UvA-DARE ( Digital Academic Repository ) Dualities in the q-Askey Scheme and Degenerate DAHA

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Introduction
The Askey-Wilson (briefly AW) polynomials [1] are a four-parameter family of orthogonal polynomials which are eigenfunctions of a second-order q-difference operator L, and which are explicitly expressed as (terminating) basic hypergeometric series [13]. We will write them as symmetric Laurent polynomials R n [z] of degree n. As orthogonal polynomials they satisfy a three-term recurrence relation. In other words, R n [z] is also an eigenfunction with eigenvalue z + z −1 of a second-order difference operator in the variable n. The operator L and the operator of multiplication by z+z −1 , both acting on symmetric Laurent polynomials f [z], generate the Zhedanov or AW algebra [42], which can be presented by generators and simple relations.
The idea of non-symmetric special functions, which yield (usually orthogonal) symmetric special functions by symmetrization, started with the introduction of the Dunkl operators [11], which are differential-reflection operator associated with a root system. These were generalized to Dunkl-Cherednik operators Y , which are q-difference-reflection operators associated with root systems, and which appear in the basic (or polynomial) representation of the double affine Hecke algebra (DAHA) [7]. Non-symmetric Macdonald polynomials arose as eigenfunctions of these operators Y . A more general DAHA [36] yielded non-symmetric Macdonald-Koornwinder polynomials. In the rank one case this is the DAHA of type (Č 1 , C 1 ) (the AW DAHA), which yields the non-symmetric AW polynomials [34]. Furthermore, the AW algebra and the AW DAHA are closely connected. A central extension of the AW algebra can be embedded [26], [38] in the AW DAHA, while conversely the AW algebra is isomorphic [27] to the spherical subalgebra of the AW DAHA.
A nice property of AW polynomials R n [z], directly visible in the q-hypergeometric expression, is that the variables z and n occur in a similar and to some extent exchangable way (completely exchangable in the corresponding discrete q-Racah case [22,Section 14.2] and in the case of the Askey-Wilson functions [24], see also Remark 8 and §8). This property is called duality. It returns in the non-symmetric case and in the underlying algebraic structures of the AW algebra and the AW DAHA. Therefore the duality extends to the operators occurring in the basic representations of these algebraic structures and having the symmetric or non-symmetric AW polynomials as eigenfunctions.
The AW polynomials are on top of the q-Askey scheme and (by letting q → 1) the Askey scheme [22,Ch. 9,14]. The orthogonal polynomials in the "lower" families are limits of AW polynomials (or q-Racah polynomials). There are also [15] corresponding limits of the AW algebra. The duality property of the AW polynomials then may also have a limit, but usually as a duality between two different families. In general, two different families of special functions φ λ (x) and ψ µ (y), both occurring as eigenfunctions of a certain operator, are dual when φ λ (x) = ψ σ(x) (τ (λ)) for certain functions σ and τ (possibly only for a restricted set of values of λ and x). When the equality holds for all spectral values of the two operators then this property is called bispectrality. It was first emphasized in the context of differential (rather than q-difference) operators by Duistermaat and Grünbaum [10]. In that seminal paper, motivated by the need to analyze the relation between amount of data and image quality in limited angle tomography, the authors classified all possible potentials in the Schrödinger equation such that the wave functions would be an eigenfunctionof a difference operator in the spectral parameter as well. Since then, the bispectrality has been key for the determination of special solutions of the KdV equation and of the KP hierarchy [43], [44], [40] and of many other integrable equations including integrable systems of particles [21], [18], [41].
This line of research produced further links with Huygens' principle of wave propagation [4], representation of infinite dimensional Lie algebras, and isomonodromic deformations of differential equations [19], [9]. In the latter context, the second author of the current paper discovered a link between the theory of the Painlevé differential equations and some families in the q-Askey scheme [32]. Let us briefly explain what this link consists of. The Painlevé differential equations are eight non-linear ODE's whose solutions are encoded by points in the so-called monodromy manifolds (a different manifold for each Painlevé equation). Each of these monodromy manifolds carries a natural Poisson struc-AW polynomials in Section 2, for contnuous dual q-Hahn and big q-Jacobi in Section 3, and for Al-Salam-Chihara and little q-Jacobi in Section 4. Next Sections 5-7 treat nonsymmetric polynomials, their duals, and the corresponding (degenerate) DAHA'a. The non-symmetric AW case is in Section 5, its 2D vector-valued realization in Section 6, and the degenerate cases in Section 7. Finally Section 8 gives a summary of other related work and offers perspectives for further research.
Notation. In this paper we denote the variable of a Laurent polynomial by z and the one of a standard polynomial by x. To emphasize the type of polynomials under consideration, we also use square brackets for Laurent polynomials and round brackets for ordinary polynomials. Correspondingly, when dealing with DAHA and its degenerations, we will denote the generators in "standard presentation" by T 0 , T 1 , Z ±1 when Z is invertible, T 0 , T 1 , X, X ′ when X is not invertible.
For q-hypergeometric series we use notation as in [13, Section 1.2], but we will usually relax the conditions on q.

Definition of Askey-Wilson polynomials and eigenvalue equations
In this paper we will use the following standardization and notation for Askey-Wilson polynomials (in short AW polynomials) R n [z] = R n [z; a, b, c, d | q] := 4 φ 3 q −n , q n−1 abcd, az, az −1 ab, ac, ad ; q, q , and we will work in the following assumptions: The polynomials (1) are related to the AW polynomials p n (x; a, b, c, d | q) in usual notation [22, (14.1.1)] by p n 1 2 (z + z −1 ); a, b, c, d | q = a −n (ab, ac, ad; q) n R n [z; a, b, c, d | q].
While p n (x; a, b, c, d | q) is symmetric in its four parameters a, b, c, d, R n [z; a, b, c, d | q] is only symmetric in b, c, d. But the larger symmetry involving a is lost anyhow with the duality to be discussed later, see (26).
The polynomials R n [z] are eigenfunctions of the operator L z acting on the space of symmetric Laurent polynomials f The eigenvalue equation is Under condition (2) all eigenvalues in (5) are distinct. The three-term recurrence relation [22, (14.1.4)] for the AW polynomials can be interpreted as an eigenvalue equation if we consider R n [z] for fixed z in its dependence on n. Then R n [z] is an eigenfunction of the operator M n , acting on functions g(n) of n (n = 0, 1, 2, . . .), defined by The eigenvalue equation is Under stricter conditions than (2), namely 0 < q < 1 and |a|, |b|, |c|, |d| ≤ 1 such that pairwise products of a, b, c, d are not equal to 1 and such that non-real parameters occur in complex conjugate pairs, the AW polynomials are orthogonal with respect to a non-negative weight function on x = 1 2 (z + z −1 ) ∈ [−1, 1]. For convenience we give this orthogonality in the variable z on the unit circle, where the integrand is invariant under z → z −1 : where h 0 = 1 and where the explicit expression for h n (omitted here) can be obtained from [22, (14.1.2)] together with (3).

Zhedanov algebra
The Zhedanov algebra or AW algebra AW(3) (see [42]) is the algebra with two generators K 0 , K 1 and with two relations Here the structure constants B, C 0 , C 1 , D 0 , D 1 are fixed complex constants. Remark 1. The relations for AW(3) were originally given in [42] in terms of three generators (which explains the notation AW(3)): K 0 , K 1 , and in addition K 2 which is given in terms of K 1 and K 2 by the q-commutator This presentation is in particular suitable for computations in computer algebra, since the three relations can be written in PBW form. In this paper we prefer the two generators version because it makes the duality we plan to discuss more transparent.
There is a Casimir element Q commuting with K 0 , K 1 : Remark 2. As observed in [27,Remark 2.3], for the five structure constants B, C 0 , C 1 , D 0 , D 1 in the relations (9) two degrees of freedom are caused by scale transformations K 0 → c 0 K 0 and K 1 → c 1 K 1 of the generators. These induce the following transformations on the structure constants: These also result into a transformation Q → c 2 0 c 2 1 Q of the Casimir element (10). So there are essentially only three degrees of freedom for the structure constants (and one more freedom to fix the value of Q in the basic representation, seeRemark 4). A nice way of presenting this symmetrically was emphasized by Terwilliger [37, (1.1)]. In slightly different notation this is done as follows. Put Then we can equivalently describe AW(3) as the algebra generated by where α 0 , α 1 , α 2 are structure constants which can be expressed in terms of B, C 0 , C 1 , D 0 , D 1 by Terwilliger [37] considers α 0 , α 1 , α 2 as central elements. He calls the resulting algebra the universal Askey-Wilson algebra. He also identifies a Casimir element ω, which is closely related to Q in (10): Then he proves in [37,Corollary 8.3] that the four central elements α 0 , α 1 , α 2 , ω generate the center of the universal Askey-Wilson algebra. Therefore, in our presentation, Q generates the center of AW(3).
To go back from relations (12) to (9) we need two arbitrary rescaling constants c 0 , c 1 = 0 and then put: Remark 3. In connection with the relations (9) defining AW(3) let K 1 , K 2 denote the free algebra generated by K 1 and K 2 . Note that the algebra isomorphism τ : K 1 , K 2 → K 1 , K 2 op which reverses the order of the factors in the terms of the elements of K 1 , K 2 , leaves invariant the ideal generated by the relations (9) (each of the two relations separately is even left invariant). So τ induces an algebra isomorphism τ : AW(3) → AW(3) op . It can be shown that the Casimir element Q, given by (10), is invariant under τ . However, in the set-up with generators A 0 , A 1 , A 2 and relations (12) there is no invariance of the relations after reversion of the order of the factors.
Then express the structure constants in (9) in terms of a, b, c, d by means of (15): Note that, for given C 0 = (q − q −1 ) 2 and for given values of B, C 1 , D 0 , D 1 we can solve (2.11) as a system of equations in e 1 , e 2 , e 3 , e 4 . This system is uniquely solvable. Next, e 1 , e 2 , e 3 , e 4 determine a, b, c, d up to permutations. There is a representation (the basic representation or polynomial representation) of the algebra AW(3) with structure constants (16) on the space of symmetric Laurent polyno-mials as follows: where L z is the operator (4) having the AW polynomials as eigenfunctions and Z ±1 is the operator of multiplication by z ±1 . The Casimir element Q becomes constant in this representation: where Remark 4. The basic representation (17) gives rise to a one-parameter family of representations of AW(3) by using a scale transformation K 0 → λK 0 , K 1 → K 1 in (9). Compare with the beginning of Remark 2: we now take c 0 = λ, c 1 = 1. Now we have to solve e 1 , e 2 , e 3 , e 4 from the system of equations and we get a, b, c, d (depending on λ) from e 1 , e 2 , e 3 , e 4 . Then, for each value of λ we have a representation Here L z depends on a, b, c, d, and hence on λ. Then Q takes the value λ −2 Q 0 with Q 0 given by (19), where e 1 , e 2 , e 3 , e 4 depend on λ.
If, conversely, we do not pick λ but fix Q 0 then a very complicated system of five equations in e 1 , e 2 , e 3 , e 4 , λ has to be solved.
For the relations (12) a one-parameter family of representations can be obtained by first passing to the relations (9) as we specified in Remark 2, obtaining the representations there, and rewriting everything again in terms of the relations (12). Definition 1. The centerfree Zhedanov or Askey-Wilson algebra AW(3, Q 0 ) is the algebra generated by K 0 , K 1 with three relations, namely the two relations (9), where the structure constants are expressed in terms of a, b, c, d, q by (16) and (15), and the relation where Q and Q 0 are given by (10) and (19).
This follows because the AW polynomials are the overlap coefficients connecting the two representations: in the following sense: the AW polynomials form a complete system of orthogonal polynomials with respect to a suitable orthogonality measure µ. Then the Fourier-Askey-Wilson intertwines between the two representations. In fact: where in the last step we have used the fact that L is a self-adjoint operator on the real Hilbert space L 2 (dµ). Similarly, More generally, if p(K 0 , K 1 ) ∈ K 0 , K 1 then Hence, Λ, M satisfy the same relations (9), (20) as L, Z + Z −1 . Thus (21) generates a representation of AW a,b,c,d;q (3, Q 0 ). This was already observed in [42] and [15], more concretely for the q-Racah case, where the representations are finite-dimensional. By faithfulness we have

Duality for AW polynomials
Define dual parametersã,b,c,d in terms of a, b, c, d bỹ a = (q −1 abcd) Jumping from one branch to the other branch in the square root in the formula forã implies thatã,b,c,d move to −ã, −b, −c, −d. This corresponds to the following trivial symmetry that follows immediately from (1): Repetition of the parameter transformation recovers the original parameters up to a possible common multiplication of a, b, c, d by −1, while the branch choice forã is irrelevant: From (1) we have the duality relation By (25) the two sides of (27) are invariant under common multiplication by −1 of a, b, c, d, There is a duality corresponding to (27) for the operators L z and M n defined by (4) and (6), respectively: whereM m is the difference operator M m with respect to dual parameters. For f [z] := R n [z] both sides of (28) yield (q −n + abcdq n−1 )R n [a −1 q −m ]. Similarly to (28) there is a duality between the multiplication operators Z + Z −1 given by (17) and Λ n given by (21): whereΛ m is the multiplication operator Λ m with respect to dual parameters. Formulas (21) and (29) are instances of operatorsǍ acting on functions on Z ≥0 which are induced by restriction of operators A acting on functions Clearly (AB)ˇ=ǍB. By (21) and (29) Corresponding to the trivial symmetry we see that, by (4), Remark 5. By [34, §5.7, §8.5] our dual parameters (24) match with the dual parameters in [34]: just interchange k 0 and u 1 in [34, §5.7].

Duality for
There are several symmetries of AW a,b,c,d;q (3, Q 0 ; K 0 , K 1 ). We already observed that it is invariant under permutations of a, b, c, d.
There is an isomorphism [42, §2], [27, (2.11)] 1 (both an algebra and an anti-algebra isomorphism): whereã,b,c,d are given in (24), andQ 0 denotes Q 0 in terms of the dual parameters. Indeed, ifB,C 0 , . . . denote B, C 0 , . . . in terms of the dual parameters theñ Hence relations (9) with dual parameters and with K 0 , K 1 replaced by aK 1 ,ã −1 K 0 are equivalent to the original relations (9). Furthermore replacement of K 0 , K 1 , a, b, c, d in the right-hand side of (10) by aK 1 ,ã −1 K 0 ,ã,b,c,d, respectively, yields the old expression multiplied by a 2ã−2 , and, by (32), the same is true if we replace a, b, c, d byã,b,c,d in the right-hand side of (19). So the algebras on the left and right of (31) satisfy equivalent relations.
Remark 7. The duality (31) takes a particularly simple and elegant form for the algebra generated by A 0 , A 1 , A 2 with relations (12) together with ω = ω 0 , where ω is given by (14) and ω 0 is a constant. By (11), (13) and (32) we see that the duality for that algebra amounts to an anti-isomorphism which interchanges A 0 and A 1 and keeps A 2 fixed, while α 0 and α 1 are interchanged and α 2 and ω 0 are kept fixed. It follows from [37, Lemma 6.1] that ω then does not change. Since the parameters a, b, c, d are no longer involved in this formulation of the duality, the symmetry breaking in (31) seems to be absent now. The price to be paid for this is that there is less immediate contact with the AW polynomials. It is also clear that the duality in this setting is part of an S 3 symmetry acting simultaneously on A 0 , A 1 , A 2 and α 0 , α 1 , α 2 . The reparametrization of the Askey-Wilson parameters in Huang [20, §3.2] seems to behave nicely under this action of S 3 . It is not clear what would be the effect on the basic representation by the action of the full S 3 symmetry group.
There is also an algebra isomorphism 3. Duality for continuous dual q-Hahn and big q-Jacobi polynomials

Limits to Continuous Dual q-Hahn and Big q-Jacobi
Limit from AW to Continuous Dual q-Hahn. The continuous dual q-Hahn polynomials are the limit case d → 0 of the AW polynomials (1): The polynomials (34) are related to the continuous dual q-Hahn polynomials p n (x; a, b, c | q) in usual notation [22, (14.3.1)] by . The corresponding limits of (4), (5), (6), (7) for the operators L, M and its eigenvalue equations are: The obtained q-difference equation and recurrence relation agree with [22, (14.3 Limit from AW to Big q-Jacobi. The big q-Jacobi polynomials [22, (14.5.1)] are obtained as a more tricky limit case [22, (14.1.18)] of AW polynomials (1): The corresponding limits of (4), (5), (6), (7) for the operators L, M and its eigenvalue equations are: When taking the limit in (6), (7) we have to substitute The obtained q-difference equation and recurrence relation agree with [22, (14.5.5),

Duality between Continuous Dual q-Hahn and Big q-Jacobi
From the q-hypergeometric expressions (34) and (39) we see that This duality turns out to be a limit case of the Askey-Wilson duality (27). Indeed, by (24) we have So, by (27), Now let λ → 0 in the above equality. By the limits (34) and (39) we obtain the duality (44). Similarly to, and as a limit case of (28) there is a duality corresponding to (44) for the operators L z and M n defined by (35) and (42), respectively: where L a,b,c z is the operator L z given by (35), M a,b,c n is the operator M n given by (42), and There is also a duality corresponding to (44) for the operators L x and M n defined by (40) and (37), respectively: where L a,b,c x is the operator L x given by (40), M a,b,c n is the operator M n given by (37), and a,b,c are as in (48).

Corresponding degenerations of AW(3, Q 0 ) and their duality
As d → 0 the Zhedanov algebra AW(3, Q 0 ) = AW a,b,c,d;q (3, Q 0 ; K 0 , K 1 ) tends to the algebra with two generators K 0 , K 1 with relations (9) and (18), where C 1 = 0 and B, C 0 , D 0 , D 1 , Q 0 depend on a, b, c and on d = 0 as in (16) and (19) and e 1 , e 2 , e 3 are the elementary symmetric polynomials in a, b, c: In the expression (10) for Q also put C 1 = 0 and substitute (50). We denote the resulting algebra by where, if needed, the two generators can be added to the notation. The representations (17) and (21) also hold for AW CDqH a,b,c;q (3, Q 0 ), but now with L z and M n given by (35) and (37), and with Λ n given by The argumentation by (22) and (23) for the equivalence of the two representations also remains valid if we take (34) for R n and if we put λ n = q −n and if we use (36) and (38). Now consider the Zhedanov algebra AW λ,qaλ −1 ,qcλ −1 ,bc −1 λ;q (3, Q 0 ; K 0 , λ −1 K 1 ), rescale Q = Q 0 as λ 2 Q = λ 2 Q 0 and take the limit as λ → 0, where the new Q and Q 0 are the limits of λ 2 Q and λ 2 Q 0 , respectively. This produces the algebra with two generators K 0 , K 1 and with relations (9) and (18), where C 0 = 0 and B, C 1 , D 0 , D 1 are given as follows: (We omit the quite lengthy explicit expressions of Q and Q 0 .) Denote the resulting algebra by For AW BqJ a,b,c;q (3, Q 0 ; K 0 , K 1 ) the representations (17) and (21) take the form and with L x and M n defined by (40) and (42). The argumentation by (22) and (23) for the equivalence of the two representations also remains valid after slight but obvious adaptations.
Proposition 1. There is an isomorphism (both an algebra and an anti-algebra isomorphism) Proof: By substitution of (45) in (31) we obtain Here, at each of the two sides, Q 0 is in terms of the parameters given at that side. Then Q = Q 0 on the right means a 2 λ −2 Q = a 2 λ −2 Q 0 or, after rescaling, a 2 Q = a 2 Q 0 in terms of Q and Q 0 on the left. Now apply the limits (51) and (54) to the left and right side of (58), respectively.
The polynomials (59) are related to the Al-Salam-Chihara polynomials Q n (x; a, b | q) in usual notation [22, (14.8 The corresponding limits of (35)- (38) for the operators L, M and its eigenvalue equations are: The obtained q-difference equation and recurrence relation agree with [22, (14.8 By (34) they can be obtained as a limit case of continuous dual q-Hahn polynomials: Then the orthogonality relations [23, (2.14)] for the functions J q n γ follow, under suitable constraints on the parameters, from the orthogonality relations [22, (14.3.2)] for continuous dual q-Hahn polynomials. If we rescale L a,b,c;q in (37) and next take a limit for N → ∞ of (38) in the form Limits from Big q-Jacobi to Little q-Jacobi. The little q-Jacobi polynomials are defined as follows (see [22, (14.12.1)]): or, equivalently (by [25, (3.38)]): Little q-Jacobi polynomials appear as limits of big q-Jacobi polynomials in two ways: and where The limits of the operators L, M and its eigenvalue equations in (40)-(43) which correspond to the limit (70) are as follows: The obtained q-difference equation and recurrence relation agree with [22, (14.12.5), (14.12.3)] if we take into account (69).
This also follows by comparing the q-hypergeometric expression in (59) and (70). This identification is less immediate from the q-hypergeometric expressions than (75) (it uses an additional q-hypergeometric transformation formula), but it has the nice property that it also encodes a duality of infinite discrete orthogonality relations. It is also a limit case of a similar identification (to be compared with our duality (44)) between dual big q-Jacobi polynomials and continuous dual q −1 -Hahn polynomials, see [2,Section 4.3]. Again it needs an additional transformation formula for its derivation and again it encodes a duality of orthogonality relations. In fact, as pointed out in [2], this identification is a limit case for N → ∞ of the duality for q-Racah polynomials on a set of N + 1 points. On the level of the AW algebra and its limit cases one should use, beside (31), also (33) in connection with the formulas just discussed.
In order to take the limit for c → ∞ in (44), first substitute n → N −n and c → q −N γ −1 . Then Here m, N, n are integers with m, N ≥ 0 and n ≤ N . Now apply (68) to the right-hand side with c = −q −N −1 γ −1 a and N → ∞ and apply (64) to the left-hand side. We obtain a duality between Askey-Wilson q-Bessel functions and little q-Jacobi polynomials: J q n γ a −1 q −m ; a, b | q = p m (−q n γa −1 ; q −1 ab, ab −1 ; q).
and let (c, λ) → (0, 0). Then we arrive directly from the AW duality at the duality (75), as is seen from (1), (59) and (70). However, if we fix one of c, λ at a nonzero value and let the other one go to 0 then we arrive at the duality (44).
Similarly to, and as a limit case of (47), there is a duality corresponding to (75) for the operators L z and M n defined by (60) and (73), respectively: where L a,b z is the operator L z given by (60), M a,b n is the operator M n given by (73), and Note that the map (a, b) → (ã,b) is inverse to the map (a, b) → (q −1 ab, ab −1 ). There is also a duality corresponding to (44) for the operators L x and M n defined by (71) and (62), respectively: where L a,b x is the operator L x given by (71), M a,b n is the operator M n given by (62), and a,b are as in (77).

Corresponding degenerations of the Zhedanov algebra and duality
As c → 0, the Zhedanov algebra AW CqDH a,b,c;q (3, Q 0 ; K 0 , K 1 ), see (51), tends to the algebra with two generators K 0 , K 1 and with relations (9) and (18) where C 1 = D 1 = 0 and B, C 0 , D 0 , Q 0 are defined as in (50) with c = 0: In the expression (10) for Q also put C 1 = 0 and substitute (79). We denote this algebra by AW ASC a,b;q (3, Q 0 ; K 0 , K 1 ). Similarly, as c → 0, the Zhedanov algebra AW BqJ a,b,c;q (3, Q 0 ; K 0 , K 1 ), see (54), tends to the algebra with two generators K 0 , K 1 and with relations (9) and (18) where C 0 = D 0 = 0 and B, C 1 , D 1 are given by formula (53) with c = 0, and also Q 0 with c = 0 has a simple expression: Furthermore Q is now obtained by following the procedure described for big q-Jacobi just before (53) and then putting c = 0. We denote this algebra by AW LqJ a,b;q (3, Q 0 ; K 0 , K 1 ). The representations of the algebras AW ASC a,b;q (3, Q 0 ; K 0 , K 1 ) and AW LqJ a,b;q (3, Q 0 ; K 0 , K 1 ) can be obtained from the representations of AW CDqH a,b,c;q (3, Q 0 ) and AW BqJ a,b,c;q (3, Q 0 ), respectively, by putting c = 0 in all formulae. Similarly, the duality formula is then simply obtained by putting c = 0 in (57): There is an isomorphism (both an algebra isomorphism and an algebra anti-isomorphism) Just as in Remark 9 we may replace λ by c 1 2 λ in (58), and then take the limit as (c, λ) → (0, 0). Then we will arrive at the duality (57) directly from the duality of AW(3).
Remark 10. Corresponding to the limit (64) we can consider AW CDqH a,b,c;q (3, Q 0 ) with structure constants (50) and there raplace K 0 by cK 0 . Then by (50) the relations (9) have a limit for c → ∞. Similarly, corresponding to the limit (68) we can consider AW BqJ a,b,c;q (3, Q 0 ; K 0 , K 1 ) with stucture constants (53) and there replace K 1 by cK 1 . Then by (53) the relations (9) have a limit for c → ∞. Representations of the algebras and duality could be considered. We omit the details.

Definition of the Askey-Wilson DAHA
The DAHA of type (C ∨ 1 , C 1 ), also called the Askey-Wilson DAHA, occurs as the rank one case of Sahi's more general construction [36]. It was studied in much detail by Noumi and Stokman [34].
Replace in the relations (82) the generators T 0 ,Ť 1 ,Ť 0 by Z, Z −1 , Y by puttinǧ (or equivalently replace in the relations (84) the generator T 0 by Y by putting T 0 = T −1 1 Y ). The substitutions (85) can be written conversely as The resulting relations are (T 1 + ab)(T 1 + 1) = 0, We denote H H in this presentation by H H a,b,c,d;q T 1 , Y, Z −1 .

Note in (82) the trivial algebra isomorphism
In terms of the generators in (84) this algebra isomorphism is generated by (T 1 , T 0 , Z, Z −1 ) → (T 1 , T 0 , −Z, −Z −1 ), and in terms of the generators in (87) by T 1 , Y, Z −1 → T 1 , Y, −Z −1 . In (82) we also recognize the following straightforward algebra isomorphism: For the main duality property we need the dual parameters (24). Observe that As a consequence, there is an anti-isomorphism Indeed, substitution in relations (82) of generators and parameters according to (89) and reversion of the order of multiplication (only needed in the last relation) interchanges the second and third relation, while it preserves the other relations. This also implies that the anti-isomorphism Φ is involutive.
In terms of the presentation (84) we can write the duality as and in terms of the presentation (87) as Note also the following anti-isomorphism in terms of the presentation (87):

DAHA representation on the Laurent polynomials and non-symmetric Askey Wilson polynomials
The algebra H H in presentation (87) has a faithful representation, the so-called basic representation, on the space A of Laurent polynomials f [z] as follows (see [26, §3] and (83) and use that Put D := Y + q −1 abcdY −1 .
D commutes with T 1 , T 0 and Y . If we compare its explicit expression [26, (3.14)] with (4) then we see that In particular, if we apply D to the AW polynomial R n [z] given by (1) then we obtain that DR n = λ n R n with λ n given by (5).

Duality of nonsymmetric AW polynomials
First observe that the trivial symmetry (30) for AW polynomials extends to a symmetry for non-symmetric AW polynomials. This is clear from (97) and (30). Compare also with the DAHA algebra isomorphism (88). Next we pass to the main duality result.
Proof: For the case that n or m = 0 use that E n [a −1 ; a, b, c, d | q] = 1. For the other cases we have to show that for m, n ∈ Z >0 . From (27) we see that By (97), (27), (105) and by the identities ab =ãb, ac =ãc, ad =ãd we see that (102), (103), (104) will respectively follow from the identities These indeed hold by (24).

Recurrence relation for the nonsymmetric AW polynomials
The duality (101) for non-symmetric AW polynomials E n (n ∈ Z) can be applied to the eigenvalue equation (98) in order to obtain a recurrence relation for the E n . First we consider (Y F )[z], given by (95), at z = z a,q (m) −1 (m ∈ Z), with z a,q (m) given by (100). Note that Note also that the terms in (95) with f [qz] and with f [z −1 ] vanish if z = a −1 = z a,q (0) −1 . Thus we can specialize (95) as follows: where the second and third term on the right in (106) vanish if m = 0. Now take f = E n and observe that the eigenvalue equation (98) can be written in a unified way as Thus, by the duality (101) we can rewrite (107) for z = z a,q (m) −1 as a zã ,q (n)Ẽ m zã ;q (n) −1 = AẼ m zã ;q (n) −1 + BE m−1 zã ;q (n) −1 where and whereẼ m means E m for dual parameters. Because we are dealing with Laurent polynomials, the equality (108) will remain valid if we replace zã ;q (n) −1 by arbitrary complex z. Next, replace in (108) a, b, c, d byã,b,c,d, and replace m by n. We obtain: Finally put Then we obtain the recurrence relation for the nonsymmetric AW polynomials: where M n is an operator acting on functions g(n) of n (n ∈ Z) which is given by M n g(n) := νã ,q (n)(1 + ab − ab(q −1 cd + 1)νã ,q (n))(q(c + d) − cd(a + b)νã ,q (n)) (q − abcdνã ,q (n) 2 )(q − q −1 abcdνã ,q (n) 2 ) g(n) Note that by the symmetry (99) there also follows a recurrence formula which expands zE n [z]. We omit the explicit expression.
Remark 15. In [34, §10.9] it is just observed that a recurrence relation for nonsymmetric AW polynomials can be derived from the Y -eigenvalue equation by duality, but no further derivation or explicit formula is given. Neither we have found such a formula elsewhere in the literature.

The dual of the basic representation
It follows from the derivation of (110), (111) in §5.5 that where, as usual,M m means the operator M m with respect to dual parameters. Define a multiplication operator N m , acting on functions g(m) (m ∈ Z), by Then, by (109), Also, in terms of T 1 acting on f [z] by (94), let T m be the operator acting on g(m) such that Since 6. The basic representation of the Askey-Wilson DAHA in a 2D realization

Definitions and explicit formulas
We use results and notation from [28, (4.7) and following] except for a slight rescaling: in (116) below we have an additional factor a in the second term on the right, which will also have impact on formulas further down. This will facilitate the limit to Big q-Jacobi, which we will consider later in the paper.
The set-up is to associate with a Laurent polynomial f a column vector where f 1 , f 2 are symmetric Laurent polynomials such that Then Then (116) and (117) can be written more succinctly as The non-symmetric AW polynomials E ±n [z], as defined in (97), already have the decomposition (116). Thus they have vector-valued form (see also [28, (4.10), (4.11) and where σ(n)R n−1 = const. (1 − q n )R n−1 := 0 for n = 0. Let A be an operator acting on the space of Laurent polynomials. Then we can write where the A i j are operators acting on the space of symmetric Laurent polynomials. So we have the identifications In the identification A ↔ A composition of operators corresponds to matrix multiplication together with entrywise composition of operators. Indeed, we can express (122) as In this way T 1 , given by (94), acts as a 2 × 2 matrix-valued operator: The very simple form of T 1 as a diagonal matrix with constant coefficients was the motivation for the decomposition (116). Next we describe the 2 × 2 matrix-valued operator corresponding to Y given by (95). Below we give the explicit expressions for the Y ij , see (4.12)-(4.15) in [28]. The expressions for Y 11 and Y 22 involve the operator L = L a,b,c,d;q as given in (4): We give Y 12 and Y 21 as operators acting on a symmetric Laurent polynomial g[z]: Note an error in the formula [28, (4.14)] for (Y 21 g)[z], which we have corrected above: In the first term on the right we have replaced the denominator factor 1 − qz 2 by q − z 2 . Then the eigenvalue equation (98), with Y in matrix-valued form and E ±n in vectorvalued form as given above, still holds: By [26, (3.7), (3.6)] we can express Y −1 in terms of Y and T −1 1 . Indeed, Then the matrix realization of Y −1 follows from (130), (125)-(128) and (123).
As a final example the multiplication operator Z, given by (93), corresponds to a 2 × 2 matrix-valued operator Z with matrix entries acting as multiplication operators: Note that det(Z) = 1. Hence Z −1 can be diagonalized by where S and S −1 are given in (118). Now consider (119) for f := E n : On combination with (133) and (110) where the operator M n , acting on functions of n, is given by (111).

Orthogonality and equivalence of representations
In addition to the conditions on a, b, c, d, q at the end of §2.1 assume that a, b are real and ab < 0. Put Finally define an operator U n acting on functions of n which extends the action of T on #» E ±n . This operator U can easily be given explicitly by using (120) and (123). Then So, in view of (92) we have settled that We have achieved this by working with 2-vector-valued functions of z and without using the DAHA duality, an approach quite different from the one in §5.6.

Duality for the 2D non-symmetric AW polynomials
While many formulas turn out to be very satisfactory in the 2D presentation, this is less so for the 2D version of the duality (101) for non-symmetric AW polynomials. Here we will give a "mixed" duality formula, with scalar-valued polynomials on the left side and 2-vector-valued polynomials on the right side, since this is most suitable when taking limits to Continuous Dual Hahn and Big q-Jacobi. We will compare E n z a,q (m) −1 and # » E m zã ,q (n) −1 (m, n ∈ Z), where z a,q (n) is given by (100) On the right-hand side we have a matrix product of a row vector and a column vector, which yields a scalar. The limits d → 0 from AW to Continuous dual Hahn and d, c → 0 from AW to Al-Salam-Chihara (see (34) and (59)) have corresponding DAHA degenerations which were discussed in [32]. Here we recall the main points of that study and construct other degenerations corresponding to the limits (39) and (70) from AW to Big and Little q-Jacobi. We will give the degenerations for the DAHA presentations (84) and (87). Because certain rescalings have to be emphasized, we now use the DAHA notation H H a,b,c,d;q [T 1 , T 0 , T −1 0 , Z, Z −1 ] in connection with (84) and H H a,b,c,d;q T 1 , Y, Y −1 , Z, Z −1 in connection with (87).
From AW to Continuous Dual Hahn. Just as in (34) and (51) we have to take the limit as d → 0. However, before taking the limit it is convenient to introduce a rescaling T ′ 0 := q −1 cdT −1 0 in (84) and a rescaling Y ′ := q −1 cdY −1 in (87) before letting d → 0. Then (84) can be equivalently written as In the limit for d → 0 we get the algebra with generators T 1 , T 0 , T ′ 0 , Z, Z −1 and relations (T 1 + ab)(T 1 + 1) = 0, Since T ′ 0 = −T 0 − 1 by the second relation, this may be substituted in the other relations in (138) which involve T ′ 0 , after which T ′ 0 can be dropped as a generator. The resulting relations are (T 1 + ab)(T 1 + 1) = 0, The presentation (138) is the same as for the algebra H V in [32, (1.6)-(1.10)].
As for the recurrence relation (110) with M n given by (111) involving formula (109) for νã ,q (n), one can see from (111) and (109) that the limit of M n for d → 0 exists, where one has to distinguish between the cases n ≥ 0 and n < 0. We do not give the explicit formulas here.
Similar results for non-symmetric Al-Salam-Chihara polynomials will simply follow by putting c = 0 in the above formulas.

Non-symmetric big and little q-Jacobi polynomials
When taking the limit (39) to big q-Jacobi (and consequently the limits (68), (69) to little q-Jacobi), one produces true polynomials rather than Laurent ones. By taking the same limits of the non-symmetric AW polynomials (97), one obtains families of polynomials that are no longer functionally independent. To overcome this difficulty we need to deal with the 2-D non-symmetric AW polynomials (120) and take limits of those. Thus define the non-symmetric big q-Jacobi polynomials where (1 − q n )P n−1 := 0 for n = 0. We will deduce the basic representation of H H BqJ a,b,c;q by 2 × 2 matrix-valued operators from the one for H H. We need to impose the substitution defined in (39), in the 2D realization of the basic representation of H H. However, when taking the limit as λ → 0, we see that to obtain well defined matrix operators we need to conjugate all operators by the diagonal matrix with entries 1, 1 λ . Moreover, because z → λ −1 x, we need to multiply Z by λ. Therefore, we introduce the following rescalings of (131), (132) and (123): (1 − qa)x Z 11 = x 2 + λ 2 − x(λ 2 + qa), (1 − qa) Z 21 = −1; where Z ±1 sub , T 1sub and Y sub denote the operators in which we have performed the substitution (157). Denote the limits for λ → 0 by X, X ′ , T 1 , Y, respectively. Then we obtain: For Y, which we have not given explicitly, we obtain from (129) that Y # » E −n (x) = q −n # » E −n (x) (n = 1, 2, . . .), Y # » E n (x) = q n+1 ab # » E n (x) (n = 0, 1, 2, . . .).
This is proved straight from (134) by substitution. Similar results for non-symmetric little q-Jacobi polynomials will simply follow by putting c = 0 in the above formulas.
Remark 17. Clearly, from (39) there is a symmetry P n (x; a, b, c) = P n (x; c, ab/c, a). Hence, from (156), for n ≥ 0 respectively n > 0. Note that the q-shifts in the parameters of the big q-Jacobi polynomials occurring in the second coordinate in (166) are different from the ones in (156). The q-shifts in (166) are more in agreement with the q-shifts in the vectorvalued little q-Jacobi polynomials discussed in [28, §6]. In fact, the limit for c → 0 of 1 0 0 c # » E n (x; c, ab/c, a; q) essentially gives these polynomials [28, (6.4), (6.5)].
The polynomials (166) will also be eigenfunctions of the Y operator in the basic representation of H H BqJ c,ab/c,a;q . A limit for c → 0 will be possible in this eigenvalue equation (a little q-Jacobi case). However, it is not clear at all if some decent algebra will result from taking the limit for c → 0 of H H BqJ c,ab/c,a;q .
In a different line of development the paper [39] introduced analogues of Askey-Wilson polynomials which are orthogonal on the unit circle, and constructed a DAHA associated with them.
An evident perspective for further work is to describe a full (q-)Askey scheme of nonsymmetric orthogonal polynomials and the associated degenerate DAHA's. Important questions here will be when it is necessary to work with vector-valued polynomials rather than Laurent polynomials, whether the orthogonality relations (135) for vector-valued AW survive in the limit cases (for a few special cases positively answered in [28]), and what the consequences are when limits of nonsymmetric AW are taken with permuted parameters (see Remark 17). The "non-symmetric" (q-)Askey scheme should also be extended to non-polynomial cases (cf. [23]). All such work should finally get analogues in the higher rank (BC n ) case.