Quadratic transformations for a function that generalizes 2F1 and the ...

Ramanujan J (2007) 13:339–364 DOI 10.1007/s11139-006-0257-x

Quadratic transformations for a function that generalizes 2 F1 and the Askey-Wilson polynomials S. N. M. Ruijsenaars

Dedicated to Richard Askey on the occasion of his 70th birthday. Received: 7 June 2004 / Accepted: 9 December 2004 ! C Springer Science + Business Media, LLC 2007

Abstract In previous papers we introduced and studied a ‘relativistic’ hypergeometric function R(a+ , a− , c; v, v) ˆ that satises four hyperbolic difference equations of AskeyWilson type. Specializing the family of couplings c ∈ C4 to suitable two-dimensional subfamilies, we obtain doubling identities that may be viewed as generalized quadratic transformations. Specically, they give rise to a quadratic transformation for 2 F1 in the ‘nonrelativistic’ limit, and they yield quadratic transformations for the AskeyWilson polynomials when the variables v or vˆ are suitably discretized. For the general coupling case, we also study the bearing of several previous results on the AskeyWilson polynomials. Keywords Relativistic hypergeometric function . Quadratic transformations . Askey-Wilson difference operators . Askey-Wilson polynomials . Parameter shifts 2000 Mathematics Subject Classification Primary—33D45, 39A70 1 Introduction The following may be viewed as a continuation of our work reported in [1–4] and recently surveyed in [5]. The subject of study is a function R(a+ , a− , c; v, v) ˆ generalizing both the hypergeometric function 2 F1 (a, b, c; w) and the Askey-Wilson polynomials [6, 7]. As shown in [1] and [2] (henceforth referred to as I and II, resp.), it has a long list of symmetry properties, both manifest and hidden. These all concern the full four-dimensional family of couplings c = (c0 , c1 , c2 , c3 ). Here we add further hidden symmetries to this list, which however refer to suitably chosen two-dimensional subfamilies. Since the new identities are related by symmetries already established in S. N. M. Ruijsenaars (!) Centre for Mathematics and Computer Science, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands e-mail: [email protected] Springer

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I and II, we only mention one of them to begin with, viz., R(a+ , 2a− , (s, 0, t, 0); 2v, v) ˆ = R(a+ , a− , 2−1 (s, s, t, t); v, v) ˆ

(1.1)

As detailed in I (cf. also [5]), the connection with 2 F1 can be made as follows. First dene ψrel (λ, c; v, v) ˆ ≡ R(π, λ, λc; v, v/2) ˆ ˜ v, v) ψnr (d, d; ˆ ≡ 2 F1 ((d + d˜ + i v)/2, ˆ (d + d˜ − i v)/2, ˆ d + 1/2; − sinh2 v)

(1.2) (1.3)

Then one has lim ψrel (λ, c; v, λv) ˆ = ψnr (c0 + c2 , c1 + c3 ; v, v) ˆ λ↓0

(1.4)

Therefore the identity (1.1) gives rise to ψnr (s + t, 0; 2v, v) ˆ = ψnr (s + t, s + t; v, 2v) ˆ

(1.5)

Rewriting this in terms of 2 F1 , we obtain 2 F1 (a, b, a

+ b + 1/2; 4w(1 − w)) = 2 F1 (2a, 2b, a + b + 1/2; w)

(1.6)

This is a well-known quadratic transformation, cf. e.g. [8], p. 125. In the main text we detail the specializations of the doubling identity (1.1) and its cousins to the Askey-Wilson polynomials, comparing them to the quadratic transformation obtained by Askey and Wilson, cf. Section 3 in [6]. The term ‘doubling identities’ is inspired by our starting point for obtaining them. To explain this, we recall that R(a+ , a− , c; v, v) ˆ is a joint eigenfunction of four independent analytic difference operators (A$Os) of Askey-Wilson type. The identities now arise precisely when at least one of these four A$Os differs from the square of an Askey-Wilson type A$O by a constant. Moreover, the identities then imply that each of the two R-functions involved is also an eigenfunction of the pertinent ‘square root A$O’. (We mention in passing that for reectionless second order A$Os this doubling phenomenon admits a quite general formulation, cf. [9, 10].) The only general representation for the R-function that is known to date is an integral representation generalizing the Barnes representation of 2 F1 , cf. I. (That is, no generalization of the Gauss series or Euler integral representation has been found.) The building block for the representation (or—perhaps more accurately—denition) of the R-function is the hyperbolic gamma function G(a+ , a− ; z) from [11] (also known as the Kurokawa double sine function or Faddeev’s quantum dilogarithm). In Section 6 we obtain another version of (1.1) (namely (6.20)), whose proof involves doubling identities for G(a+ , a− ; z) following from a general multiplication formula established in [11]. But the integral representation itself seems utterly useless to understand why (1.1) holds true, and accordingly we do not even recapitulate it in the main text. Springer

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We do invoke various other results from I, II and [4]. We need them to prove the doubling identities for the R-function and to derive the doubling identity (6.20) for the function E(a+ , a− , γ ; v, v). ˆ The latter function may be viewed as a renormalized and similarity transformed R-function, cf. (6.3) and (3.3). Since we have not previously done so, we establish its relation to the Askey-Wilson polynomials (for general parameters). We also take this opportunity to consider the action of the parameter shifts obtained and studied in [4] on the Askey-Wilson polynomials, and relate the outcome to previous results in the literature. We have tried to present our results in such an order that the doubling identity (1.1) and its relative (3.16) can already be understood by readers who are willing to take some results from I for granted. Indeed, the gist of the matter can be gleaned from Section 3, which only involves elementary calculations. But we have not found a complete proof that bypasses various ndings from II and [4]. The latter results are reviewed before embarking on the proof. The doubling identity (6.20) and the results concerning Askey-Wilson polynomials involve even more material from previous work, which we also summarize at the point it is needed. In keeping with this line of exposition, we have organized the paper as follows. In Section 2 we summarize the eigenfunction properties and manifest symmetries of the R-function, all of which can be found in I. We also recall how the AskeyWilson polynomials arise via a discretization of v, ˆ and hence derive the quadratic transformation in Section 3 of [6] from the identity (1.1). In Section 3 we rst answer the question: When can the square of an Askey-Wilson A$O be viewed as an Askey-Wilson A$O plus an additive constant C? With our choice of additive normalization constant (which has a non-trivial dependence on the 6 parameters a+ , a− , c0 , c1 , c2 , c3 ), we obtain a 4-dimensional family for which this holds true with C = 2, cf. (3.11). We then explain how this key observation bears on the eigenfunction properties of the R-function, rendering the doubling identities (1.1) and (3.16) eminently plausible. The full proof is however postponed to Section 5. Section 3 is concluded by deriving direct consequences of (1.1) and (3.16). From the viewpoint of multi-variable Askey-Wilson (Koornwinder-Macdonald) polynomials [12], the one-variable Askey-Wilson case corresponds to the Lie algebra BC1 . Specializing the A N −1 multi-variable Macdonald polynomials to the one-variable (N = 2) case, the q-Gegenbauer (or q-ultraspherical) polynomials arise, which form a 1-coupling subfamily of the 4-coupling Askey-Wilson family [13]. In Section 4 we specialize the ndings of Section 3 to four corresponding 1-coupling families of Rfunctions. As we demonstrate in Section 4, they can be expressed in terms of a single function R A1 (a+ , a− , b; v, v) ˆ ≡ R(a+ , a− , (b, 0, 0, 0); v, v/2) ˆ

(1.7)

of A1 type. For b = ka+ + la− with k, l ∈ Z, this function was already introduced and studied in [14]. (More precisely, Eq. (3.75) in [14] yields a renormalized version of R A1 , as can be deduced from [4].) Section 5 is devoted to the proof of the doubling identity (3.16). Our proof hinges on previous ndings obtained via parameter shifts [4]. As already suggested above, the R-function and the E-function may be viewed as avatars of the same object. Even so, the renormalization, different parametrization, Springer

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and similarity transformation connecting the two functions give rise to certain features being considerably simpler for one of them. In Section 6 we have occasion to invoke the D4 invariance of the E-function (proved in II), whose translation to the R-function is rather awkward to work with. In the other direction, one might have expected that the simplicity of the doubling identities (1.1) and (3.16) for the R-function is no longer present for the E-function. But actually the factors connecting R to E have their own doubling identities, and so we wind up with the equally simple version (6.20). This is the main result of Section 6, but as a spin-off we also obtain a short and more illuminating derivation of the key formulas (3.11)–(3.13). The nal Sections 7 and 8 are mainly concerned with connections of our previous results to the Askey-Wilson polynomials. Here we are dealing with the general 6parameter case, hence shifting our focus away from the 4-parameter doubling context. In Section 7 we demonstrate that the E-function admits 32 discretizations yielding 32 distinct sets of monic Askey-Wilson polynomials (for generic parameters (a+ , a− , γ )). This is readily understood for the E-function, whereas a reformulation of this nding for the R-function would be quite clumsy. Indeed, for the latter one needs to keep track of the similarity factors and moreover the pertinent analytic continuation would involve not only one of the parameters a+ , a− , but also the couplings c in a case-dependent way. By contrast, for E(a+ , a− , γ ; v, v) ˆ we can deal with a xed γ ∈ R4 , cf. (3.3). In Section 8 we start from the results in [4] concerning shifts in the parameters γ0 , . . . , γ3 of the Askey-Wilson A$Os and renormalized R-function Rr (a+ , a− , γ ; v, v). ˆ We then determine the analogs of the shift and factorization identities for the similarity-transformed A$Os and E-function. The results are new and surprisingly simple. As an unexpected bonus, a suitable specialization yields the squaring identity (6.21) from the shift perspective. Finally, we turn to the consequences of the shift formulas for the Askey-Wilson polynomials. As it turns out, this yields relations that can mostly be found in previous literature. In particular, a paper by Kalnins and Miller [15] is quite relevant in this connection. 2 The R-function and the Askey-Wilson polynomials In this section we recall the eigenfunction properties and manifest symmetries of the R-function and its relation to the Askey-Wilson polynomials. We begin by dening the (hyperbolic Askey-Wilson) A$Os Aδ (c; z) ≡ Cδ (c; z)[exp(−ia−δ d/dz) − 1] + Cδ (c; −z)[exp(ia−δ d/dz) − 1] + 2 cos(π[c0 + c1 + c2 + c3 ]/aδ ),

δ = +, −

(2.1)

Here the coefcient Cδ is dened by Cδ (c; z) ≡

sinh(π [z − ic0 ]/aδ ) cosh(π[z − ic1 ]/aδ ) · sinh(π z/aδ ) cosh(π z/aδ ) sinh(π[z − ia−δ /2 − ic2 ]/aδ ) cosh(π[z − ia−δ /2 − ic3 ]/aδ ) · sinh(π[z − ia−δ /2]/aδ ) cosh(π[z − ia−δ /2]/aδ )

Springer

(2.2)

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The function R(a+ , a− , c; v, v) ˆ is a joint eigenfunction of the four A$Os A+ (c; v), A− (I c; v), A+ (ˆc; v), ˆ A− (I cˆ ; v) ˆ

(2.3)

with eigenvalues 2 cosh(2π v/a ˆ + ), 2 cosh(2π v/a ˆ − ), 2 cosh(2π v/a+ ), 2 cosh(2π v/a− )

(2.4)

respectively. Here, I is the transposition I c ≡ (c0 , c2 , c1 , c3 )

(2.5)

and the dual couplings cˆ are given by cˆ ≡ J c

(2.6)

with

J≡



1

1

1

1 1  2 1

1 −1 −1

−1 1 −1

1

1



−1    −1  1

(2.7)

From the integral representation for the R-function (cf. I) one reads off various symmetry properties, including evenness, R(a+ , a− , c; v, v) ˆ = R(a+ , a− , c; δv, δ & v), ˆ δ, δ & = +, −

(2.8)

scale invariance, R(a+ , a− , c; v, v) ˆ = R(λa+ , λa− , λc; λv, λv) ˆ

(2.9)

R(a+ , a− , c; v, v) ˆ = R(a− , a+ , I c; v, v) ˆ

(2.10)

R(a+ , a− , c; v, v) ˆ = R(a+ , a− , cˆ ; v, ˆ v)

(2.11)

modular invariance,

and self-duality,

We recall that when one takes one of the four eigenfunction properties (cf. (2.3)–(2.4)) for granted, the remaining three follow from (2.10) and (2.11). The R-function is meromorphic in its eight arguments, as long as a+ /a− stays away from (−∞, 0]. Moreover its singular locus and (maximal) pole multiplicities are explicitly known, cf. I Theorem 2.2. Springer

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Proceeding to the relation with the Askey-Wilson polynomials, we set vˆn ≡ −i cˆ0 − ina− , n ∈ Z

Rn (a+ , a− , c; v) ≡ R(a+ , a− , c; v, vˆn ), n ∈ Z

(2.12) (2.13)

The key point is the identity R0 (a+ , a− , c; v) = 1

(2.14)

Indeed, upon combining the A+ (ˆc; v)-eigenvalue ˆ equation with (2.14), we wind up with a recurrence relation implying Rn (a+ , a− , c; v) = Pn (a+ , a− , c; cosh(2π v/a+ )), n ∈ N

(2.15)

where Pn (a+ , a− , c; x) is a polynomial in x of degree n. As a caveat, we should add that the summary in the previous paragraph pertains to generic parameters. When the v-values ˆ (2.12) with n ∈ N belong to the singular locus of the R-function (cf. I (2.35)), then (2.14)–(2.15) need not hold. Furthermore, even when the latter v-values ˆ are regular, it may happen that the degree of Pn is smaller than n. Next, we take a+ equal to −2iπ, so that we obtain polynomials in cos v. With the reparametrization q = e−a− , α = e−c0 , β = −e−c1 , γ = e−c2 −a− /2 , δ = −e−c3 −a− /2

(2.16)

the latter polynomials are the Askey-Wilson polynomials, normalized as Pn(φ) (q, α, β, γ , δ; cos v) ≡ 4 φ3 (q −n , αβγ δq n−1 , αeiv , αe−iv , αβ, αγ , αδ; q, q) (2.17) (Cf. Eqs. (1.6), (1.7), (3.32)–(3.39), (4.61) and (4.62) in I.) Specializing the doubling identity (1.1) to ˜ vˆ = −i(g + g)/2 − ina− , n ∈ N

(2.18)

it therefore yields ˜ − /2 Pn(φ) (e−a− , e−g , −1, e−g−a , −e−a− /2 ; cos 2v)

˜ ˜ − /4 − /4 = P2n (e−a− /2 , e−g/2 , −e−g/2 , e−g/2−a , −e−g/2−a ; cos v) (φ)

(2.19)

cf. (2.15). When we use (2.17) and substitute ˜ − /2 e−a− /2 → q, v → θ, e−g/2 → a, e−g/2−a →b

this amounts to the quadratic 4 φ3 -transformation (3.1) in [6]. Springer

(2.20)

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3 Squared Askey-Wilson A$Os The general structure of the A$O A (2.1) reads A = C(z) exp(α∂z ) + C(−z) exp(−α∂z ) + Vb (z)

(3.1)

Therefore, its square can only be of the same form when Vb (z) vanishes identically. Then it is given by A2 = C(z)C(z + α) exp(2α∂z ) + C(−z)C(−z + α) exp(−2α∂z ) + C(z)C(−z − α) + C(−z)C(z − α)

(Vb = 0)

(3.2)

So when do we have Vb = 0? Using well-known hyperbolic identities, it is easy to see that this holds true for A+ (c; z) when c2 = c3 = 0, cf. (2.1)–(2.2). It can also be directly established that in that case A2+ is again of Askey-Wilson form. But for a systematic study of this issue and various other purposes it is crucial to trade the coupling vector c for a parameter vector γ dened by c(a+ , a− , γ ) ≡ (γ0 + a+ /2 + a− /2, γ1 + a− /2, γ2 + a+ /2, γ3 )

(3.3)

and to make the dependence of the A$Os on the parameters a+ and a− explicit. Accordingly, we switch to the A$O A(a+ , a− , γ ; z) ≡ V (a+ , a− , γ ; z) exp(−ia− d/dz)+V (a+ , a− , γ ; −z) exp(ia− d/dz) + Vb (a+ , a− , γ ; z),

(3.4)

where V (a+ , a− , γ ; z) ≡ −

4

'3

µ=0

cosh(π[z − iγµ − ia− /2]/a+ )

sinh(2π z/a+ ) sinh(2π[z − ia− /2]/a+ )

(3.5)

Vb (a+ , a− , γ ; z) ≡ − V (a+ , a− , γ ; z) − V (a+ , a− , γ ; −z)

− 2 cos(π[γ0 + γ1 + γ2 + γ3 + a− ]/a+ )

(3.6)

Then the four A$Os (2.3) amount to A(a+ , a− , γ ; v),

A(a− , a+ , γ ; v),

A(a+ , a− , γˆ ; v), ˆ

A(a− , a+ , γˆ ; v) ˆ

(3.7)

(with γˆ = J γ ), resp. It is clear from (3.4)–(3.6) that A(a+ , a− , γ ; z) is even, scale invariant, and invariant under permutations of γ0 , γ1 , γ2 and γ3 . We need a second representation for Vb Springer

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revealing D4 symmetry in γ (invariance under permutations and even sign changes): Vb (a+ , a− , γ ; z) =

2

'3

µ=0

cos(π γµ /a+ )

sinh(π[z − ia− /2]/a+ ) sinh(π[z + ia− /2]/a+ ) ' 2 3µ=0 sin(π γµ /a+ ) − cosh(π[z − ia− /2]/a+ ) cosh(π[z + ia− /2]/a+ )

(3.8)

(This can be readily checked from (3.5)–(3.6) by comparing periodicity, asymptotics and residues.) From this it is immediate that Vb vanishes identically if and only if γµ1 = ka+ , γµ2 = (l + 1/2)a+ , k, l ∈ Z

(3.9)

for some µ1 , µ2 ∈ {0, 1, 2, 3}. We now consider the choice γ2 = −a+ /2,

γ3 = 0

(3.10)

yielding Vb (a+ , a− , γ ; z) = 0. Then the square of (3.4) is not only of the form (3.2), but also of the form (3.4) with different parameters. Specically, we have the squaring identity A(a+ , a− , η; v)2 = A(a+ , 2a− , τ ; v) + 2

(3.11)

where we have introduced η ≡ (γ0 , γ1 , −a+ /2, 0), τ ≡ (γ0 − a− /2, γ1 − a− /2, γ0 + a− /2, γ1 + a− /2) (3.12) This key identity can be checked by a long, but routine calculation. (In Section 6 we rederive it in a more conceptual way.) Now it is easy to see from (3.5)–(3.6) that there is a second identity A(a− , a+ , η; v) = A(2a− , a+ , τ ; v)

(3.13)

Therefore, the rst two eigenvalue equations satised by R(a+ , 2a− , (c0 , c1 , c0 , c1 ); v, 2v) ˆ hold true for R(a+ , a− , (c0 , c1 , 0, 0); v, v), ˆ too. Turning to the last two (dual) eigenvalue equations, we rst point out that we have A(a+ , a− , η; ˆ v) ˆ = A(a+ , 2a− , τˆ ; 2v) ˆ

(3.14)

Indeed, just as for (3.13) this amounts to checking that the two coefcients given by (3.5) are equal. Now we also have

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A(a− , a+ , η; ˆ v) ˆ = A(2a− , a+ , τˆ ; 2v) ˆ 2−2

(3.15)

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(This can be checked directly, but also follows from (3.11) by suitable substitutions.) Hence the Aˆ ± -eigenvalue equations obeyed by R(a+ , a− , (c0 , c1 , 0, 0); v, v) ˆ are valid for R(a+ , 2a− , (c0 , c1 , c0 , c1 ); v, 2v), ˆ as well. We are not aware of any general argument entailing that these four shared eigenfunction properties imply that the two R-functions must be proportional. Using a lot more information, however, we will prove R(a+ , a− , (c0 , c1 , 0, 0); v, v) ˆ = R(a+ , 2a− , (c0 , c1 , c0 , c1 ); v, 2v) ˆ

(3.16)

in Section 5. Here we only point out that proportionality readily yields equality. Indeed, when we substitute vˆ = −i(c0 + c1 )/2 in the two functions, then we obtain 1 in both cases, cf. (2.12)–(2.14). Taking (3.16) for granted, we conclude this section by deriving some direct consequences. First, combining (3.16) with the modular invariance (2.10), we obtain (1.1); alternatively, we can obtain (1.1) from (3.16) by using the self-duality property (2.11) and a simple reparametrization. Second, when we take the nonrelativistic limit of (3.16), we obtain a tautology. Third, the specialization to the Askey-Wilson polynomials is obtained by choosing vˆ = −i(c0 + c1 )/2 − ina− ,

n∈N

(3.17)

and proceeding just as for (1.1), cf. (2.12)–(2.19). Specically, we nd Pn(φ) (e−a− , e−c0 , −e−c1 , e−a− /2 , −e−a− /2 ; cos v)

= Pn(φ) (e−2a− , e−c0 , −e−c1 , e−c0 −a− , −e−c1 −a− ; cos v)

(3.18)

Taking e−a− → q, e−c0 → a, e−c1 → b

(3.19)

this yields the identity (cf. (2.17)) 4 φ3 (q

−n

, abq n , aeiv , ae−iv , −ab, aq 1/2 , −aq 1/2 ; q, q)

= 4 φ3 (q −2n , a 2 b2 q 2n , aeiv , ae−iv , −ab, a 2 q, −a 2 q; q 2 , q 2 )

(3.20)

which we have not encountered in the literature. 4 The A1 subfamilies In this section we are concerned with four 1-parameter families that are subfamilies of the 2-parameter families featuring in the previous section. They are characterized by the couplings (b, 0, 0, 0), 2−1 (b, b, b, b), (b, 0, b, 0),

(b, b, 0, 0)

(4.1) Springer

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To appreciate their special properties, we consider successively the four A$Os (2.3) associated to each of the couplings (4.1). First, when we square the A$Os A± associated to the rst family, we may view the resulting A$Os as the A$Os A± + 2 of the second family with z → z/2. Setting η A ≡ (b, 0, 0, 0) − (a+ /2 + a− /2, a− /2, a+ /2, 0)

(4.2)

this may be expressed as A(a+ , a− , η A ; z)2 = A(a+ , a− , ηˆ A ; z/2) + 2

(4.3)

A(a− , a+ , η A ; z)2 = A(a− , a+ , ηˆ A ; z/2) + 2

(4.4)

where ηˆ A ≡ J η A , cf. (2.7) To relate this to (3.11)–(3.13), some care is needed. First, we write the parameters (a+ , 2a− , τ ) arising from (a+ , a− , η A ) as (a˜ − , a˜ + , η), ˜ η˜ ≡ (γ0 − a− /2, γ0 + a− /2, −a˜ + /2, 0)

(4.5)

Since this η˜ is of the form occurring in (3.12), we may invoke (3.13) and scale invariance to deduce (4.3). Now A(a+ , a− , γ ; z) is invariant under permutations of γ , and the interchange of a+ and a− yields a permutation of η A and ηˆ A . Therefore, (4.4) follows from (4.3). Turning to the third family, we can relate its A$O pair (A+ , A− ) to the pair (A2+ − 2, A− ) of the rst one, as follows from (2.1)–(2.2), or (by suitable specialization) from (3.11)–(3.13). But since the coupling vector is self-dual (invariant under J (2.7)), this is also true for the A$O pair ( Aˆ + , Aˆ − ) of the third family. Finally, for the fourth family we have the same state of affairs as for the third one, up to an interchange of + and − in the previous paragraph. The upshot is that for each of the four families two of the four associated AskeyWilson A$Os may be viewed as squares of Askey-Wilson A$Os. A priori, this does not imply that the pertinent R-functions are also eigenfunctions of the two square root A$Os, but actually this is the case. As will become clear shortly, this is a consequence of (3.16) (whose proof we still take for granted in this section). First, we introduce the A1 -type R-function by (1.7). Then we obtain from (3.16) R A1 (a+ , a− , b; v, v) ˆ = R(a+ , 2a− , (b, 0, b, 0); v, v) ˆ

(4.6)

Using modular invariance (2.10), this entails R A1 (a+ , a− , b; v, v) ˆ = R(2a− , a+ , (b, b, 0, 0); v, v) ˆ

(4.7)

Invoking once more (3.16) for the rhs of (4.7), and scaling the result by a factor 1/2, we get

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R A1 (a+ , a− , b; v, v) ˆ = R(a+ , a− , 2−1 (b, b, b, b); v/2, v) ˆ

(4.8)

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Now from (1.7) or (4.8) it follows that R A1 is modular invariant, R A1 (a+ , a− , b; v, v) ˆ = R A1 (a− , a+ , b; v, v) ˆ

(4.9)

whereas (4.6) or (4.7) entails self-duality, R A1 (a+ , a− , b; v, v) ˆ = R A1 (a+ , a− , b; v, ˆ v)

(4.10)

As a consequence, R A1 is a joint eigenfunction of the A$Os A+ (b; v),

A− (b; v),

A+ (b; v), ˆ

A− (b; v) ˆ

(4.11)

with eigenvalues 2 cosh(π v/a ˆ + ), 2 cosh(π v/a ˆ − ), 2 cosh(π v/a+ ), 2 cosh(π v/a− )

(4.12)

resp. Here, we have introduced the A1 A$Os Aδ (b; z) ≡ Aδ ((b, 0, 0, 0); z) =

sinh(π[z − ib]/aδ ) exp(−ia−δ d/dz) + (i → −i), sinh(π z/aδ )

δ = +, − (4.13)

From the relations just detailed, it can be read off that for each of the four families (4.1) the R-function satises two extra Askey-Wilson analytic difference equations of ‘square root’ type, as announced. It should be noted, however, that the relations encode far more information. To conclude this section, we specify the properties of the polynomials associated to the above A1 type R-functions and the corresponding 4 φ3 identities. To begin with, combining (2.14)–(2.17) with (4.8), we obtain R A1 (a+ , a− , b; v, −ib − ina− ) = Pn (a+ , a− , 2−1 (b, b, b, b); cosh(π v/a+ )) (4.14) When we set a+ = −2iπ, q = e−a− , β = e−b , v = 2θ

(4.15)

this yields the Askey-Wilson polynomials (cf. (2.16)–(2.17)) Pn(φ) (q, β 1/2 , −β 1/2 , β 1/2 q 1/2 , −β 1/2 q 1/2 ; cos θ)

= 4 φ3 (q −n , β 2 q n , β 1/2 eiθ , β 1/2 e−iθ , −β, βq 1/2 , −βq 1/2 ; q, q)

(4.16)

Comparing to Eq. (4.2) in [6], we deduce that these are basically the (continuous) q-ultraspherical polynomials. Springer

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From (4.14) and (4.7) we now deduce Pn (a+ , a− , 2−1 (b, b, b, b); cosh(π v/a+ )) = Pn (a+ , a− /2, 2−1 (b, b, 0, 0); cosh(π v/a+ ))

(4.17)

The associated 4 φ3 identity amounts to a special case of (3.20), cf. (3.18). Likewise, from (4.14) and (4.6) we obtain P2n (a+ , a− , 2−1 (b, b, b, b); cosh(π v/a+ )) = Pn (a+ , 2a− , (b, 0, b, 0); cosh(2π v/a+ )) (4.18) which gives rise to a special case of (2.19) (hence of Eq. (3.1) in [6])). The third relation P2n (a+ , a− , 2−1 (b, b, b, b); cosh(π v/a+ )) = Pn (a+ , a− , (b, 0, 0, 0); cosh(2π v/a+ )) (4.19) following from (4.14) and (1.7) yields the 4 φ3 identity 4 φ3 (q

−2n

, β 2 q 2n , β 1/2 eiθ , β 1/2 e−iθ , −β, βq 1/2 , −βq 1/2 ; q, q)

= 4 φ3 (q −n , βq n , βe2iθ , βe−2iθ , −β, βq 1/2 , −βq 1/2 ; q, q)

(4.20)

It can also be obtained by combining the above two special cases. 5 The proof of the doubling identity (3.16) We now turn to the proof of (3.16). Thus far, we have only shown that the R-functions on its lhs and rhs satisfy four identical Askey-Wilson type second order analytic difference equations (henceforth A$Es). Moreover, we have already pointed out (below (3.16)) that it sufces to prove proportionality. As mentioned in the introduction, we are only able to prove proportionality by bringing in various new ingredients. We proceed to collect these. To begin with, it is convenient to switch to a renormalized R-function dened by Rr (a+ , a− , γ ; v, v) ˆ ≡ ρ(a+ , a− , γ )R(a+ , a− , c(a+ , a− , γ ); v, v) ˆ

(5.1)

Here, c(a+ , a− , γ ) is given by (3.3) and ρ is dened by (

ρ(a+ , a− , γ ) ≡ 1

Springer

3 )

j=1

G(a+ , a− , ; i(γ0 + γ j + a+ /2 + a− /2))

(5.2)

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351

Next, we introduce a dense subset /el of the parameter set / ≡ {(a, b, p) ∈ R6 | a, b > 0}

(5.3)

To this end we rst dene a subset Z of Z4 × Z4 by requiring that for (M, N ) ∈ Z the four pairs (Mµ , Nµ ), µ ∈ {0, 1, 2, 3}, are distinct mod(2); equivalently, the pairs are of the form (even, even), (odd, odd), (even, odd), (odd, even). Then /el can be dened by + 3 + 1, + /el ≡ (a, b, p) ∈ /+ p = (Mν a + Nν b)eν , (M, N ) ∈ Z + 2 ν=0 *

(5.4)

where e0 , . . . , e3 are the canonical basis vectors of R4 . The subscript ‘el’ stands for ‘elementary’. Indeed, for (a+ , a− , γ ) ∈ /el the function Rr is the sum of two functions Rr(+) and Rr(−) that are elementary in the sense that they are of the form ˆ + ˆ − Rr(σ ) (a+ , a− , γ ; v, v) ˆ = ρ (σ ) (eπ v/a+ , eπ v/a− , eπ v/a , eπ v/a )

× exp(2πiσ v v/a ˆ + a− ),

σ = +, −

(5.5)

where ρ (σ ) is rational in each of its four arguments. The functions Rr(σ ) are related by Rr(+) (a+ , a− , γ ; v, v) ˆ = Rr(−) (a+ , a− , γ ; −v, v) ˆ = Rr(−) (a+ , a− , γ ; v, −v), ˆ (a+ , a− , γ ) ∈ /el

(5.6)

and they are a basis for the space of joint solutions to the Askey-Wilson A$Es A(a+ , a− , γ ; v)R = 2 cosh(2π v/a ˆ + )R,

A(a− , a+ , γ ; v)R = 2 cosh(2π v/a ˆ − )R (5.7)

provided a+ /a− is irrational and Re vˆ is sufciently large. The assertions in the previous paragraph are proved in [4] (cf. also Section 11 in [5]). To exploit these results, consider the renormalized R-function corresponding to the rhs of (3.16). It reads Rr (a+ , 2a− , τ ; v, 2v), ˆ τ = (γ0 − a− /2, γ1 − a− /2, γ0 + a− /2, γ1 + a− /2) (5.8) Now we choose a+ , a− > 0 with a+ /a− irrational and x γ0 = a+ /2 + ka+ + 2la− , γ1 = a− + ma+ + 2na− , k, l, m, n ∈ Z

(5.9)

Then the parameters (a+ , 2a− , τ ) belong to /el , and Rr (5.8) is the even linear combination of the basis functions Rr(±) for Re vˆ large enough. Since the function Rr (a+ , a− , η; v, v), ˆ η = (γ0 , γ1 , −a+ /2, 0)

(5.10) Springer

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corresponding to the lhs of (3.16) is a joint solution to the same eigenvalue A$Es (5.7), it is also a linear combination. Moreover, in view of (5.6) and evenness, the coefcients of Rr(+) and Rr(−) are equal. Hence it follows that the ratio of the two ˆ functions given by (5.8)–(5.10) is a function that can only depend on a+ , a− and v. Next, we note that with (5.9) in effect, the parameters (a+ , a− , η) belong to /el as well. Since /el is invariant under the action of J (2.7) (cf. [4])), we can repeat the reasoning for the basis functions Rr(±) associated with (5.10), with the v-A$Es ˆ for (5.10) now playing the role of the v-A$Es for (5.8). Thus it follows that the ratio of the Rr -functions (5.8) and (5.10) can only depend on a+ , a− and v. Combining this with our previous conclusion, we deduce that the ratio can only depend on a+ and a− . In other words, the two Rr -functions are proportional. As a consequence, the R-functions on the lhs and rhs of (3.16) are proportional as well and hence equal, provided a+ /a− is irrational and γ0 , γ1 are given by (5.9). Since numbers (γ0 , γ1 ) of the form (5.9) are dense in R2 for a+ /a− ∈ / Q, we deduce (3.16) for arbitrary γ0 , γ1 ∈ C from analyticity in γ . Likewise, positive a+ , a− with irrational quotient are dense in (0, ∞)2 , so (3.16) holds true without restrictions on the parameters. Hence the proof of (3.16) is complete. 6 A doubling identity for the E-function The R-function has a non-manifest D4 covariance in the couplings that is best understood in terms of the parameters γ and a function E(a+ , a− , γ ; v, v) ˆ we proceed to dene (cf. II). It involves the c-function c(a+ , a− , γ ; z) ≡

'3

µ=0

G(a+ , a− ; z − iγµ )

G(a+ , a− ; 2z + ia+ /2 + ia− /2)

(6.1)

and the normalizing factor 2 2 χ(a+ , a− , γ ) ≡ exp(2πi[γ · γ /4 − (a+ + a− + a+ a− )/8]/a+ a− )

(6.2)

Specically, we have E(a+ , a− , γ ; v, v) ˆ ≡

χ (a+ , a− , γ ) 1 Rr (a+ , a− , γ ; v, v) ˆ c(a+ , a− , γ ; v) c(a+ , a− , γˆ ; v) ˆ

(6.3)

This function is D4 invariant, in the sense that E(a+ , a− , γ ; v, v) ˆ = E(a+ , a− , w(γ ); v, v), ˆ ∀w ∈ W

(6.4)

where W is the D4 Weyl group (permutations and even sign changes). The D4 invariance can be understood from E being a joint eigenfunction of the four A$Os A(a+ , a− , γ ; v), A(a− , a+ , γ ; v), A(a+ , a− , γˆ ; v), ˆ A(a− , a+ , γˆ ; v) ˆ

Springer

(6.5)

Quadratic transformations for a function that generalizes 2 F1 and the Askey-Wilson polynomials

353

with eigenvalues (2.4), resp. Here we have A(a+ , a− , γ ; z) ≡ c(a+ , a− , γ ; z)−1 A(a+ , a− , γ ; z)c(a+ , a− , γ ; z)

= e−ia− d/dz + Va (a+ , a− , γ ; z)eia− d/dz + Vb (a+ , a− , γ ; z) (6.6)

with Vb given by (3.8) and Va by Va (a+ , a− , γ ; z) ≡ V (a+ , a− , γ ; −z)V (a+ , a− , γ ; z + ia− ) ' ' 16 3µ=0 δ=+,− cosh(π[z + iδγµ + ia− /2]/a+ ) = sinh(2π z/a+ ) sinh(2π[z + ia− /2]/a+ )2 sinh(2π[z + ia− ]/a+ )

(6.7)

Indeed, the D4 invariance of the A$Os (6.5) follows from the manifest D4 invariance of the coefcients Va and Vb . The formulas (6.6) and (6.7) can be understood from the A$Es G(z + iaδ /2) = 2 cosh(π z/a−δ ), δ = +, − G(z − iaδ /2)

(6.8)

satised by G(a+ , a− ; z). Indeed, from (6.8) with δ = − it follows that V (a+ , a− , γ ; z) = c(a+ , a− , γ ; z)/c(a+ , a− , γ ; z − ia− )

(6.9)

cf. (3.5). This relation between the coefcient V and the c-function is also useful to rederive the squaring identity (3.11), as will be done shortly. But rst we aim to obtain a doubling identity for the E-function. The pivotal tools for doing so are the duplication formula G(a+ , a− ; 2z) =

)

δ,δ & =+,−

G(a+ , a− ; z + i(δa+ + δ & a− )/4)

(6.10)

and the doubling identities G(a+ , a− ; z) = G(a+ , 2a− ; z + ia− /2)G(a+ , 2a− ; z − ia− /2) G(a+ , 2a− ; 2z) = G(a+ , a− ; z + ia+ /4)G(a+ , a− ; z − ia+ /4)

(6.11) (6.12)

(These formulas are easy consequences of the general multiplication formula obeyed by the hyperbolic gamma function, cf. [11] (3.25).) For brevity, we use the notation ˜ G(z) ≡ G(a+ , a− ; z), G(z) ≡ G(a+ , 2a− ; z)

(6.13)

for the remainder of this section. Springer

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We now employ (6.10)–(6.13) to show that the doubling identity (3.16) entails a doubling identity for the E-function. We begin by showing (recall (3.12)) Rr (a+ , a− , η; v, v) ˆ = Rr (a+ , 2a− , τ ; v, 2v) ˆ

(6.14)

In view of (3.16) and (5.1)–(5.2), we need only prove that the renormalizing factors are equal. Now we have ρ(a+ , a− , η) = 1/G(i(γ0 + γ1 + a+ /2 + a− /2))G(i(γ0 + a− /2)) × G(i(γ0 + a+ /2 + a− /2))

˜ = 1/G(i(γ0 + γ1 + a+ /2 + a− /2))G(i(2γ 0 + a+ /2 + a− )) (6.15) where we used (6.12) in the second step. Likewise, using (6.11) we infer that ˜ ˜ ρ(a+ , 2a− , τ ) = 1/G(i(γ 0 + γ1 + a+ /2))G(i(2γ0 + a+ /2 + a− )) ˜ × G(i(γ 0 + γ1 + a+ /2 + a− ))

(6.16)

is equal to (6.15). Hence we obtain (6.14). Next, we calculate G(z − iγ0 )G(z − iγ1 ) G(z + ia+ /2 + ia− /2)G(z + ia− /2) ' ˜ ˜ δ=+,− G(z − iγ0 + iδa− /2)G(z − iγ1 + iδa− /2) = ˜ + ia+ /2 + ia− )G(z ˜ + ia+ /2)G(z ˜ + ia− )G(z) ˜ G(z

c(a+ , a− , η; z) =

= c(a+ , 2a− , τ ; z)

(6.17)

where we used (6.10) in the rst and last step and (6.11) in the second one. Similarly, we obtain the dual equality c(a+ , a− , η; ˆ z) = c(a+ , 2a− , τˆ ; 2z)

(6.18)

from (6.10) and (6.12). (Alternatively, this follows from (6.17) by suitable substitutions.) Finally, it is straightforward to verify χ (a+ , a− , η) = χ (a+ , 2a− , τ )

(6.19)

cf. (6.2). On account of (6.3) and (6.14), we now obtain the doubling identity

Springer

E(a+ , a− , η; v, v) ˆ = E(a+ , 2a− , τ ; v, 2v) ˆ

(6.20)

Quadratic transformations for a function that generalizes 2 F1 and the Askey-Wilson polynomials

355

where η and τ are given by (3.12). From (6.17) and (3.11) we also deduce the squaring identity A(a+ , a− , η; v)2 = A(a+ , 2a− , τ ; v) + 2

(6.21)

A(a− , a+ , η; v) = A(2a− , a+ , τ ; v)

(6.22)

while (3.13) entails

(Indeed, the c-function is invariant under interchange of its rst and second argument, since the G-function is.) Likewise, we obtain A(a+ , a− , η; ˆ v) ˆ = A(a+ , 2a− , τˆ ; 2v) ˆ

(6.23)

A(a− , a+ , η; ˆ v) ˆ = A(2a− , a+ , τˆ ; 2v) ˆ 2−2

(6.24)

To conclude this section, we derive (3.11)–(3.13) in a more telling fashion. We note rst that the lhs of (3.11) is of the form (3.2) with α = −ia− and C(z) = V (a+ , a− , η; z) = c(a+ , a− , η; z)/c(a+ , a− , η; z − ia− )

(6.25)

cf. (6.9). Thus we have in (3.2) C(z)C(z + α) = c(a+ , a− , η; z)/c(a+ , a− , η; z − 2ia− )

= c(a+ , 2a− , τ ; z)/c(a+ , 2a− , τ ; z − 2ia− ) = V (a+ , 2a− , τ ; z)

(6.26)

where we used (6.17). Hence the equality of the shift coefcients in (3.11) follows. Next, we use the D4 invariance of Vb to write Vb (a+ , 2a− , τ ; z) = Vb (a+ , 2a− , w(τ ); z)

= −V (a+ , 2a− , w(τ ); z) − V (a+ , 2a− , w(τ ); −z) − 2

(6.27)

where w is the even sign change w(τ ) ≡ (γ0 − a− /2, γ1 − a− /2, −γ0 − a− /2, −γ1 − a− /2)

(6.28)

Thus we need only verify V (a+ , 2a− , w(τ ); z) = −V (a+ , a− , η; z)V (a+ , a− , η; −z + ia− )

(6.29)

This identity easily follows from (3.5), and so the proof of (3.11) is complete. Springer

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Finally, (3.13) follows from V (a− , a+ , η; v) =

c(a− , a+ , η; v) c(2a− , a+ , τ ; v) = c(a− , a+ , η; v − ia+ ) c(2a− , a+ , τ ; v − ia+ )

= V (2a− , a+ , τ ; v)

(6.30)

where we used (6.9), (6.17), and symmetry of c(a+ , a− , γ ; z) in its rst two arguments. 7 The E-function and the Askey-Wilson polynomials In Section 2 we recalled how the Askey-Wilson polynomials arise via a suitable discretization of the R-function. In this section we elaborate on this theme, focusing however on the function E(a+ , a− , γ ; v, v). ˆ We already had occasion to exploit the relation of the latter to one specic set of Askey-Wilson polynomials in Section 4 of [3]. Here we briey discuss the general state of affairs. To begin with, we should detail the singular locus of the E-function. Indeed, in order to connect it to the Askey-Wilson polynomials, we need to discretize v or v. ˆ Whenever these discretizations intersect the singular locus, the generic formulas may lose their validity. (This does happen in special cases, as transpires already from [3] Section 4.) The singular locus of the function Rr (a+ , a− , γ ; v, v) ˆ is given by a union of hyperplanes in C8 , cf. I (2.34)–(2.35). Since E is connected to Rr by (6.3), its singular locus can be readily determined. It is the union of the singular loci of 1/c(a+ , a− , γ ; v) and 1/c(a+ , a− , γˆ ; v), ˆ which are given by v = −i[la+ + ma− ]/2, v = i[(l − 1/2)a+ + (m − 1/2)a− + γµ ],

l, m ∈ N∗ (7.1)

vˆ = −i[la+ + ma− ]/2, vˆ = i[(l − 1/2)a+ + (m − 1/2)a− + γˆµ ], l, m ∈ N∗ (7.2) resp., and the hyperplanes v = i[(l − 1/2)a+ + (m − 1/2)a− − γµ ],

vˆ = i[(l − 1/2)a+ + (m − 1/2)a− − γˆµ ], l, m ∈ N∗

(7.3)

Thus E is holomorphic in its eight arguments away from (7.1)–(7.3), it being understood that a+ /a− stays away from (−∞, 0], cf. I. We proceed to show that for xed generic (a+ , a− , γ ) in the set / (5.3), the function E(a+ , a− , γ ; v, v) ˆ admits specializations to 32 distinct sets of functions that are essentially monic Askey-Wilson polynomials. The sets are associated to the 32 discretizations vˆn = −i(σ γˆν + a+ /2 + a− /2 + naδ ), σ, δ = +, −, ν = 0, 1, 2, 3 vn = −i(σ γν + a+ /2 + a− /2 + naδ ), σ, δ = +, −, ν = 0, 1, 2, 3

Springer

(7.4) (7.5)

Quadratic transformations for a function that generalizes 2 F1 and the Askey-Wilson polynomials

357

where n ∈ N. For generic parameters, these vˆ and v-values are not in the singular locus (7.1)–(7.3), even when we allow n ∈ Z. It sufces for example to require that a+ , a− , γ0 , γ1 , γ2 , γ3 be rationally independent. To ease the exposition, we do so from now on. We also restrict attention to (7.4) with δ = −, since the δ = + case amounts to interchanging + and −, and (7.5) can be treated by using self-duality. To start with, we consider E(a+ , a− , γ ; v, vˆ0 ). Since vˆ0 is not in the singular locus of E for each of the 8 choices of (σ, ν), this is a well-dened meromorphic function of v. Taking rst σ = +, ν = 0, we see from I (2.35) that vˆ0 is not in the singular locus of Rr (a+ , a− , γ ; v, v) ˆ either. Recalling (2.12)–(2.14), we obtain the explicit formula E(γ ; v, −i((J γ )0 + a+ /2 + a− /2)) = ρ(γ ) ·

χ(γ ) 1 · c(γ ; v) c(J γ ; −i((J γ )0 + a+ /2 + a− /2))

(7.6)

Due to our rational independence assumption, we have ρ(γ ) ∈ R∗ , so it follows that (7.6) is not identically zero. Let us now consider one of the remaining 7 choices for (σ, ν). We can nd w in the D4 Weyl group such that σ γˆν = (J w(γ ))0

(7.7)

Moreover, vˆ0 is not in the singular locus of Rr (a+ , a− , w(γ ); v, v), ˆ so we obtain as before (7.6) with γ → w(γ ). From the D4 invariance of the E-function (cf. (6.4)), we now get E(γ ; v, −i(σ γˆν + a+ /2 + a− /2)) =

ρ(w(γ ))χ (w(γ )) c(w(γ ); v)c(J w(γ ); −i(σ γˆν + a+ /2 + a− /2))

(7.8)

for all w obeying (7.7). This implies in particular that for each of the 8 choices of vˆ0 , the function E(γ ; v, vˆ0 ) does not vanish identically. As a consequence, the functions Q n (v) ≡ E(γ ; v, vˆn )/E(γ ; v, vˆ0 ), n ∈ Z

(7.9)

are well-dened meromorphic functions of v. On account of the third eigenvalue A$E satised by E (cf. (6.5)–(6.6) and (2.4)), they are related by Q n+1 (v) + Va (a+ , a− , γˆ ; vˆn )Q n−1 (v) + Vb (a+ , a− , γˆ ; vˆn )Q n (v) = (e2π v/a+ + e−2π v/a+ )Q n (v)

(7.10)

Now we have Va (a+ , a− , γˆ ; vˆ0 ) = 0 for all of the 8 choices, cf. (6.7). Since also Q 0 (v) = 1 by (7.9), it follows from (7.10) that we have Q n (v) = Pn (cosh(2π v/a+ )), n ∈ N

(7.11) Springer

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where Pn may be viewed as a monic Laurent polynomial of degree n in the variable z = exp(2π v/a+ ). Inspecting the three-term recurrence (7.10), we see that we obtain 8 distinct sets of polynomials, as announced. We proceed to relate them to the Askey-Wilson polynomials (2.17), renormalized so that they are monic in eiv . To this end we take a+ → −2iπ (keeping the real numbers a− , γ0 , . . . , γ3 xed) in the polynomials (7.11). Denoting the monic Askey-Wilson polynomials by Pn(m) (q, η0 , η1 , η2 , η3 ; cos v), we now obtain Pn (cos v) = Pn(m) (q, η0 , η1 , η2 , η3 ; cos v), q = e−a−

(7.12)

for 8 distinct vectors η ∈ (−∞, 0)4 . Specically, for the case σ = +, ν = 0 we have ηµ = −q 1/2 e−γµ , µ = 0, 1, 2, 3

(7.13)

whereas the other 7 choices of (σ, ν) correspond to an even number of η-involutions ηµ → q/ηµ = −q 1/2 eγµ

(7.14)

For the choice σ = +, ν = 0, this assertion follows from (2.16) and (3.3), noting that in the present context we view γ (and not c) as a xed vector in R4 . The remaining choices give rise to a D4 transformation on γ , and since the monic Askey-Wilson polynomials are symmetric in η0 , . . . , η3 , we need only encode the even sign ips via (7.14). The precise correspondence can be established by comparing the monic recurrence coefcient Mn(σ,ν)

=

16

'3

µ=0

'

δ=+,−

sinh([σ γˆν − δ γˆµ + na− ]/2)

sinh(σ γˆν + [n − 1/2]a− ) sinh(σ γˆν + na− )2 sinh(σ γˆν + [n + 1/2]a− )

(7.15)

obtained from Va (a+ , a− , γˆ ; vˆn ) (cf. (7.10) and (6.7)) to the pertinent Askey-Wilson coefcient Mn =

(1 − q n )(1 − q n−2 pη ) (1 − q 2n−3 pη )(1 −

'

n−1 ηµ ην ) 0≤µ