Charting the $q$-Askey scheme. II. The $q$-Zhedanov scheme

This is the second in a series of papers which intend to explore conceptual ways of distinguishing between families in the $q$-Askey scheme and uniform ways of parametrizing the families. For a system of polynomials $p_n(x)$ in the $q$-Askey scheme satisfying $Lp_n=h_np_n$ with $L$ a second order $q$-difference operator the $q$-Zhedanov algebra is the algebra generated by operators $L$ and $X$ (multiplication by $x$). It has two relations in which essentially five coefficients occur. Vanishing of one or more of the coefficients corresponds to a subfamily or limit family of the Askey-Wilson polynomials. An arrow from one family to another means that in the latter family one more coefficient vanishes. This yields the $q$-Zhedanov scheme given in this paper. The $q$-hypergeometric expression of $p_n(x)$ can be interpreted as an expansion of $p_n(x)$ in terms of certain Newton polynomials. In our previous paper arXiv:2108.03858 we used Verde-Star's clean parametrization of such expansions and we obtained a $q$-Verde-Star scheme, where vanishing of one or more of these parameters corresponds to a subfamily or limit family. The actions of the operators $L$ and $X$ on the Newton polynomials can be expressed in terms of the Verde-Star parameters, and thus the coefficients for the $q$-Zhedanov algebra can be expressed in terms of these parameters. There are interesting differences between the $q$-Verde-Star scheme and the $q$-Zhedanov scheme, which are discussed in the paper.


Introduction
Jaap Korevaar has worked on a wide range of topics in analysis, in particular complex and harmonic analysis and applications to number theory. As for special functions, he likes in particular the special functions of number theory, with the zeta function on top [15], [16], but he does certainly not ignore the classical orthogonal polynomials. In [14] he used Hermite polynomials, and in work with his PhD student Meyers [17], [18] he used Legendre and ultraspherical polynomials. The paper [17] was a key reference in [3], a paper coauthored by Maryna Viazovska who was a 2022 Fields Medal winner.
After pioneering work of Hahn q-analogues of classical orthogonal polynomials got during 1975-1985 an increasing interest. This culminated in the introduction of the Askey-Wilson polynomials [1], a 4-parameter family of q-hypergeometric orthogonal polynomials which stand on top of the q-Askey scheme [9,Ch. 14]. In that scheme there are boxes representing families of q-hypergeometric orthogonal polynomials and arrows connecting them. An arrow stands for a limit transition or specialization and it diminishes the number of parameters by at least one. This paper is the second in a series of papers which intend to explore conceptual ways of distinguishing between families in the scheme and uniform ways of parametrizing the families. First of all we do this for the q-Askey scheme, and tentatively later for its q = 1 limit (the original Askey scheme [9,Ch. 9]) and its q = −1 limit [26], [22]. The present and the previous paper concentrate on the case of general q. The previous paper [12] was inspired by Verde-Star's paper [27] (see also Vinet and Zhedanov [29] and, for finite systems, Terwilliger [24], cf. [12,Remark 2.3]), which classifies the families in the q-Askey scheme in combination with a basis of Newton polynomials in terms of which they have nice expansions. There are uniform parameters describing this. The vanishing of one or more of the parameters corresponds to a subfamily.
The present paper classifies the q-Zhedanov algebras [6, §3] associated with families in the q-Askey scheme. Such an algebra is generated by the second order q-difference operator L for which the polynomials in the family are eigenfunctions, and by the operator of multiplication by x. Denote these two operators by K 1 , K 2 , respectively. Then the algebra is described by two relations which expand certain cubic forms in K 1 , K 2 (repeated q-commutators) as linear combinations of K 1 , K 2 and the identity operator. The vanishing of one or more of the coefficients in these linear combinations indicates a subfamily in the q-Askey scheme. The prototypical algebra of this type is the Askey-Wilson algebra, associated with the Askey-Wilson polynomials. See Zhedanov [30].
In Verde-Star's approach the action of L on the space of polynomials is conceptually described by its action on the corresponding Newton polynomials, while multiplication by x immediately yields an explicit action on the Newton polynomials. This makes it possible to express the coefficients of the q-Zhedanov algebra in terms of Verde-Star's parameters. From that it can be read off for which constraints on Verde-Star's parameters certain q-Zhedanov coefficients vanish. Since we already expressed in [12] Verde-Star's parameters in terms of the parameters of the corresponding family in the q-Askey scheme, we can associate vanishing properties of the q-Zhedanov parameters with families in the q-Askey scheme, and thus the q-Zhedanov scheme occurs. One can find these results in Sections 3, 4 and 5, which form the heart of this paper. It should be observed that our work was anticipated, in the case of finite systems and by a different approach, by Terwilliger and Vidunas [25,Theorem 4.5 and 5.3] and Vidunas [28]. More limited q-Zhedanov schemes, not related to q-Verde-Star parameters, were earlier given by Mazzocco [19], see also [13].
Filling in this q-Zhedanov scheme gave some surprises. Several families in the q-Askey scheme which usually do not get much attention attain an independent status in the present scheme. We mention symmetric cases of Askey-Wilson, Al-Salam-Chihara and big q-Jacobi polynomials, and also q 2 versions of the continuous q-Jacobi polynomials and its subfamilies, which give interesting q-polynomials by quadratic transformation. In particular, continuous q 2 -Hermite polynomials get after quadratic transformation a q-hypergeometric expression which allows to describe this family by Verde-Star's approach, something which was not possible in [27] for the continuous q-Hermite polynomials.
Very relevant for this paper is the fact that the Askey-Wilson algebra in its dependence on the Askey-Wilson parameters a, b, c, d is not only symmetric in these parameters, but also remains invariant if two parameters e, f from a, b, c, d are sent to q/e, q/f , respectively. Such a symmetry under (a, b, c, d) → (a, b, qd −1 , qc −1 ) for the Askey-Wilson DAHA [23], [20], [13, §5.1] was communicated to me by Marta Mazzocco. She pointed for this to the DAHA symmetry (a, b, c, d) → (a, b, qc −1 , d) observed by Oblomkov [21, (2.11)]. Since Askey-Wilson polynomials are not invariant under these transformations (however, see Section 7), it means that a possible charting of the q-Zhedanov scheme cannot be a charting of the q-Askey scheme which completely distinguishes between individual systems of polynomials.
Duality is also an important theme in this paper, as it was also in the author's earlier paper [13] with Mazzocco. From the graphical q-Zhedanov scheme in Section 5 the dualities between the occurring families can be nicely seen.
It should be emphasized that, although the families from the q-Askey scheme are first of all known as orthogonal polynomials, we completely disregard orthogonality aspects in this paper. Only those algebraic aspects which give rise to a q-Verde-Star description and a nice associated q-Zhedanov algebra matter here for us.
The contents of this paper are as follows. In Section 2 we treat some (but not all) families from the q-Askey scheme. We concentrate on the families which play a particularly important role later in the paper. Duality properties are already discussed here. This section may be skipped on first reading. As already said, Section 3 on Verde Star's framework, Section 4 on using this framework in the context of the q-Zhedanov algebra, and Section 5 which presents the q-Zhedanov scheme in Figure 1, form the heart of this paper. Some subsections of Section 4 consider examples of q-Zhedanov algebras in connection with q-Verde-Star parameters. Here the further symmetries just observed also enter. But these symmetries appear already in Subsection 2.3 in connection with duality, although not yet in connection with the Askey-Wilson algebra. In Section 6 the duals of continuous q 2 -Hermite polynomials and of discrete q-Hermite I polynomials are discussed. Finally Section 7 gives some recapitulation, open questions and thoughts about an optimal q-Askey scheme. Note For definition and notation of q-shifted factorials and q-hypergeometric series we follow [5, §1.2]. We will only need terminating series: 2 Preliminaries about polynomials in the q-Askey scheme
In this form they are symmetric in a, b, c, d (see [1, p.6]). In the normalization (2.1) the polynomials are still symmetric in b, c, d, but for a ↔ b we have where we assume in addition that b = 0 and bc, bd = 1, q −1 , q −2 , . . . . For c = −a, d = −b we have the symmetric Askey-Wilson polynomials [1, p.7] with the symmetry The polynomials R n (z) given by (2.1) satisfy the eigenvalue equation [9, (14.1.7)] They also satisfy the three-term recurrence relation (for n = 1, 2, . . .) (z + z −1 )R n (z) = (a + a −1 )R n (z) The starting values R 0 (z) = 1 and R 1 (z) as given by (2.1) satisfy (2.6) in its reduced form Note that here the factors (1 − q −1 abcd) in numerator and denominator in (2.6) cancel, which even turns out to be correct if abcd = q. In the following we mention a few special cases of Askey-Wilson which will play a remarkable role in the q-Zhedanov scheme. These special cases have their own, generally used, notations (see [9,Ch. 14]), but we will just denote them as a restricted R n (z).
Continuous q-Jacobi polynomials [9, (14.10.1)]   This is no longer a special case of Askey-Wilson, but a limit case. There is the symmetry H n (−x) = (−1) n H n (x).
together with (2.3), allows to express the polynomials (2.10)-(2.13) for base q 2 in a different way: We call the polynomials R n (z) on the left-hand sides the continuous q 2 -Jacobi, continuous q 2 -Laguerre, continuous q 2 -ultraspherical and continuous q 2 -Hermite polynomials, respectively. The continuous q 2 -Jacobi polynomials as represented by the left-hand side of (2.14) were first observed by M. Rahman, see [9, p.468].

Big q-Jacobi polynomials and some subfamilies
Assume ab, a, c = q −1 , q −2 , . . . . Define big q-Jacobi polynomials [9, (14.5.1)] by They are limit cases [9, (14.1.18)] of Askey-Wilson polynomials: The S 4 symmetry of Askey-Wilson in its parameters reduces to an S 2 × S 2 symmetry for big q-Jacobi. Here the S 3 symmetry which is obvious from (2.1) reduces to a symmetry obvious from (2.18): The other symmetry is a limit case of (2.20) under the limit (2.19): c −n (qac, qc; q) n P n (cx; ac, b, c; q) = b −n (qab, qb; q) n P n (bx; ab, c, b; q) (equivalenty), where we assume that b, c = 0 and (in the first equality) a, b, c, abc −1 = q −1 , q −2 , . . . . In particular, we have the symmetric big q-Jacobi polynomials They also satisfy the three-term recurrence relation (for n = 1, 2, . . .) The starting values P 0 (x) = 1 and P 1 (x) as given by (2.18) satisfy (2.25) in its reduced form We mention some further subfamilies of the big q-Jacobi polynomials.

Duality
In our charting of the q-Askey scheme we consider systems of polynomials {p n (x)} in the scheme up to dilation of x by a nonzero constant. In particular, taking −1 for this constant and noting from (2.1) that So for charting purposes this is harmless.
with both sides equal to Formula (2.32) gives a duality between the two Askey-Wilson polynomials R n (z; a, b, c, d | q) and R m (w;ã,b,c,d | q).
Remark The above concept of duality is rather weak. It certainly does not involve in general that orthogonality relations for the original system imply orthogonality for the dual system. See [2] for a treatment with many examples of dual systems in this stronger sense. Still our weak notion of duality, when occurring in the q-Askey scheme, usually makes it possible to relate the three-term recurrence relation and the second order q-difference eigenvalue equation for the one system in a formal way with the eigenvalue equation and the recurrence relation, respectively, for the dual system. For instance, the recurrence relation (2.6) for Askey-Wilson polynomials can be rewritten as Compare this with (2.5) with z replaced by q nã in the coefficients on the right-hand side.
Note that the definition of the dual parameters depends on a choice of one of the original parameters. In (2.31) this is a. The duals of the dual parametersã,b,c,d with respect toã are the original parameters a, b, c, d again, up to common multiplication by −1. (In the following this identification of parameters with their opposites will be silently assumed.) Now let a ′ , b ′ , c ′ , d ′ be dual parameters of a, b, c, d with respect to b: Then the dual parameters of a ′ , b ′ , c ′ , d ′ with respect to a ′ are b, a, qd −1 , qc −1 . This result can be alternatively stated by the commutative diagram From this we have: be a symmetric function which is also invariant under common multiplication of a, b, c, d by −1. Then f is symmetric inã,b,c,d iff f is invariant under any mapping which sends an even number of its variables a, b, c, d to q/a, q/b, q/c, q/d, respectively.
The Askey-Wilson polynomials, although invariant under permutations of a, b, c, d, are no longer invariant under the larger group which also involves mappings sending two parameters e, f out of a, b, c, d to qe −1 , qf −1 , respectively, but such symmetries will play a role later in this paper, see Subsection 4.2. The larger group is isomorphic to the semidirect product S 4 ⋉ (Z/2Z) 3 (where (Z/2Z) 3 is the group of an even number of sign changes of 1, 2, 3, 4), which is also the Weyl group for the root system D 4 , see [7, §12.1].
Continuous dual q-Hahn ←→ big q-Jacobi [13, (44)] Note that there is a one-to-one correspondence ±(a, b, c) ↔ (q −1 ab, ab −1 , q −1 ac). By (2.2) there is a symmetry (a, b, c) ↔ (b, a, c) for suitably normalized continuous dual q-Hahn. By the duality (2.35) this leads to a symmetry (a, b, c) ↔ (a, b −1 , b −1 c) for the big q-Jacobi parameters a, b, c, although it does not leave P n (x; a, b, c; q) invariant. Conversely, the symmetry (2.21) for big q-Jacobi polynomials leads to a symmetry (a, b, c) ↔ (a, qc −1 , qb −1 ) for the continuous dual Hahn parameters a, b, c, which, however, does not leave R n (z; a, b, c, 0 | q) invariant.
3 Verde-Star's description of the q-Askey scheme Monic polynomials u n in the q-Askey scheme can be described by the data [27], [12, (3.1)-(3.4)]: 3) Define a linear operator L on the space of polynomials by Lv 0 = 0, Lv n = h n v n + g n v n−1 , n > 0, (3.5) or equivalently, in view of (3.1), by For specific u n in the q-Askey scheme the eigenvalue equation (3.5), (3.6) can be rewritten as a second order q-difference equation (the generalized Bochner property [8]).
In [12, (3.2)] x k had an additional term b 0 and h k had an additional term a 0 , but without loss of generality these can be omitted, because the term b 0 only leads to a translation of the x-variable and the term a 0 only leads to adding a constant to the operator L in (3.5).
The polynomials u n are determined by the 9 parameters a 1 , a 2 , b 1 , b 2 , d 0 , d 1 , d 2 , d 3 , d 4 under the 3 constraints (3.3). There are two other operations on these parameters which do not lead to essential changes. Simultaneous multiplication of a 1 , a 2 and d 0 , d 1 , d 2 , d 3 , d 4 by µ = 0 only leads to multiplication of L by µ. Simultaneous multiplication of b 1 , b 2 and d 0 , d 1 , d 2 , d 3 , d 4 by ρ = 0 leads to replacing u n (x) by ρ n u n (ρ −1 x). So essentially there are only 4 parameters, just as the Askey-Wilson polynomials (corresponding to the generic case of these parameters) have 4 parameters.
As observed in [12, §3], there are two further remarkable operations which can be performed on the 11 parameters: This relates u n given by (3.1) to its dual u n given by If we put If U n orŨ m are written in a special case with (3.2) substituted then they will appear as qhypergeometric series.

q-Zhedanov algebra with q-Verde-Star parameters
For {u n } a system of polynomials in the q-Askey scheme consider operators K 1 , K 2 acting on the space of polynomials in one variable x, K 1 being the second order q-difference operator which has the u n as eigenfunctions and K 2 the operator of multiplication by x. Then for certain constants C 1 , C 2 , D, G 1 , G 2 . This is a rewriting of [6, (3.2)] by eliminating there K 3 and by substituting R = 1 − 1 2 (q + q −1 ). From [6, (3.2)] there would have been additional terms A 2 K 2 2 and A 1 K 2 1 on the right-hand sides of the two equalities in (4.1), respectively. But these can be removed by adding suitable constants to K 1 and K 2 . There is also a Casimir operator Q, commuting with K 1 and K 2 and given by [6, (3.4)]: Then, for a certain constant ω, Q = ω. Note the duality symmetry for (4.1), (4.2): The operators K 1 and K 2 act on the Newton polynomials v n (see (3.1) and (3.5)) by So by the correspondence f = {f n } ∞ n=0 ←→ n f n v n (x) these operators also act on sequences f by Substitution of (4.5) on the left-hand sides of (4.1) leads to the right-hand sides with the constants explicitly expressed in terms of the parameters a 1 , a 2 , b 1 , b 2 , d 0 , d 1 , d 2 . We get We can also express ω in (4.3) in terms of these parameters: Since multiplication of K 1 by a nonzero constant µ and multiplication of K 2 by a nonzero constant ρ does not essentially change the algebra generated by K 1 and K 2 , it follows from (4.1), (4.2) and (4.3) that this algebra is essentially determined by with µ, ρ arbitrarily nonzero. By (4.6), (4.7) this precisely matches with the allowed multiplication by constants of the q-Verde-Star parameters in Section 3. So again there are essentially four parameters coming from the q-Zhedanov coefficients, just as the Askey-Wilson polynomials have four parameters. The constants C 1 , C 2 and D are invariant under q ↔ q −1 , but not so G 1 , G 2 and ω. The property for G 1 or G 2 to be zero for all q is clearly invariant under , then all constants in (4.6), (4.7) remain unchanged.
By (4.6), (4.7) the duality symmetry In the following subsections we consider some special cases of the q-Zhedanov algebra in connection with the q-Verde-Star parameters: the general case associated with the Askey-Wilson polynomials, followed by a discussion of the (c, d) → (qd −1 , qc −1 ) symmetry, and finally the dual cases associated with the continuous dual q-Hahn and the big q-Jacobi polynomials.

The Askey-Wilson algebra
The generic case of the q-Zhedanov algebra is the algebra associated with the Askey-Wilson polynomials. Put U n (z + z −1 ) = R n (z; a, b, c, d | q). Then this can be written in the form (3.9) with Then, by (4.6) and (4.7), and In [13, (16)] the expressions (4.12) for C 1 , C 2 , D, G 1 , G 2 were also obtained, but starting from the second order q-difference operator L given in (2.5), by which K 1 is represented in the polynomial representation [13, (17)] of the Askey-Wilson algebra. Note that indeed h n as given by (4.9) equals the eigenvalue of the eigenfunction R n (z) in (2.4).

A further symmetry of the Askey-Wilson algebra
As observed in Section 2, nothing essentially changes for the q-Verde-Star parameters if L, h n and g n are simultaneously multiplied by a nonzero constant µ, or equivalently, if a 1 , a 2 , g 1 , g 2 are multiplied by µ. In the context of the q-Zhedanov algebra this means, by (4.8), that D, G 1 are multiplied by µ and C 2 , G 2 , Q, ω are multiplied by µ 2 . Now, in the case of the Askey-Wilson algebra, take µ = (q/(abcd)) 1 2 =ã −1 (withã given by (2.31). Then it follows for the q-Zhedanov algebra generated by K 1 , K 2 with K 1 =ã −1 L that, by (4.12) and (4.13), C 1 ,ã −2 C 2 ,ã −1 D,ã −1 G 1 ,ã −2 G 2 andã −2 ω are not only invariant under permutations of a, b, c, d, but also invariant if two parameters e, f out of a, b, c, d are sent to qe −1 , qf −1 , respectively (the symmetries already discussed in Subsection 2.3). Such symmetries for the Askey-Wilson DAHA were pointed out to me by Marta Mazzocco. As was observed in Subsection 2.3, these symmetries generate a group which is isomorphic to the Weyl group of the root system D 4 . For some version of the Askey-Wilson algebra a symmetry under the same group was earlier observed in [4,Section 4].
These further symmetries are relevant for the q-Zhedanov scheme ( Figure 1 in the next section). If in (4.12) some of the C 1 , C 2 , D, G 1 , G 2 vanish under certain constraints on a, b, c, d then they will also vanish under transformed constraints by application of a symmetry. For instance, the constraints c = −a, d = −b give symmetric Askey-Wilson (2.3) and cause D and G 2 to vanish. Then also the constraints c = −qa −1 , d = −qb −1 make D and G 2 vanish. Furthermore, these symmetries induce symmetries on limit cases of the Askey-Wilson algebra.

The dual algebras for continuous dual q-Hahn and big q-Jacobi
For obtaining the continuous dual q-Hahn algebra put d = 0 in (4.9)-(4.13). Then e 4 = 0 and e 1 , e 2 , e 3 can be considered as elementary symmetric polynomials in a, b, c. We obtain See also [13, (50)]. Note that U n (z + z −1 ) = R n (z; a, b, c, 0 | q) can be written in the form (3.9) with x k , h k , g k given by (4.14), and that h n = q −n is the eigenvalue of L for d = 0 in (2.4).
Just as for the Askey-Wilson algebra nothing essentially changes if we multiply h n , g n by µ, and a 1 , a 2 , g 1 , g 2 by µ, and D, G 1 by µ, and C 2 , G 2 by µ 2 (here we have omitted ω). If we now take µ = (q/(abc)) 1 2 then the resulting C 1 , µ 2 C 2 , µD, µG 1 , µ 2 G 2 are not only invariant under permutations of a, b, c, but also invariant if two parameters e, f out of a, b, c, are sent to qe −1 , qf −1 , respectively.
With x k , h k , g k given by (4.14) we get from (3.9) thatŨ m (y) = P m (y; q −1 ab, ab −1 , q −1 ac; q), a big q-Jacobi polynomial as we saw it already in the duality (2.35).
Below we give data associated with the big q-Jacobi polynomial U n (x) = P n (x; a, b, c; q).
If in (4.17)-(4.19) we multiply h k , g k , a 1 , a 2 , d 1 , d 2 , D, G 1 by (qab) − 1 2 and C 2 , G 2 by (qab) −1 , and if we multiply x; a, b, c; q . Moreover the big q-Jacobi algebra then has the invariances corresponding to the ones we observed for the continuous dual Hahn algebra. We see that

The q-Zhedanov scheme
It will turn out that vanishing of some of the coefficients C 1 , C 2 , D, G 1 , G 2 in the q-Zhedanov algebra with relations (4.1) is a characterizing property of corresponding polynomials in the q-Askey scheme. An arrow then will indicate that one more of the five constants becomes zero. We arrange the five coefficients in an array By (4.4) we can pass to the dual q-Zhedanov algebra by reflection of the array with respect to the vertical axis. We will replace coefficients in this array by symbols or , where denotes any value (including zero) and denotes zero. From the explicit data for the q-Verde-Star parameters of the families in the q-Askey scheme as given in [12, Appendix A] we can obtain the corresponding vanishing pattern for the array (5.1) by means of (4.6). Then connect the arrays by arrows such Figure 1: The q-Zhedanov scheme. In referring to this rows are numbered top to bottom, and in each row the elements are labeled by the letters a,b,c,. . . from left to right. that in the direction of the arrow at least one is turned into . The resulting graph is in Figure 1.
Let us number the rows in the scheme from top to bottom by 1 to 6. In each row list the successive arrays from left to right by a, b, . . . . Note the following dualities in Figure 1: 1a, 3d, 4b, 5c, 6a are self-dual. 2a ↔ 2b, 3a ↔ 3b, 3c ↔ 3e, 4a ↔ 4e, 4c ↔ 4g, 4d ↔ 4f are dual pairs. The duals of 5a and 5b are not included in the scheme, but will be discussed in Section 6.
The arrays in Figure 1 correspond with families in the q-Askey scheme as given in the list below. Usually we mention only one family for an array, but also subfamilies still corresponding to that array. There may be more families for the same array, by q ↔ q −1 exchange, and there may be discrete families. If one wants to add these, one can use [9, Ch. 14]. We do not bother at all about orthogonality properties. The only things that matter are that the polynomials are eigenfunctions of a second order q-difference operator and satisfy a three-term recurrence relation.
As much as possible formulas for the occurring families in the list will be given in terms of Askey-Wilson R n (z; a, b, c, d | q), x = z + z −1 (see (2.1)) or big q-Jacobi P n (x; a, b, c; q) (see (2.18)). Otherwise a formula is given as a limit case of R n or P n or as some r φ s q-hypergeometric function. Everything is up to constant factors.
5c. x n (x −1 ; q) n 6a. x n Remarks 1. For obtaining this list we started with the q-Verde-Star data given in [12,Appendix A]. From these data, with a 0 and b 0 being put to zero, the q-Zhedanov coefficients C 1 , C 2 , D, G 1 , G 2 can be computed by (4.6), and it can be read off which of these five coefficients vanish. Moreover, for a given family, it can be seen for which constraints on the parameters a further coefficient will vanish.

2.
It is interesting to compare the scheme in Figure 1 with the q-Verde-Star scheme [12, Figure 1]: • First, the present scheme is more compact because it does not discern between q and q −1 , and also because it does not bother about the Newton polynomials v k in which u n is expanded, so each family occurs in at most one place in the scheme.
• Furthermore, some separate families in [12, Figure 1] here merge as subfamilies or limit families with other families (continuous big q-Hermite, q-Bessel and Stieltjes-Wigert).
• On the other hand, the present scheme is richer because many subfamilies here get an independent status which they did not have in the q-Verde-Star scheme. We mention symmetric Askey-Wilson, symmetric Al-Salam-Chihara, symmetric big q-Jacobi, continuous q 2 -Jacobi, continuous q 2 -Laguerre, continuous q 2 -ultraspherical, discrete q-Hermite and q-Krawtchouk with p = q −N .
• Continuous q-Hermite could not be included in the q-Verde-Star scheme, because it cannot be written in the form (3.1), but a q-Zhedanov algebra can be associated with it. Therefore, in the present scheme it occurs, but only as a limit family of symmetric Al-Salam-Chihara. However, continuous q 2 -Hermite, obtained by the quadratic transformation (2.17), has its own place in our scheme. We also see from (2.17) that continuous q 2 -Hermite will live in the q-Verde-Star scheme as a subfamily of Askey-Wilson, but not with a separate place in that scheme. To some extent this overcomes the defect that continuous q-Hermite could not be handled by Verde-Star [27].
3. The cases 5c and 6 are degenerate. They are eigenfunctions of a first order q-difference operator and they satisfy a two-term recurrence relation.

4.
It turns out that with each of the 32 possible arrays a family of polynomials can be associated, but we have not included all of these in the scheme, because some families did not seem to be of sufficient interest. The following cases were missing in the scheme.
idem with the previous two constraints.
big q-Jacobi P n (x; a, b, c; q) with a + c + a(b + c) = 0.

5.
In Subsections 4.2, 4.3 we already observed symmetries for q-Zhedanov algebras which are not satisfied by the corresponding Askey-Wilson, continuous dual q-Hahn and big q-Jacobi polynomials. Such symmetries will also be present in algebras associated with families in the third and fourth row of Figure 1.
6 Duals of continuous q 2 -Hermite and discrete q-Hermite I In this section we will phrase the various dualities in the notation of (3.8)-(3.10).
Continuous q 2 -Laguerre ←→ special big q-Jacobi For obtaining the dual of continuous q-Hermite we first consider a suitable dual of continuous q 2 -Laguerre and then specialize the parameter. Put x k = q k+ 1 2 + q −k− 1 2 , h k = q −k , g k = (q 1 2 − aq k )(q −2k − 1). Then by (4.6) we have C 1 = (q − q −1 ) 2 , C 2 = D = G 1 = 0, G 2 = (1 + q −1 )(1 − q) 2 a. Also (3.8)-(3.10) take the form (6.1) By (2.1), (2.15) and (2.18) we can identify the dual polynomials U n (z + z −1 ) andŨ m (y) as continuous q 2 -Laguerre and special big q-Jacobi: It is precisely this dual pair which allows the specialization a = 0 which we need. TheŨ m belong to the same array 4e in Figure 1 as symmetric big q-Jacobi. In fact, with the choice we would have obtained the same values for C 1 , C 2 , D, G 1 , G 2 , but U n (z + z −1 ) = R n (z; a, q 1 2 , −q 1 2 , 0 | q) and forŨ m (y) the symmetric big q-Jacobi polynomial P m (y; q − 1 2 a, q − 1 2 a, −q − 1 2 a; q). It is also remarkable that for both specializations of big q-Jacobi belonging to array 4e the specialization of the three-term recurrence relation (2.25) gives (for n = 1, 2, . . .) xP n (x) as a linear combination of P n+1 (x) and P n−1 (x) without a term for P n (x). Still only for symmetric big q-Jacobi P n (x; a, a, −a; q) we have P n (−x) = (−1) n P n (x), but for P n (x; −1, −1, c; q) the starting value P 1 (x) already lacks this symmetry.
Continuous q 2 -Hermite ←→ very special big q-Jacobi Just put a = 0 in the above case.
Note that the "dual continuous q 2 -Hermite polynomials" P n (x) = P n (x; −1, −1, 0; q) cannnot be orthogonal with respect to a positive orthogonality measure. This is seen from the signs of the coefficients in the recurrence relation (a specialization of (2.25))

Al-Salam-Carlitz I ←→ q-Charlier
For obtaining the dual of discrete q-Hermite I we first consider the dual families of Al-Salam-Carlitz I and q-Charlier and then specialize the parameter. Discrete q-Hermite I ←→ special q-Charlier Just put a = 1 in the above case. Then D will also vanish and, by [9, (14.28.1), (14.23.1)], the expressions in terms of discrete q-Hermite I and q-Charlier are: U n (x) = q − 1 2 n(n−1) h n (x; q), U m (y) = C m (y; 1; q).. (6.6) 7 Charting the q-Askey scheme: conclusions and questions From Sections 4 and 5 we can conclude that a system {p n }, belonging to a family in the q-Askey scheme for some values of the parameters for that family, and considered up to dilation of the independent variable, gives rise to values of C 1 , C 2 , D, G 1 , G 2 , ω up to multiplication by certain powers of nonzero constants µ and ρ as in (4.8). In view of Subsections 4.2, 4.3 and Remark 5 in Section 5 the system {p n } will not be uniquely determined by the equivalence class of C 1 , C 2 , D, G 1 , G 2 , ω. If the p n are Askey-Wilson polynomials then we may conjecture from Subsection 4.2 that its parameters ±(a, b, c, d) are determined by the equivalence class of C 1 , C 2 , D, G 1 , G 2 , ω up to permutations and up to sending any e, f from a, b, c, d to q/e, q/f , but even that is not yet clear.
Still it is of interest to let the equivalence classes (4.8) form a complex manifold. Here we impose that ω and at least one of the other five coordinates are nonzero. Then we can put ω and another nonvanishing coordinate equal to 1. This will fix µ and ρ, possibly up to sign. Then use the other four coordinates as local coordinates. There will still be point pairs which have to be identified.
Another question, more a matter of taste and of convenience than of mathematical rigor, is whether the usual version [9, p.414] of the q-Askey scheme should be adapted in view of the q-Verde-Star scheme [12] and the q-Zhedanov scheme Figure 1. I would say that an adapted scheme should not distinguish between q and q −1 variants of the same family. Neither should it distinguish between two cases of the same family with expansions in terms of different Newton polynomials. Interesting as these features of the q-Verde-Star scheme may be, they make the graphical display too large and too detailed. For the rest the adapted scheme should be the union of the q-Verde-Star and the q-Zhedanov scheme, since each of the schemes highlights some families which deserve special attention but are not in the other scheme. It would be convenient to have one scheme which starts with Askey-Wilson and another scheme which starts with q-Racah. In the lower rows these schemes would have several families in common.
Transcendental functions might also be associated with the q-Zhedanov scheme, as we will argue now. The q-Zhedanov algebra yielding the coefficients C 1 , C 2 , D, G 1 , G 2 by (4.1) is generated by a second order q-difference operator K 1 and the operator K 2 which is multiplication by x. Polynomials associated with the algebra occur as eigenfunctions of K 1 for special eigenvalues. However, the algebra and its coefficients are independent of the choice of the eigenvalues. Other eigenfunctions of K 1 on another domain may yield generalized orthogonal systems leading to integral transforms. Notably, in addition to Askey-Wilson polynomials, there are Askey-Wilson functions, see Koelink and Stokman [10]. The same authors have in [11, Figure 1.2] a scheme, somewhat analogous to the q-Askey scheme, with limit cases of the Askey-Wilson functions. It would be interesting to match our scheme with a possibly extended version of the scheme in [11]. Passage to the transcendental case will also be needed for realizing some symmetries of the q-Zhedanov algebra. For instance, the symmetry (a, b, c, d) → (a, b, qd −1 , qc −1 ) of the Askey-Wilson algebra (see Subsection 4.2) is, by [11, (3.2)], visible for Askey-Wilson functions, while this symmetry is not present if we restrict ourselves to the polynomial case.
Schemes for nonsymmetric Askey-Wilson polynomials and their subfamilies, and for the corresponding DAHAs might be another follow-up topic. A Verde-Star type approach for nonsymmetric polynomials may be promising. However, in view of the cases already considered in [13], there does not seem to be a uniform approach for the DAHA as we do have by (4.1) for the q-Zhedanov algebras.