Raising and Lowering Operators for Askey-Wilson Polynomials

0.18 MB·English

by  Siddhartha Sahi

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3(2007), 002, 11 pages
Raising and Lowering Operators
⋆
for Askey–Wilson Polynomials
Siddhartha SAHI
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA
E-mail: [email protected]
7 Received September 20, 2006,in final form December 27, 2006; Published online January 04, 2007
0 Original article is available at http://www.emis.de/journals/SIGMA/2007/002/
0
2
Abstract. In this paper we describe two pairs of raising/lowering operators for Askey–
n
a Wilson polynomials, which result from constructions involving very different techniques.
J The first technique is quite elementary, and depends only on the “classical” properties of

4 thesepolynomials,viz.theq-difference equationandthe threetermrecurrence. Thesecond
techniqueislesselementary,andinvolvestheone-variableversionofthedoubleaffine Hecke
]
A algebra.
Q Keywords: orthogonalpolynomials;Askey–Wilsonpolynomials;q-differenceequation;three
. term recurrence; raising operators; lowering operators; root systems; double affine Hecke
h
algebra
t
a
m 2000 Mathematics Subject Classification: 33D45; 33D52; 33D80
[

1
Dedicated to the memory of Vadim Kuznetsov
v
4
3
1 1 Introduction
1
0
7 One of the approaches to proving integrality of the (q,t)-Kostka coefficients is the idea, due to
0
Kirillov–Noumi [9, 10] and Lapointe–Vinet [16, 17], of using raising operators for Macdonald
/
h polynomials. (See also [6, 7, 11, 21] for other approaches.) In their proof Kirillov and Noumi
t
a give an explicit construction of such raising operators for the Macdonald polynomials J (x;q,t)
λ
m
for the root system of type A . They also pose the problem of finding analogous operators
n−1
: for the six-parameter Koornwinder corresponding to the BC root system.
v n
i ThisquestionwasalsoraisedbyTomKoornwinderattheEdinburghconferenceonsymmetric
X
functions organized by Vadim Kuznetsov. The case n = 1 corresponds to the celebrated Askey–
r
a Wilson polynomials and Koornwinder’s paper [14] from that conference contains partial results
in this direction as well as a survey of earlier results.
In this paper we such construct raising/lowering operators for Askey–Wilson polynomials.
In fact we describe two such pairs of operators, which result from constructions involving very
different techniques.
The first technique is quite elementary, and depends only on the “classical” properties of
these polynomials, viz. the q-difference equation and the three term recurrence. Therefore it
can be applied to all the polynomials in the Askey scheme. After this work was completed, we
obtained a recent preprint by T. Koornwinder [15], the main result of which is very close to this
approach. Also through [15] we discovered still earlier work of G. Bangerezako [2] which obtains
similar operators based on an ad-hoc factorization of the Askey–Wilson operator. Our approach
however is more direct and quite short.
⋆
ThispaperisacontributiontotheVadimKuznetsovMemorialIssue“IntegrableSystemsandRelatedTopics”.
The full collection is available at http://www.emis.de/journals/SIGMA/kuznetsov.html
2 S. Sahi
The second technique is less elementary and involves the one-variable version of the powerful
Hecke algebra method as described in [18, 19, 20, 22, 23, 24]. This approach is related to a
fairly remarkable mathematical object – the double affine Hecke algebra (see [3, 4, 5, 22]).
The calculations, while non-trivial to carry out, are conceptually rather straightforward. The
raising/lowering operators so obtained are different from those coming from the “classical”
method. This method also provides a new factorization of the Askey–Wilson operator described
in Lemma 3, which is much simpler than that of Bangerezako.
In subsequent work, we hope to extend these methods to construct raising operators for
Koornwinder polynomials [13, 25, 22, 23].
2 The classical approach
2.1 Askey Wilson polynomials
The q-hypergeometric series is given by the formula
rφs a1,...,ar q;y = (a1,...,ar)k (−1)(1+s−r)kq(1+s−r)(k2) yk ,
b ,...,b (b ,...,b ) (q)
(cid:18) 1 s(cid:12) (cid:19) k≥0 1 s k k
(cid:12) X
(cid:12)
where the “q-Pochham(cid:12) mer symbols” are defined by
(a,b,c,...) := (a) (b) (c) ··· ,
k k k k
(a) := (1−a)(1−aq)··· 1−aqk−1 .
k
(cid:0) (cid:1)
The Askey–Wilson polynomials [1] are defined by the formula
(ab,ac,ad) q−n,abcdqn−1,az,az−1
P (z;a,b,c,d|q) = n φ q;q .
n an(abcdqn−1) 4 3 ab,ac,ad
n (cid:18) (cid:12) (cid:19)
(cid:12)
Since (q−n) vanishes for k >n, we have (cid:12)(cid:12)
k
n abcdqn−1 (q−n) qk
φ (···) = k k (az) az−1 .
4 3 (ab,ac,ad) (q) k k
k=0"(cid:0) (cid:1)k#(cid:20) k (cid:21)
X (cid:0) (cid:1)
It follows that P is a Laurent polynomial of degree n, which is moreover symmetric in z
n
and z−1 and is of the form
P (z;a,b,c,d|q) = zn +z−n + lower terms.
n
(cid:0) (cid:1)
It is also symmetric in {a,b,c,d} although this is not entirely obvious from the formula above.
We have chosen to normalize P in order to make it monic. Of course there are several other
n
possible normalizations, and we discuss some of these below.
First of all, we remark that formula (3.1.7) of [12] considers the polynomial
(ab,ac,ad) q−n,abcdqn−1,az,az−1
n φ q;q
an 4 3 ab,ac,ad
(cid:18) (cid:12) (cid:19)
(cid:12)
(cid:12)
which is abcdqn−1 n times our Pn. (cid:12)
Next, since the Askey–Wilson polynomial is symmetric in z, z−1, it can be expressed as an
(cid:0) (cid:1)
(ordinary) polynomial of degree n in
x = z+z−1 /2.
(cid:0) (cid:1)
Raising and Lowering Operators for Askey–Wilson Polynomials 3
The function p (x) considered in (3.1.5) of [12], is monic in x, and hence is related to our
n
normalization P by the formula
n
z+z−1
p = 2−nP (z).
n n
2
(cid:18) (cid:19)
Finally, the polynomials P are orthogonal with respect the inner product h·,·i defined in
n
(3.1.2) of [12]. If we define
Q = γ P , (1)
n n n
where
(abqn,acqn,adqn,bcqn,bdqn,cdqn,qn+1)
γ = ∞ abcdqn−1 .
n (abcdq2n) n
∞
(cid:0) (cid:1)
Then Q is dual to P in the sense that
n n
hP ,Q i =δ .
m n m,n
2.2 Raising and lowering operators
The main result of this section are the following raising and lowering operators for the Askey–
Wilson polynomials:
Theorem 1. For all n> 1, the Askey–Wilson polynomials satisfy the relations
D z+z−1 −λ z+z−1 −α (λ −λ ) P = (λ −λ )P ,
n−1 n n n−1 n n+1 n−1 n+1
D z+z−1 −λ z+z−1 −α (λ −λ ) Q = (λ −λ )Q ,
(cid:2) (cid:0) (cid:1) n+1(cid:0) (cid:1) n n n+1 (cid:3) n n−1 n+1 n−1
where(cid:2)D,(cid:0)λ and (cid:1)α are as(cid:0)in (3) a(cid:1)nd (6) below. (cid:3)
n n
Proof. The proof involves two key properties of the Askey–Wilson polynomials.
The first property is the ‘q-difference equation’ from (3.1.7) ) of [12] which asserts that P
n
is an eigenfunction for the Askey–Wilson operator, i.e.
DP (z) = λ P (z). (2)
n n n
The operator and its eigenvalue are defined by
D = A(z)(Tq −1)+A z−1 Tq−1 −1 , (3)
λ = q−n−1 1−abcdqn−1 = q−n+abcdqn−1 − 1+abcdq−1 ,
n (cid:0) (cid:1)(cid:0) (cid:1)
(cid:0) (cid:1)(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
where A(z) is the rational function
(az,bz,cz,dz) (1−az)(1−bz)(1−cz)(1−dz)
A(z) = 1 = (4)
(z2) (1−z2)(1−qz2)
2
and T is the shift operator
q
T f(z) = f (qz).
q
(To forestall possible confusion we emphasize that, in accordance with custom, we think of f (z)
as a Laurent polynomial rather than as a function of z. This means that we have T zk = qkzk
q
rather than T zk = q−kzk.)
q
(cid:0) (cid:1)
(cid:0) (cid:1)
4 S. Sahi
The second key property of these polynomials is the ‘normalized recurrence relation’ from
(3.1.5) of [12] which can be rewritten in the form
γ
z+z−1 P = P +α P + n−1P for n > 1, (5)
n n+1 n n n−1
γ
n
(cid:0) (cid:1)
where
a bcqn−1,bdqn−1,cdqn−1,qn abqn,acqn,adqn,abcdqn−1
α = a+1/a− 1 − 1. (6)
n (abcdq2n−2) a(abcdq2n−1)
(cid:0) 2 (cid:1) (cid:0) 2 (cid:1)
We combine these two properties as follows:
First apply the operators D−λ and D−λ , respectively, to the recurrence relation to
n−1 n+1
get
(D−λ ) z+z−1−α P = (λ −λ )P ,
n−1 n n n+1 n−1 n+1
γ
(D−λ )(cid:0)z+z−1−α (cid:1)P = n−1 (λ −λ )P .
n+1 n n n−1 n+1 n−1
γ
n
(cid:0) (cid:1)
Finally simplify, using the q-difference equation (2), to get
D z+z−1 −λ z+z−1 −α (λ −λ ) P = (λ −λ )P ,
n−1 n n n−1 n n+1 n−1 n+1
D z+z−1 −λ z+z−1 −α (λ −λ ) Q = (λ −λ )Q
(cid:2) (cid:0) (cid:1) n+1(cid:0) (cid:1) n n n+1 (cid:3) n n−1 n+1 n−1
as des(cid:2)ired(cid:0). (cid:1) (cid:0) (cid:1) (cid:3) (cid:4)
3 The Hecke algebra approach
In this section we provide raising/lowering operators for Askey–Wilson polynomials based on
Hecke algebra considerations [22, 23]. Once again the main idea is quite straightforward, al-
though the calculations are a little more intricate. The resulting formulas are different and
perhaps slightly simpler.
3.1 The Hecke algebra
The key to this approach are the involutions s , s which act on Laurent polynomials as follows:
1 0
s f (z) = f z−1 and s f(z)= f qz−1 .
1 0
Once again we re(cid:0)gard(cid:1)these operators as acting(cid:0)on po(cid:1)lynomials, rather than functions, so that
we have
s zk = z−k and s zk = qkz−k.
1 0
(cid:16) (cid:17) (cid:16) (cid:17)
These operators provide a factorization of the q-shift operators, and one has
s1s0 = Tq and s0s1 = Tq−1.
TheaffineHeckealgebra[22,23]isthealgebraofoperatorsgeneratedbythetwooperatorsT
0
and T defined as
1
T := t +r (s −1),
i i i i
where
(z−c)(z−d)
t = −cd/q, r = , (7)
0 0 (z2−q)
(1−az)(1−bz)
t = −ab, r = .
1 1 (1−z2)
Raising and Lowering Operators for Askey–Wilson Polynomials 5
1/2
Remark 1. The operator T as defined here is t times the corresponding operator from [22,
i i
23]. This accounts for the slight difference between the formulas here and in [23].
From the definition of T it follows that a polynomial f is symmetric in z, z−1, if and only if
1
T f = t f. (8)
1 1
Consequently, if g is any polynomial, then the quadratic relation (9) implies that (T +1)g
1
is a symmetric polynomial. The operators T are deformations of s and satisfy a quadratic
i i
relation. For the convenience of the reader unfamiliar with the Hecke algebra, we give a proof
this relation.
Lemma 1. The operators T satisfy the relation
i
(T −t )(T +1) = 0. (9)
i i i
Proof. Define s (r ) = r′, then we claim that
i i i
r +r′ = t +1 (10)
i i i
To see this, we calculate for i = 0,
(z−c)(z−d) qz−1−c qz−1−d
r +r′ = +
0 0 (z2−q) (q2z−2−q)
(cid:0) (cid:1)(cid:0) (cid:1)
(z−c)(z−d) (q−cz)(q−dz)
= −
(z2−q) q(z2−q)
q z2−cz−dz+cd − q2−qcz−qdz+cdz2
=
q(z2−q)
(cid:0) (cid:1) (cid:0) (cid:1)
qz2+qcd−q2−cdz2 (q−cd) z2−q
= =
q(z2−q) q(z2−q)
(cid:0) (cid:1)
cd
= 1− = 1+t .
0
q
The calculation for i= 1 is similar and simpler.
Now the quadratic relation can be proved as follows:
(T −t )(T +1) = r (s −1)[t +1+r (s −1)]
1 1 1 i i i i i
= r [(t +1)(s −1)+(s r )(s −1)−r (s −1)]
i i i i i i i i
= r (t +1)(s −1)+ r′s (s −1)−r (s −1)
i i i i i i i i
= r (t +1)(s −1)+r′(1−s )−r (s −1)
i(cid:2) i i (cid:0)i (cid:1) i i i (cid:3)
= r (cid:2)t +1−r −r′ (s −1) by=(10) 0. (cid:3) (cid:4)
i i i i i
The following result is an(cid:0)immediate con(cid:1)sequence
Corollary 1. The operators T are invertible, with
i
t T−1 = T −t +1. (11)
i i i i
We will also need a number of commutation results between the T and the multiplication
i
operator by z. They follow directly from the definition, and we leave the (easy) proof to the
reader.
Lemma 2. The operators T satisfy the following commutation relations
i
zt T−1 = qT z−1+c+d,
0 0 0
(T +1)z−1 = t z−1+zT +a+b,
1 1 1
(T +1)z = z+t z−1T−1−(a+b).
1 1 1
6 S. Sahi
3.2 Nonsymmetric Askey–Wilson polynomials
The next ingredient in the Hecke algebra method are the nonsymmetric Askey–Wilson polyno-
mials. These are certain Laurent polynomials, E , n ∈ Z, which can be characterized up to
n
multiples as eigenfunctions of the operator
Y = T T .
1 0
More precisely, one has
YE = µ E , (12)
n n n
where
qn for n < 0,
µ =
n qnt t = qn−1abcd for n ≥ 0.
1 0
(cid:26)
The symmetric Askey–Wilson polynomials P are closely related to E . Up to normaliza-
|n| ±n
tion, one has P = E = 1, while for n > 0 we have up to a scalar
0 0
P ∼ (T +1)E = c±E +c± E . (13)
|n| 1 ±n n n −n −n
The explicit formula for the coefficients c± and c± is known, but will not be needed in what
n −n
follows.
We now define a slight variant of the Askey–Wilson operator, as follows:
D′ = A(z)(Tq −s1)+A z−1 Tq−1s1−1 . (14)
(cid:0) (cid:1)(cid:0) (cid:1)
Observe that D and D′ have the same action on functions which are symmetric z and z−1,
thus the Askey–Wilson polynomials satisfy
D′P = DP = λ P .
n n n n
Justastheoperators ands factorize theq-shiftoperator, itturnsoutthattheoperatorsT
0 1 0
and T provide a factorization of D′.
1
Lemma 3. The operator D′ of formula (14) admits the following factorization:
D′ = (T +1)(T −t ).
1 0 0
Proof. To prove this, we calculate as follows
(T +1)(T −t )= (t +r (s −1)+1)(r (s −1)) by=(10) r′ +r s (r s −r )
1 0 0 1 1 1 0 0 1 1 1 0 0 0
= r′r (s −1)+r r (s s −s ) = r′r T−1s −1 +r r (T −s ),
1 0 0 1 0 1 0 1 1 0 (cid:0)q 1 (cid:1) 1 0 q 1
(cid:0) (cid:1)
where r′ = r z−1 and r = r z−1 .
1 1 0 0 e e
Comparing the formulas for r (7) and A(z) (4), we conclude that
(cid:0) (cid:1) (cid:0)i (cid:1)
e
A(z) = r r and A z−1 = r′r
1 0 1 0
(cid:0) (cid:1)
which completes the proof. (cid:4)
e
Raising and Lowering Operators for Askey–Wilson Polynomials 7
3.3 Raising and lowering operators
To state our main result we need some notation. We write
e = a+b+c+d, e = abc+abd+acd+bcd. (15)
1 3
Also recall that for n ≥ 0, λ is the symmetric Askey–Wilson eigenvalue as in (3). For n < 0
n
we define
λ = λ
n |n|
and for all integral n we set
λ +1−µ 1−µ
n n−1 n−1
β = e − e . (16)
n 1 3
µ −µ µ −µ
n−1 −n n−1 −n
Theorem 2. The Askey–Wilson polynomials satisfy the following relations:
D′z+ 1−q1−n z+z−1 +β P = qnabcd−q1−n P , n ≥ 0, (17)
−n n n+1
D′z+(1−qnabcd) z+z−1 +β Q = q1−n−qnabcd Q , n > 0. (18)
(cid:2) (cid:0) (cid:1)(cid:0) (cid:1) n(cid:3) n (cid:0) (cid:1) n−1
(cid:2) (cid:0) (cid:1) (cid:3) (cid:0) (cid:1)
Thekey for the proof is the “affine intertwiner” for thenonsymmetric Askey–Wilson polyno-
mials from [23]. This involves the additional parameters u and u , which satisfy the relations
0 1
a = t1/2u1/2, b= −t1/2u−1/2, c= q1/2t1/2u1/2, d = −q1/2t−1/2u1/2.
1 1 1 1 0 0 0 0
Now from Theorem 1.2 of [23] we have, up to a multiple,
E ∼ (a U +b )E , (19)
n n 0 n −n−1
where
a = qn−q−n−1 and b = qn u−1/2−u1/2 +q−1/2 u−1/2−u1/2
n n 0 0 1 1
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
with
qn = µ t1/2t1/2
n 1 0
and U is the operator
0
U = q−1/2t1/2T−1z.
0 0 0
We will derive Theorem 2 from formula (19); however some remarks are in order before we
proceed:
1. Thereisatypointhestatementofformula(19)inTheorem1.2of[23],namelynand−n−1
have been inadvertently switched. This is easily seen by comparison with Theorem 4.1
from which Theorem 1.2 is derived.
2. The formula for U here is slightly different from that in [23] because of the difference
0
in T – see Remark 1).
0
3. Although Theorem 1.2 in [23] is only stated (and needed) for n ≥ 0, it is easy to see that
after the correction above it holds for all integer n.
8 S. Sahi
4. Finally, we note that the ideas of [22] and [23] work in the more general setting of Koorn-
winder polynomials, and they involve the affine intertwiner S , which can also be written
0
as
S = Y,z−1T−1 .
0 1
(cid:2) (cid:3)
It is expected that this operator will play a key role in the raising operators for Koorn-
wonder polynomials.
We now give the proof of Theorem 2.
Proof. We first simplify (19) by multiplying through by q1/2t1/2t . This gives
1 0
E ∼ (µ −µ )t T−1z−µ (c+d)−t (a+b) E
n n −n−1 0 0 n 0 −n−1
µ (c+d)+t (a+b)
∼ (cid:2)t T−1z− n 0 E . (cid:3)
0 0 µ −µ −n−1
(cid:20) n −n−1 (cid:21)
Replacing n by n−1, we get
µ (c+d)+t (a+b)
E ∼ t T−1z− n−1 0 E .
n−1 0 0 µ −µ −n
(cid:20) n−1 −n (cid:21)
For ease in subsequent calculations, we write this as
E ∼ t T−1z−κ E , (20)
n−1 0 0 n −n
(cid:0) (cid:1)
where
µ y+t x
n−1 0
κ = , x= a+b, y = c+d. (21)
n
µ −µ
n−1 −n
ThekeyideatoobtainaraisingoperatorforP isasfollows: Byformula(13),forn > 1,P is
n |n|
a combination of E and E . We first kill off the E component. This can be accomplished
n −n n
by applying the operator Y −µ to P . However it is more convenient (and equivalent) to apply
n n
the operator
t t
t t Y−1−µ−1 = t t T−1T−1− 1 0
1 0 n 1 0 0 1 µ
n
(cid:0) (cid:1)
For n 6= 0 we have t1t0 = µ . Thus since P is symmetric, formula (8) implies that up to
µn −n n
a non-zero multiple, one has
t T−1−µ P ∼E . (22)
0 0 −n n −n
(cid:0) (cid:1)
Although the argument given above only applies for n 6= 0, it is easy to see that formula (22) is
true (up to a non-zero multiple) for n = 0 as well! Now combining formulas (13), (20), and (22)
we conclude that up to a multiple, we have
P ∼ R P ,
|n−1| n |n|
where
R = (T +1) t T−1z−κ t T−1−µ .
n 1 0 0 n 0 0 −n
(cid:0) (cid:1)(cid:0) (cid:1)
The main problem now is to simplify the expression of the operator R using properties
n
of P .
|n|
Raising and Lowering Operators for Askey–Wilson Polynomials 9
We first calculate using Lemma 2 and (11), as follows
t T−1z−κ t T−1−µ
0 0 n 0 0 −n
(cid:0) = t0T0−1(cid:1)zt(cid:0)0T0−1−t0T0−1(cid:1)(µ−nz+κn)+κnµ−n
= t T−1 qT z−1+y −t T−1(µ z+κ )+κ µ
0 0 0 0 0 −n n n −n
= t T−1(y−µ z−κ )+qt z−1+κ µ
0 0 (cid:0) −n (cid:1) n 0 n −n
= (T −t +1)(y−µ z−κ )+ qt z−1+κ µ .
0 0 −n n 0 n −n
Applying T +1 to this, we get by Lemma 3 (cid:0) (cid:1)
1
R = D′+T +1 (−µ z+y−κ )+(T +1) qt z−1+κ µ .
n 1 −n n 1 0 n −n
To simpli(cid:0)fy this furthe(cid:1)r we note that thecommutatio(cid:0)n relations of Lem(cid:1)ma 2 can berewritten
as and hence imply
(T +1)z−1 = t z−1+zT +x,
1 1 1
(T +1)z = z+t z−1T−1−x.
1 1 1
Also on P , D′ acts by λ while T acts by t . Therefore R acts by the operator
|n| n 1 1 n
−µ D′z+z+z−1−x +(λ +t +1)(y−κ )
−n n 1 n
+qt t z+z−1 +x +(t +1)κ µ .
(cid:2) 0 1 (cid:3) 1 n −n
Dividing by −(cid:2)µ (cid:0), we see(cid:1)that(cid:3)up to a multiple
−n
qt t
D′z+ 1− 1 0 z+z−1 +β′ P ∼ P , (23)
µ n |n| |n−1|
(cid:20) (cid:18) −n (cid:19) (cid:21)
(cid:0) (cid:1)
where
qt λ +t +1
β′ = −x− 0 x− n 1 (y−κ )−(t +1)κ .
n µ µ n 1 n
−n −n
We now show that β′ is equal to β from formula (16). For this we simplify the expression,
n n
substituting for κ , using (20) above to get
n
qt (λ +t +1) µ y+t x µ y+t x
β′ = −x− 0 x+ n 1 −n 0 −(t +1) n−1 0
n µ µ µ −µ 1 µ −µ
−n −n n−1 −n n−1 −n
λ +1−µ 1−µ
n n−1 n−1
= y+ t y+c x+c (qt x),
1 1 2 0
µ −µ µ −µ
n−1 −n n−1 −n
where c and c are moderately complicated expressions which can be computed explicitly.
1 2
However, wecan save some computation by observingthat since theresultis a priori symmetric
in {a,b,c,d}, c and c must reduce to the coefficients of y and −t y respectively.
1 2 1
It follows then that we have
λ +1−µ 1−µ
β′ = n n−1 (x+y)+ n−1 (qt x+t y)
n µ −µ µ −µ 0 1
n−1 −n n−1 −n
λ +1−µ 1−µ
n n−1 n−1
= e − e = β .
1 3 n
µ −µ µ −µ
n−1 −n n−1 −n
Replacing n by −n, formula (23) becomes
qt t
D′z+ 1− 1 0 z+z−1 +β P ∼ P .
−n |n| |−n−1|
µ
(cid:20) (cid:18) n (cid:19) (cid:21)
(cid:0) (cid:1)
10 S. Sahi
For n ≥ 0, we have µ = qnt t and this becomes
n 1 0
D′z+ 1−q1−n z+z−1 +β P ∼ P (24)
−n n n+1
which(cid:2)is (17)(cid:0)up to a m(cid:1)u(cid:0)ltiple. (cid:1) (cid:3)
For n ≥ 1, we have µ = q−n and formula (23) becomes
−n
D′z+(1−qnabcd) z+z−1 +β P ∼ P (25)
n n n−1
which(cid:2)is (18) up to a mult(cid:0)iple. (cid:1) (cid:3)
It remains then only to calculate the multiples in (24), (25).
To determine the multiple (24), it suffices to calculate the coefficient of zn+1 on the left. For
this we divide the left side of (24) by zn+1 and take the limit as z → ∞. This gives
1
lim A(z)qn+1zn+1−A z−1 zn+1+ 1−q1−n zn+1
z→∞zn+1
abc(cid:0)d (cid:0) (cid:1) (cid:0) (cid:1) (cid:1)
= qn+1−1+1−q1−n =qnabcd−q1−n
q
which proves formula (17).
To determine the multiple in (25), we rewrite (25) in the form
γ
D′z+(1−qnabcd) z+z−1 +β P = c q1−n−qnabcd n−1P , n> 0
n n n n−1
γ
n
(cid:2) (cid:0) (cid:1) (cid:3) (cid:0) (cid:1)
for some unknown constant c . It then suffices to show that c = 1. Subtracting this from (17)
n n
we have
qnabcd−q1−n z+z−1 +β −β P
−n n n
γ
(cid:2)(cid:0) = qnabcd−(cid:1)(cid:0)q1−n P(cid:1) +c n−1(cid:3)P
n+1 n n−1
γ
(cid:18) n (cid:19)
(cid:0) (cid:1)
or
β −β γ
z+z−1 P = P + n −n P +c n−1P .
n n+1 qnabcd−q1−n n n γ n−1
n
(cid:0) (cid:1)
Comparing this with the recurrence relation (5) we deduce c = 1, which proves formula (18)
n
and completes the proof of the theorem.
We note in passing that comparison with (5) also proves the following identity (which can be
verified independently):
β −β
α = n −n . (cid:4)
n qnabcd−q1−n
Acknowledgements
We would like to thank the (anonymous) referee for several insightful suggestions which have
improved the paper considerably. The referee has also pointed out that one can give an alterna-
tive proofof formulas (17)and (18) by combiningTheorem 1 with thefollowing identity relating
the operators D and D′ :
(1−q2)D′z+q2D(z+z−1)−q(z+z−1)D f
= (1−q) (e −e )−(1−abcd)(z +z−1) f,
(cid:2) 1 3 (cid:3)
which holds for all s(cid:2)ymmetric Laurent polynomials f(cid:3).