (j ≤ r) have zeros on WAW,rby the conditions on the parameters. Hence the
maps z7→ φ±AW,j(z) are in L1(W
Let j≥ r. In this case, the factors in the denominator of φ−AW,j(z) resp. φ+AW,j(z) which
have zeros on WAW,r−1are (1− zj2) and (1− zl±1zj) with l ≥ r and l 6= j. The factors
(1−zl±1zj) cancel out with factors in the numerator of δk(z), and (1−zj2) cancels out with
a factor in the numerator of wAW,c(zj) (the factor (1− zj2) in the numerator of wAW,c(zj)
is not needed to compensate a factor in the denominator of wAW,c(zj) by the conditions on
the parameters). Hence the maps z7→ φ±AW,j(z) are in L1(W
AW,r−1) for j≥ r.
In particular, we may change order of (q-)integration when dealing with the multidimen-sional (q-)integrals
hDF
rmλ, mµir− hmλ, DFrmµir (4.2)
and
hmλ, DLr−1mµir−1− hDLr−1mλ, mµir−1. (4.3)
We will introduce some new notations before proving the second statement of Proposition 2. We will remove from now on the subindex AW . Let
ψ−j(z) :=− φ − j(z) (1− q)zj , ψ+j(z) := φ + j(z) (1− q)zj . (4.4)
The second order q-difference operator D can then be rewritten as
D = n X j=1 ¡ ψj−(z) (Tq,j− Id ) + ψ+j(z) ¡ Tq−1,j − Id ¢¢ . (4.5)
When summation is taken over e or over ej, it is always over the four parameters a, b, c, d,
so
X
e
g(e) := g(a) + g(b) + g(c) + g(d).
For r∈ {1, . . . , n} and z = (z1, . . . , zn)∈ Cnwe write
ˆ
zr:= (z1, . . . , zr−1, zr+1, . . . , zn)∈ Cn−1.
For q∈ (0, 1), t = qk (k∈ N) and (a, b, c, d) ∈ V (V given by Definition 1) with |e| > 1
for some e∈ {a, b, c, d}, denote
ˆ
Wr:={ˆzr| z ∈ Wr} = {ˆzr| z ∈ Wr−1} (4.6)
for r = 1, . . . , n (Wrgiven by (3.5)). In particular, we have for r = 1 that
ˆ
W1={ˆz1| z ∈ W0}, (4.7)
where W0 = Tnis the n-torus. Note that (4.7) is also well defined when (a, b, c, d)∈ V
with|e| ≤ 1 for all e ∈ {a, b, c, d}. Write for γ∈ C and z = (z1, . . . , zn),
ˆ
zr(γ) := (z1, . . . , zr−1, γ, zr+1, . . . , zn).
For (a, b, c, d)∈ V write F0for the L1functions on the torus W0= Tnwith respect to the
measure dz1 z1 . . .dzn zn , z∈ Tn.
Define a linear map I0:F0→ C by
I0g := Z .. Z z∈Tn g(z)dz1 z1 . . .dzn zn .
Note that f ∆0 ∈ F0 if f ∈ L1(W0). For (a, b, c, d) ∈ V with |e| > 1 for certain
e∈ {a, b, c, d}, the L1-functions on Wrwith respect to the measure dq,Ne1z1 (1− q)z1 . . . dq,Nerzr (1− q)zr dzr+1 zr+1 . . .dzn zn , (4.8)
z = (z1, . . . , zn) = (e1ql1, . . . , erqlr, zr+1, . . . , zn) ∈ Wr, will be denoted by Fr for r = 1, . . . , n. Define for these parameter values a linear map Ir:Fr→ C by
Irg := X e1,...,er Z e1 z1=0 .. Z er zr=0 Z .. Z (zr+1,...,zn)∈Tn−r (4.9) g(z)dq,Ne1z1 (1− q)z1 ..dq,Nerzr (1− q)zr dzr+1 zr+1 ..dzn zn .
Note that f ∆r∈ Frif f ∈ L1(Wr). We have the following lemma.
Lemma 2 Fix λ, µ∈ P+and r∈ {1, . . . , n}. Fix parameters q, a, b, c, d and k satisfying
the conditions given in Proposition 2. (I) We have hDF rmλ, mµir− hmλ, DFrmµir= 2r¡n r ¢ r (2πi)n−rIr ¡ fλ,µr ∆r ¢ (4.10) with fλ,µr ∈ L1(Wr) given by fλ,µr (z) :=¡Dr,q−mλ ¢ (z)mµ(z)φ−r(z) + ¡ Dr,+q mλ ¢ (z)mµ(z)φ+r(z) (4.11) −mλ(z) ¡ Dqr,−mµ ¢ (z)φ−r(z)− mλ(z) ¡ Dr,+q mµ ¢ (z)φ+r(z). (II) We have hmλ, DLr−1mµir−1− hDLr−1mλ, mµir−1= 2r¡nr¢r (2πi)n−r+1Ir−1 ¡ gλ,µr ∆r−1 ¢ (4.12) with gr λ,µ∈ L 1(W r−1) given by gλ,µr (z) = mλ(z) ¡ Tq−1,rmµ ¢ (z)ψr+(z)− (Tq,rmλ) (z)mµ(z)ψ−r(z). (4.13)
Remark 5 For parameters satisfying the extra conditions|e| ≤ 1 for all e ∈ {a, b, c, d},
Lemma 2 should be read as the statement that (4.12) holds for r = 1, i.e. that
hmλ, DAWmµiAW − hDAWmλ, mµiAW =
2n (2πi)nI0(g
1
λ,µ∆0).
Proof: (I) We may assume that|e| > 1 for some e ∈ {a, b, c, d} by Remark 5.
Note that for s≤ r and ν ∈ P+,
φ±s(z0) = φ±s(z0), z0∈ Wr a.e. and mν(z0) = mν(z0), ¡ Dqs,±mν ¢ (z0) = ¡ Dqs,±mν ¢ (z0), ∀z0∈ Wr. In particular, (DF rmν) (z0) = ¡ DF rmν ¢
(z0), z0∈ Wra.e. for all ν∈ P+. Hence
hDF rmλ, mµir− hmλ, DFrmµir= 2r¡n r ¢ (2πi)n−rIr ³ ˆ fλ,µr ∆r ´ with ˆ fλ,µr (z) :=¡DrFmλ ¢ (z)mµ(z)− mλ(z) ¡ DFrmµ ¢ (z).
Let w∈ Sn, and let k, l∈ {1, . . . , n} such that w(k) = l. We have
¡
wφk±¢(z) = φ±l (z), wDqk,±w−1 = Dl,q±. (4.14) Let Sr ⊂ Sn be the subgroup consisting of permutations of{1, . . . , n} which act as the
identity on{r + 1, . . . , n}. It follows from (4.14) that DrF = 1 (r− 1)! X w∈Sr ¡ (wφ−r)(z)wDqr,−w−1+ (wφ+r)(z)wDqr,+w−1¢. (4.15) Hence ˆ fλ,µr (z) = 1 (r− 1)! X w∈Sr ¡ wfλ,µr ¢ (z), (4.16) with fr
λ,µgiven by (4.11). Formula (4.10) follows, since (w∆r) (z) = ∆r(z) and Ir(wf ) = Ir(f ) for w∈ Srand f∈ Fr.
(II) If|e| ≤ 1 for all e ∈ {a, b, c, d}, then the proof should be read for r = 1 only in view
of Remark 5.
Note that for s≥ r and ν ∈ P+,
ψ+s(z0) = ψs−(z0), z0∈ Wr−1 a.e.
mν(z0) = mν(z0), ¡ Tq±1,smν ¢ (z0) = ¡ Tq∓1,smν ¢ (z0), ∀z0∈ Wr−1. In particular¡DL r−1mν ¢ (z0) = ¡ DrL−1mν ¢
(z0), z0 ∈ Wr−1a.e. for all ν∈ P+. Hence
we have hmλ, DLr−1mµir−1− hDLr−1mλ, mµir−1= 2r−1³ n r−1 ´ (2πi)n−r+1Ir−1 ¡ ˆ gλ,µr ∆r−1 ¢ with ˆgr λ,µ∈ L1(Wr−1) given by ˆ gλ,µr (z) = mλ(z) ¡ DLr−1mµ ¢ (z)−¡DLr−1mλ ¢ (z)mµ(z). Let w∈ W.
(a) If w(²r) = ²s, then (wψr±) (z) = ψs±(z) and wTq±1,rw−1= Tq±1,s.
(b) If w(²r) =−²s, then (wψ±r) (z) = ψs∓(z) and wTq±1,rw−1= Tq∓1,s.
LetWn−r+1be the subgroup ofW consisting of elements w ∈ W which acts as the identity
on{±²1, . . . ,±²r−1} (the action of W on {±²1, . . . ,±²n} as given in the beginning of
section 3). It follows from (a) and (b) that
DrL−1 = 1 2n−r(n− r)! X w∈Wn−r+1 ¡ wψr− ¢ (z)w (Tq,r− Id ) w−1 (4.17) = 1 2n−r(n− r)! X w∈Wn−r+1 ¡ wψr+ ¢ (z)w¡Tq−1,r− Id ¢ w−1. Hence we have ˆ gλ,µr (z) = 1 2n−r(n− r)! X w∈Wn−r+1 ¡ w˜grλ,µ¢(z) (4.18) with ˜gr λ,µ∈ L1(Wr−1) given by ˜ gλ,µr (z) = mλ(z) ¡ Tq−1,rmµ ¢ (z)ψ+r(z)− (Tq,rmλ) (z)mµ(z)ψ−r(z) (4.19) +mλ(z)mµ(z)ψr−(z)− mλ(z)mµ(z)ψ+r(z).
Since (w∆r−1) (z) = ∆r−1(z) and Ir−1(wg) = Ir−1(g) for w∈ Wn−r+1and g∈ Fr−1,
we get hmλ, DLr−1mµir−1− hDLr−1mλ, mµir−1 = 2r¡nr¢r (2πi)n−r+1Ir−1 ¡ ˜ grλ,µ∆r−1 ¢ . (4.20)
Let wr∈ Wn−r+1be given by wr(±²r) =∓²rand wr(±²j) =±²jfor j 6= r, then Ir−1 ¡ mλmµψ−r∆r−1 ¢ = Ir−1 ¡ wr ¡ mλmµψr+∆r−1 ¢¢ = Ir−1 ¡ mλmµψ+r∆r−1 ¢ , so we may replace ˜gr
λ,µin the right hand side of (4.20) by g r
λ,µ(given by (4.13)). This
Lemma 3 Fix λ, µ∈ P+and r∈ {1, . . . , n}. Fix parameters q, a, b, c, d and k satisfying the conditions given in Proposition 2. Define hr
λ,µ∈ L
1(W
r) by
hrλ,µ(z) := (Tq,rmλ) (z)mµ(z)ψ−r(z)− mλ(z) (Tq,rmµ) (z)ψr−(z). (4.21)
(I) For e∈ {a, b, c, d} with |e| > 1 and for ˆzr∈ ˆWr, we have
Z e x=0 fλ,µr (ˆzr(x))∆r(ˆzr(x)) dq,Nex (1− q)x = h r λ,µ ¡ ˆ zr(eqNe) ¢ ∆r ¡ ˆ zr(eqNe) ¢ . (4.22)
(II) For ˆzr∈ ˆWrwe have
1 2πi Z x∈C gλ,µr (ˆzr(x))∆r−1(ˆzr(x)) dx x = X e:|e|>1 hrλ,µ¡zˆr(eqNe) ¢ ∆r ¡ ˆ zr(eqNe) ¢ . (4.23)
Remark 6 If|e| ≤ 1 for all e ∈ {a, b, c, d}, then Lemma 3 should be read as the statement
that (4.23) is valid for r = 1, i.e. that
1 2πi Z x∈C gλ,µ1 (ˆz1(x))∆0(ˆz1(x)) dx x = 0 (4.24) for ˆz1∈ ˆW1= Tn−1.
Proof: (I) Fix an e ∈ {a, b, c, d} with |e| > 1. We will use the partial q-integration
rule (2.1) and a functional relation (which generalizes (2.16)) for the proof of (4.22). De-fine wAW,d(eq−1; e) := 0, so in particular ∆r(ˆzr(eq−1)) = 0 for ˆzr ∈ ˆWr. Define wAW,d(eqNe+1; e) by formula (2.8) with i = Ne+ 1, then wAW,d(eqNe+1; e) is well
defined in view of the conditions on the parameters.
Lemma 4 We have
ψ+r(ˆzr(eqm+1))∆r(ˆzr(eqm+1)) = ψ−r(ˆzr(eqm))∆r(ˆzr(eqm))
for ˆzr∈ ˆWrand m∈ {−1, . . . , Ne}.
Proof: The lemma is valid for m =−1, because ∆r(ˆzr(eq−1)) = 0 and ψ+r(ˆzr(e)) = 0.
For m∈ {0, . . . , Ne} the lemma follows by combining (2.16) with the functional equation Tq,r δk(z) Y l6=r (qk− zrzl)(qk− zrzl−1) (1− zrzl)(1− zrzl−1) = δk(z) Y l6=r (1− qkzrzl)(1− qkzrzl−1) (1− zrzl)(1− zrz−1l ) .
Fix ˆzr∈ ˆWrand denote
f1(zr) := mλ(z), f2(zr) :=−mµ(z)ψ−r(z)∆r(z), f3(zr) := mλ(z), f4(zr) := mµ(z)ψ+r(z)∆r(z),
and fL(x) := ¡ D−q f1 ¢ (x)f2(x) + ¡ D+qf3 ¢ (x)f4(x), fR(x) := f1(x) ¡ D+qf2 ¢ (x) + f3(x) ¡ D−qf4 ¢ (x).
Lemma 4 implies that
fL(x) + fR(x) = fλ,µr (ˆzr(x))
∆r(ˆzr(x))
(1− q)x
for all x∈ {e, . . . , eqNe} and that
−f1 ¡ eqNe+1¢f 2 ¡ eqNe¢− f 3 ¡ eqNe¢f 4 ¡ eqNe+1¢= hr λ,µ ¡ ˆ zr(eqNe) ¢ ∆r ¡ ˆ zr(eqNe) ¢ . On the other hand, applying the partial q-integration rule (2.1) twice gives that
Z e x=0 fL(x)dq,Nex =− Z e x=0 fR(x)dq,Nex + f1(e)f2(eq −1) + f 3(eq−1)f4(e) −f1(eqNe+1)f2(eqNe)− f3(eqNe)f4(eqNe+1).
Note that f2(eq−1) = 0 and f4(e) = 0 because ∆r(ˆzr(eq−1)) = 0 and ψ+r(ˆzr(e)) = 0, so
we conclude that Z e x=0 fλ,µr (ˆzr(x))∆r(ˆzr(x)) dq,Nex (1− q)x = −f1(eq Ne+1)f
2(eqNe)− f3(eqNe)f4(eqNe+1) = hrλ,µ¡zˆr(eqNe) ¢ ∆r ¡ ˆ zr(eqNe) ¢ .
(II) If|e| ≤ 1 for all e ∈ {a, b, c, d}, then the proof should be read for r = 1 only in view
of Remark 6. Fix
ˆ
zr= (z1, . . . , zr−1, zr+1, . . . , zn) = (e1qi1, . . . , er−1qir−1, zr+1, . . . , zn)∈ ˆWr−1.
Without loss of generality, we may assume that zi6= qlzjfor all l = 0, . . . , k−1 if 1 ≤ i 6= j≤ r − 1 by Lemma 1. We start with rewriting the integrand grλ,µ(z)∆r−1(z)
zr as function
of zr∈ C in a more suitable form. Define therefore three functions ∆−r−1(z), ∆
+ r−1(z) and Ψr−1(z) by ∆−r−1(z) := n Y l=r ¡ z2l; q¢∞ (azl, bzl, czl, dzl; q)∞ Y 1≤i<j≤n j6=r ¡ zizj, ziz−1j ; q ¢ k rY−1 m=1 ¡ zrzm, zrzm−1; q ¢ k, ∆+r−1(z) := ∆−r−1(z1, . . . , zr−1, z−1r , . . . , z−1n ) and Ψr−1(z) := ∆r−1(z) ∆−r−1(z)∆+r−1(z) (4.25) = rY−1 l=1 wAW,d(zl; el) Y 1≤i<j≤n i≤r−1; j6=r ¡ z−1i zj, zi−1zj−1; q ¢ k ¡ zizj, zizj−1; q ¢ k .
Note that Ψr−1(z) is independent of zr. The conditions on the parameters a, b, c, d and on ˆ zr∈ ˆWrimply that ¡ zizj, zizj−1; q ¢ k 6= 0 for i ≤ r − 1, i < j and j 6= r, so Ψr−1(z) is well defined.
Note that for the special case r = 1, we have ∆−0(z) = ∆+0(z) and ∆0(z) = ∆−0(z)∆ + 0(z)
for z∈ W0= Tn.
We can rewrite ψr±(z) in terms of ∆±r−1as
ψ±r(z) = ¡ Tq∓1,r∆±r−1 ¢ (z) ∆±r−1(z) . (4.26) Hence gλ,µr (z) ∆r−1(z) zr = q−1¡Tq−1,rgˇλ,µr ¢ (z)− ˇgλ,µr (z) (4.27) with ˇgr λ,µgiven by ˇ gλ,µr (z) = Ψr−1(z) ¡ Tq,r ¡ mλ∆−r−1 ¢¢ (z)mµ(z) ∆+r−1(z) zr (4.28) = (Tq,rmλ) (z)mµ(z)ψ−r(z) ∆r−1(z) zr .
Consider now the following closed, oriented contourD in the complex plane.
Definition 3 (ContourD)
LetD be the closed contour in C given by the image of ˜D under the map u 7→ eiu, where
˜
D is the closed contour in C given by the following union of line segments, ˜
D = [−π/2, 3π/2] ∪ [3π/2, 3π/2 + i log q] ∪ [3π/2 + i log q, −π/2 + i log q] ∪ [−π/2 + i log q, −π/2].
GiveD the orientation induced from the counterclockwise orientation on ˜D. The function x7→ ˇgr
λ,µ(ˆzr(x)) has no poles onD, and simple poles within D at the points
{eqNe,¡eqNe+1¢−1 | e ∈ {a, b, c, d}, |e| > 1} (4.29)
(one needs here the extra conditions on the parameters a, b, c and d, given in Proposition 2). Note that the poles are independent of ˆzr. So we have that
1 2πi Z x∈C gλ,µr (ˆzr(x))∆r−1(ˆzr(x)) dx x = 1 2πi Z x∈D ˇ grλ,µ(ˆzr(x))dx (4.30)
by (4.27) and that (4.24) is valid if|e| ≤ 1 for all e ∈ {a, b, c, d} by Cauchy’s Theorem. So we may assume that there exists an e ∈ {a, b, c, d} with |e| > 1. We will calculate
the residues of ˇgr
λ,µ(ˆzr(x)) for x a pole withinD (so in the set given by (4.29)). For e∈ {a, b, c, d} with |e| > 1, it follows from (2.11) that
Resx=eqNe µ ∆r−1(ˆzr(x)) x ¶ = ∆r ¡ ˆ zr(eqNe) ¢ ,
Resx=(eqNe+1)−1 µ ∆r−1(ˆzr(x)) x ¶ = −∆r ¡ ˆ zr(eqNe+1) ¢ .
Now we can rewrite the residue of ˇgr
λ,µ(ˆzr(x)) at the pole x = e−1q−Ne−1using ψ−r ¡ˆzr(e−1q−Ne−1) ¢ ∆r ¡ ˆ zr(eqNe+1) ¢ = ψr+¡zˆr(eqNe+1) ¢ ∆r ¡ ˆ zr(eqNe+1) ¢ = ψr−¡zˆr(eqNe) ¢ ∆r ¡ ˆ zr(eqNe) ¢ , and we obtain Resx=eqNe ¡ ˇ grλ,µ(ˆzr(x)) ¢
+ Resx=(eqNe+1)−1 ¡ ˇ gλ,µr (ˆzr(x)) ¢ = (4.31) = hrλ,µ¡zˆr(eqNe) ¢ ∆r ¡ ˆ zr(eqNe) ¢
for e∈ {a, b, c, d} with |e| > 1. By Cauchy’s Theorem, (4.30) and (4.31), we obtain (4.23). This completes the proof of the lemma.
Proof of Proposition 2: Fix parameters q, a, b, c, d and k, such that they satisfy the
condi-tions given in Proposition 2. Suppose that|e| ≤ 1 for all e ∈ {a, b, c, d}, then we have to proof that DAW is selfadjoint with respect toh., .i0in view of Remark 1. This follows
im-mediately by combining the statements of Lemma 2 and Lemma 3 for the case that|e| ≤ 1 for all e∈ {a, b, c, d} (see Remark 5 and Remark 6). If |e| > 1 for some e ∈ {a, b, c, d}, then we have to prove (3.15) for r = 1, . . . , n. Lemma 2 and Lemma 3 give then that
hDF rmλ, mµir− hmλ, DFrmµir = 2r¡nr¢r (2πi)n−rIr ¡ fλ,µr ∆r ¢ = 2 r¡n r ¢ r (2πi)n−r+1Ir−1 ¡ gλ,µr ∆r−1 ¢ = hmλ, DLr−1mµir−1− hDLr−1mλ, mµir−1.
The proof for the special case that|e| ≤ 1 for all e ∈ {a, b, c, d} is in essence the same as the proof given by Koornwinder [3, Lemma 5.3].
Remark 7 The proof of Lemma 3(I) uses techniques introduced in [7, section 6]. In [7],
second order q-difference operators where introduced which diagonalize the multivariable big and little q-Jacobi polynomials. The partial q-integration rule for Jackson q-integrals was used to prove selfadjointness of these second order q-difference operators with respect to the orthogonality measures of the big and little q-Jacobi polynomials. In these cases, the selfadjointness followed from a similar type of functional relation as in Lemma 4, together with the observation that the sum of the stockterms is zero.
Remark 8 Recently the author was able to extend the results in this paper to arbitrary
deformation parameter t ∈ (0, 1) using completely different methods. The methods are more in the spirit of the recent paper of Heckman and Opdam [2]. In [2] a multidimensional residue calculus was developed for the Plancherel measure associated with Yang systems. As it turns out, one can also develop a residue calculus (which is more simple then in [2]) for the measure
∆AW,0(z) dz1
z1
. . . ,dzn zn
with weight function ∆AW,0given by (3.2). Using this residue calculus, one can give an
ex-plicit positive orthogonality measure for the Askey-Wilson polynomials when (a, b, c, d)∈ VAW and t∈ (0, 1). A paper on this subject is in preparation.
Acknowledgments
The author thanks Prof. T.H. Koornwinder for valuable comments on an earlier version of the manuscript.
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