UvA-DARE (Digital Academic Repository) Some remarks on very-well-poised 8ϕ7 series

. Nonpolynomial basic hypergeometric eigenfunctions of the Askey–Wilson second order difference operator are known to be expressible as very-well-poised 8 φ 7 series. In this paper we use this fact to derive various basic hypergeometric and theta function identities. We relate most of them to identities from the existing literature on basic hypergeometric series. This leads for example to a new derivation of a known quadratic transformation formula for very-well-poised 8 φ 7 series. We also provide a link to Chalykh’s theory on (rank one, BC type) Baker–Akhiezer functions.


Introduction
In this paper we present derivations of various basic hypergeometric and theta function identities using the interpretation of very-well-poised 8 φ 7 series as eigenfunctions of the Askey-Wilson second order difference operator D. For instance, we reobtain a nonstandard type three term transformation formula for very-well-poised 8 φ 7 series [7] as the connection formula for the asymptotically free eigenfunctions of D, we investigate the eigenfunctions of D with trivial quantum monodromy and relate them to (rank one, BC type) Baker-Akhiezer functions from [4,5], we rederive the quadratic transformation formula [6, (3.5.10)] for very-well-poised 8 φ 7 series, and we obtain various theta function identities by translating symmetries of the Askey-Wilson function and of the asymptotically free eigenfunction of D in terms of the associated normalized c-functions.
The most general family of orthogonal polynomials satisfying a second order q-difference equation is the family of Askey-Wilson polynomials [1]. For our purposes it is convenient to view the associated difference equations as eigenvalue equations for the second order Askey-Wilson second order difference operator D already mentioned in the previous paragraph. The operator D depends, besides on the difference step-size and the deformation parameter q, on four additional free parameters.
Nonpolynomial basic hypergeometric eigenfunctions of D have been subject of study in various papers (see, e.g., [8,9,10,19,20,24,26]). Important examples are the Askey-Wilson function E(·, z) and the asymptotically free eigenfunction Φ(·, z) (the corresponding eigenvalue depends in an explicit way on q z + q −z ∈ C). They are defined provided that 0 < |q| < 1. They are selfdual eigenfunctions (the role of the argument and z is interchangeable). They naturally arise in harmonic analysis on the quantum SU(1, 1) group and in the study of the double affine Hecke algebra of type C ∨ C 1 (the rank one Koornwinder case), see, e.g., [11] and [24,25] respectively. From this representation theoretic viewpoint E(·, z) plays the role of the spherical function and Φ(·, z) the role of the Harish-Chandra series.
Ruijsenaars' R-function [19] is another nonpolynomial selfdual eigenfunction of D which is required to satisfy yet another second order difference equation of Askey-Wilson type. The step-direction of the two Askey-Wilson second order difference equations are allowed to be colinear (which corresponds to deformation parameter q being of modulus one). The R-function arises as matrix coefficient of representations of the quantum double of the quantized universal enveloping algebra of sl 2 , see [2].
The Askey-Wilson function E(·, z) can be explicitly expressed in terms of the asymptotically free eigenfunctions Φ(·, ±z). The elliptic function c(·, z) governing the expansion coefficients, E(·, z) = c(·, z)Φ(·, z) + c(·, −z)Φ(·, −z), is called the (normalized) c-function [10]. It is explicitly given as quotient of theta functions. The selfduality of E and Φ and the fact that E(−x, z) = E(x, z) then allow us to express Φ(−x, z) in terms of Φ(x, ±z) (connection formula). The cases when the connection coefficient formula trivializes is particularly interesting since it directly relates to the theory of Baker-Akhiezer functions [4,5]. We discuss this in Section 3.
Suitable two parameter specializations of the Askey-Wilson polynomials yield the continuous q-Jacobi polynomials. They have appeared in two guises (see [17] and [1]) which are interrelated by a quadratic transformation formula going back to Singh [22], see also [1, § 4] and [6, § 3.10]. This quadratic transformation formula was derived in [1] using the orthogonality relations of the Askey-Wilson polynomials. Ruijsenaars [20] stressed that for these parameter specializations the Askey-Wilson second order difference operator D factorizes up to an additive constant as a square of an Askey-Wilson type second order difference operator with step-size half of the stepsize of D. We use this observation to prove a quadratic transformation formula for a two parameter family of the asymptotically free eigenfunction Φ(·, z) of D. This complements Ruijsenaars' results [20], where he lifted the quadratic transformation formula for continuous q-Jacobi polynomials to a quadratic transformation formula for a two parameter subfamily of the R-function.
Using the known explicit expression of Φ(·, z) in terms of basic hypergeometric series we link the above mentioned results to various known identities for very-well-poised 8 φ 7 series. For example, the quadratic transformation formula for Φ(·, z) becomes the known quadratic transformation formula [6, (3.5.10)] for very-well-poised 8 φ 7 series after applying suitable transformation formulas to both sides of the identity, see Remark 5.3(i) (this was observed by M. Rahman).
Combining symmetries of the Askey-Wilson function with its c-function expansion yields nontrivial identities for the c-function, hence nontrivial theta function identities. For instance, the fact that E(·, z) is invariant under negating the argument yields a theta function identity which is a six parameter dependent subcase of a theta function identity [6,Exercise 5.22] due to Slater [23]. The fact that Φ(·, ±z) satisfies a quadratic transformation formula but the Askey-Wilson function E(·, z) does not, yields a nontrivial identity for the c-function and consequently a quadratic type theta function identity (7.4).
The results in the present paper play an important role in the study of the spectral problem of the trigonometric Macdonald-Ruijsenaars-Cherednik commuting family of difference operators associated to root systems. They are for instance needed in the asymptotic analysis of q-analogues of Harish-Chandra series associated to root systems (cf. [13,14,15,25]) along codimension one facets of the Weyl chamber. I will return to this topic in a future work.
2 Eigenfunctions of the Askey-Wilson second order dif ference operator We use, besides a deformation parameter 0 < q = e 2π √ −1τ < 1 (τ ∈ √ −1R >0 ) and a choice of step-size s ∈ Q >0 , four free parameters {κ, λ, υ, ς} which we call Hecke parameters (this name comes from their interpretation as multiplicity parameters in the Cherednik-Macdonald theory of the double affine Hecke algebra of type C ∨ C 1 , see [16]). We assume that the Hecke parameters κ, λ, υ and ς are real. We set q x := e 2π √ −1τ x for x ∈ C. In the paper q will be fixed throughout.
Step-sizes s and s 2 will simultaneously appear in formulas. One can restrict without loss of generality to considering step-sizes 2 and 1, but formulas are more transparent when arbitrary values s of the step size are taken into account because it makes the s-dependence of the Askey-Wilson parameters i.e. the roles of the Hecke parameters λ and υ are interchanged. The parameters a s , b s , a s and b s do not depend on s, we therefore occasionally omit the subindex s for these Askey-Wilson parameters. Interchanging λ and υ defines an involution on the set of Hecke parameters, hence also on the associated set of Askey-Wilson parameters. In addition we have We now first recall the asymptotically free eigenfunction of the Askey-Wilson [1] second order q s -difference operator, which we will regard here as a second order difference operator with step size s. Explicitly, the Askey-Wilson second order difference operator D, acting on meromorphic functions on C, is defined by Sometimes it is important to write explicitly the dependence on the parameters, in which case we write D as D (s) κ,λ,υ,ς;q .
Remark 2.1. The Askey-Wilson second order difference operator D can be interpreted as a second order q s -difference operator when acting on τ −1 -translation invariant meromorphic functions on C (which are the meromorphic functions of the form g(q x ) with g meromorphic on C * ).
There exist unique holomorphic functions Γ r on C (r ≥ 0) satisfying the following three conditions, (2) The power series Ψ(x, z) := ∞ r=0 Γ r (z)q rx is normally convergent on compacta of (x, z) ∈ where D x stands for the Askey-Wilson second order difference operator D = D (s) κ,λ,υ,ς;q acting on the x-variable.
Furthermore, Γ r is τ −1 -translation invariant and It is a well defined meromorphic function in (x, z) ∈ C × C provided that |d s q x | < 1, with possible poles at q s+sr+x+2z = d s (r ∈ Z ≥0 ). It can be expressed as sum of two 4 φ 3 series using [ leading to the expression This alternative expression provides the meromorphic continuation and shows that Ψ(x, z) is holomorphic in (x, z) ∈ C × C. The expression (2.4) of Ψ(x, z) shows that Ψ(x, z) satisfies (1) and (2). By [9] (see also [10] for notations close to the present one), the resulting meromorphic function Φ(x, z) (see (2.1)) indeed satisfies the difference equation (2.2). It remains to prove uniqueness. If a series of the form is a formal solution of (2.2) then the Γ r (z) (r ≥ 0) satisfy recursion relations of the form for some Laurent polynomials v r t . This shows that the Γ r (z) (r ≥ 1) are uniquely determined by Γ 0 (z).
If confusion may arise about the parameter dependencies then we write the asymptotically free solution Φ(x, z) as Φ(x, z; κ, λ, υ, ς; q, s) and Ψ(x, z) as Ψ(x, z; κ, λ, υ, ς; q, s). Γ r (z)q rx as eigenfunction of the second order difference operator obtained from D by gauging it with gauge factor W (·, z)/St(·). The gauged difference operator and the relevant eigenvalue are symmetric under arbitrary permutations of the Askey-Wilson parameters a s , b s , c s , d s . Since the normalization Γ 0 (z) = q s+2z ; q s ∞ of Ψ(x, z) is independent of the Askey-Wilson parameters, it follows that Ψ(x, z) is symmetric under arbitrary permutations of the Askey-Wilson parameters a s , b s , c s , d s .
Let θ(u; q) := u, q/u; q ∞ be the modified Jacobi theta function and write for products of theta functions. Define the (normalized) c-function c(x, z) = c(x, z; κ, λ, υ, ς; q, s) by for |q s−z / d s | < 1 (see [10]). Using [6, (III.36)] to express E(x, z) as sum of two 4 φ 3 series it follows that E(·, ·) has a meromorphic extension to C × C. We write E d and Φ d for the meromorphic functions E and Φ with respect to dual parameters. The following properties of E and Φ are known from [10,25] (cf. also [8,9,26]).
Proof . Remark 2.5. The selfduality of Φ(x, z) and E(x, z) ensures that Φ(x, ·) and E(x, ·) are eigenfunctions of the Askey-Wilson second order difference operator with respect to dual parameters.
Note that the Askey-Wilson second order difference operator D is invariant under x → −x, hence Φ(−x, z) is again an eigenfunction of D x with eigenvalue q z + q −z + a + a −1 . It can be expressed in terms of the eigenfunctions Φ(x, z) and Φ(x, −z) as follows.
Proof . By the previous proposition we have Compared with the c-function expansion for E(x, z) (see Proposition 2.4(iv)) we get the desired result.
Applying (2.8) twice to the right hand side of this formula implies with the function α(x, z) given by Replace in (2.10) the variable x by x + ms and consider the asymptotic behaviour as m → ∞ of both sides, using the fact that c(x, z) is s-translation invariant in both x and z and using that as m → ∞. It gives α(x, z) = 0. This is equivalent to (2.9).
Remark 2.9. Substituting in (2.9) the explicit expression (2.6) of the c-function c(x, z) gives the theta function identity where we have taken s = 1 and have written a = a 1 , . . . , d = d 1 (and similarly for the dual parameters). This is a special case of Slater's theta function identity [

The case of trivial quantum monodromy
Determining the monodromy representation for Gauss' second order hypergeometric differential equation is equivalent to deriving its connection coefficient formulas. The connection coefficient formulas explicitly relate the fundamental series expansion solutions around the three regular singularities of the hypergeometric differential equation. These formulas turn out to be directly related to well known three term transformation formulas for the Gauss' hypergeometric function 2 F 1 . From the above notion of monodromy only its incarnation in terms of connection coefficient formulas generalizes to the difference setup (see Sauloy [21] and references therein for a detailed discussion of this issue). We therefore say that the explicit connection coefficient formula (2.8) solves the quantum monodromy problem of D. In addition we say that the quantum monodromy is trivial if the coefficient of Φ(x, z) in (2.8) vanishes. The latter terminology is motivated as follows. Firstly, for trivial quantum monodromy the connection coefficient formula (2.8) reduces to a simple equivariance property of the following renormalization Φ(x, z) = Φ(x, z; κ, λ, υ, ς; q, s) of the asymptotically free eigenfunction Φ(x, z), see Proposition 3.1(iii) below. Secondly, for an important subclass with trivial quantum monodromy, a suitable renormalization of the asymptotically free eigenfunction Φ(x, z) has a terminating series expansion, see Remark 3.2(i) for further discussions on this issue. Note that the renormalization Φ(x, z) of Φ(x, z) still satisfies the eigenvalue equation x; κ, υ, λ, ς; q, s).
(i) If κ s , λ s , υ s , ς s ∈ 1 2 Z with an even number of them being integers, then c(x, z) = c d (z, x). In the remaining items of the proposition we assume that (κ, λ, υ, ς) is a four-tuple of real parameters such that c(x, z) = c d (z, x).
Proof . (i) Consider the quotient α := c(x, z)/c d (z, x) (we suppress the dependence on the variables and parameters). Using the explicit expression of the normalized c-function (2.6) in terms of theta functions it follows that α = 1 if κ s , λ s , υ s , ς s ) is taken from the set 0, 0, 0, 0 , In addition, it is easy to check that α is invariant under integral shifts of the rescaled Hecke parameters κ s , λ s , υ s , ς s . This gives the result. (ii) This follows from Proposition 2.4(iii). (iii) This follows from the connection formula (2.8) for Φ(x, z). (iv) This is immediate from the c-function expansion of the Askey-Wilson function, see Proposition 2.4(iv).

Remark 3.2.
(i) Due to [13, § 4] the function Φ(x, z) for Hecke parameters (κ, λ, υ, ς) satisfying, besides the integrality conditions from Proposition 3.1(i), suitable additional (positivity) conditions, is up to normalization Chalykh's BC 1 type normalized Baker-Akhiezer function [4, § 6]. An important property of the Baker-Akhiezer function is the fact that it has a terminating series expansion. Surprisingly this is not evident from the explicit expressions of Φ(x, z) in terms of basic hypergeometric series. The properties from Proposition 3.1 can be easily matched with the properties (see [4,5]) of the Baker-Akhiezer functions. For instance, the Askey-Wilson function E(x, z) relates to the symmetrized normalized Baker-Akhiezer function Φ + from [5, (3.20)]. In particular, formula [5, Theorem 3.9] relating the symmetrized Baker-Akhiezer function to Askey-Wilson polynomials matches with the known fact that the Askey-Wilson function E(x, z) reduces to the normalized Askey-Wilson polynomial for suitable discrete values of z, see (6.2) below.

The factorization of the Askey-Wilson second order dif ference operator
Ruijsenaars [20] analyzed when the square D + a + a −1 2 is an Askey-Wilson second order difference operator again (with doubled step-size). An important special case turns out to be when the Hecke parameters are of the form (κ, λ, 0, 0). In our notations the resulting formula [20, (3.11)] reads as Remark 4.1. Our operator D + a + a −1 = D + q 2κ + q −2κ for arbitrary Hecke parameters (κ, λ, υ, ς) is essentially the second order difference operator A δ (c; ·) in Ruijsenaars' paper [20, (2.1)] with the parameters (c 0 , c 1 , c 2 , c 3 ) in [20] related to our Hecke parameters by κ A simple derivation of (4.1) is as follows. Consider the second order difference operator L = L (s/2) κ,λ;q defined by Formula (4.1) thus is equivalent to which is an easy check (use that the Askey-Wilson parameters associated to Hecke parameters (κ, λ, κ, λ) satisfy c s = q . The eigenfunctions of D x of the form p n (q x + q −x ) (n ∈ Z ≥0 ) with p n a polynomial of degree n, are the well known Askey-Wilson polynomials [1] (see also Section 6). In the special case that the associated four Hecke parameters are taken to be (κ, λ, κ, λ) (corresponding to the condition that the Askey-Wilson parameters satisfy c s = q s 2 a s and d s = q s 2 b s ) the Askey-Wilson polynomials are called the continuous q-Jacobi polynomials by Askey and Wilson [1, § 4]. In [1, (4.22)] a quadratic transformation formula for balanced 4 φ 3 series, going back to Singh [22], is used to relate the above continuous q-Jacobi polynomials to Rahman's definition [17] of the continuous q-Jacobi polynomials (see also Section 6).

Quadratic transformation formulas
For a function Ξ(x, z; κ, λ, υ, ς; q, s) we write This is the parameter specialization which reduces the Askey-Wilson polynomials to Rahman's version of the continuous q-Jacobi polynomials [17] (in base q s 2 ). Furthermore we set Ξ d R (x, z) := Ξ(x, z 2 ; κ, 0, λ, 0; q, s 2 ) for its dual version. Since we know from the previous section that the meromorphic function Φ R (x, z) satisfies In fact, Φ R (x, z) is the unique solution to (5.1) which is of the form with Γ R,r (z) holomorphic in z ∈ C, with Γ R,0 (z) = q s 2 +z ; q s 2 ∞ and with the power series Ψ R (x, z) := ∞ r=0 Γ R,r (z)q rx converging normally on compacta of (x, z) ∈ C × C.
Proof . First of all note that the Askey-Wilson parameters associated to the Hecke parameters (κ, λ, 0, 0), deformation parameter q and step-size s 2 are given by The expression of Ψ(x, z) as a sum of two 4 φ 3 's, see (2.5), shows that We write its power series expansion in q x as The H r (z) are holomorphic in z ∈ C, the series converges normally on compacta of (x, z) ∈ C×C, and H 0 (z) = q for r ∈ Z ≥0 . Then H 0 (z) = q s+2z ; q s ∞ and with the series converging normally on compacta of (x, z) ∈ C × C. A direct computation now shows that By (5.1) and (4.2) we furthermore have Hence Φ R (x, z) satisfies the characterizing properties of Φ J (x, z).
By the proof of the theorem we in particular have Substituting the explicit expressions of Ψ R and Ψ J in this formula gives the following quadratic transformation formula for very-well-poised 8 φ 7 series.
if both |q 1 2 x| < 1 and | qαx β | < 1. (ii) The formal classical limit q ↑ 1 of (5.2) can be computed after replacing the parameters (x, z, α, β) in (5.2) by (−x, q z , q α , q β ) and moving all the infinite q-shifted factorials in (5.2) to one side. The resulting classical limit turns out to be trivial, since both sides reduce to where 2 F 1 is Gauss' hypergeometric series (in contrast to the classical limits of the quadratic transformations of very-well-poised 8 φ 7 's from [18, § 5], which reduce to nontrivial quadratic transformations for 2 F 1 ). On the polynomial level (5.2) reduces to the quadratic transformation formula for the continuous q-Jacobi polynomials, see Section 6, which is known to reduce to nontrivial quadratic transformations (see [1, (4.24)]) on the classical level when taking the limit q ↑ 1 after replacing the parameters (x, z, α, β) by (−q x , q z , q α , q β ).
There is a dual version of (5.2), which can be obtained by using the selfduality of Φ(x, z) to both sides of the quadratic transformation formula Φ J (x, z) = Φ R (x, z) before substituting the explicit expression as a 8 W 7 series. It leads to the following quadratic transformation formula.

Polynomial reduction
Both the asymptotically free eigenfunctions Φ(x, z) of the Askey-Wilson second order difference operator, and the Askey-Wilson function E(x, z), reduce to the Askey-Wilson polynomials when z is specialized appropriately (see, e.g., [10,25]). Concretely, for Φ(x, z) we have for n ∈ Z ≥0 , Formula (6.1) can also be proved by observing that P n (x) is an eigenfunction of D x with eigenvalue a(q n + 1) + a −1 (q −n + 1) (see [1]) of the form hence up to a multiplicative constant it must be equal to Φ(x, −κ − υ − ns) (this argument generalizes to the setting of Macdonald-Koornwinder polynomials, see [13,25]). Also the Askey-Wilson function E(x, −κ − υ − ns) (n ∈ Z ≥0 ) is a multiple of the Askey-Wilson polynomial, E(x, −κ − υ − ns) = a s b s , a s c s ; q s ∞ q s asds ; q s ∞ P n (x). (6.2) This can again be proved directly using transformation formulas and the expression of E(x, z) as a 8 W 7 series, see [10]. It can also be proved using the c-function expansion of E(x, z) (see Proposition 2.4(iv)) and (6.1), since c(x; κ + υ + ns) = 0 for n ∈ Z ≥0 . Specializing now z = −2κ − ns (n ∈ Z ≥0 ) in the identity Φ J (x, z) = Φ R (x, z) and using (6.1) twice gives, after straightforward simplifications, the following result (we take s = 2 without loss of generality), 4 φ 3 q −2n , aq x , aq −x , q 2n a 2 b 2 ab, qa 2 , qab ; q 2 , q 2 = 4 φ 3 q −n , aq x , aq −x , −q n ab ab, q ; q, q for n ∈ Z + . This quadratic transformation formula was first proved by Singh [22]. It was reobtained by Askey and Wilson [1, (4.22)] and interpreted as quadratic transformation formula [1, (4.22)] for the continuous q-Jacobi polynomial (to get it in the same form one has to apply Sear's transformation formula [6, (III.15)]), see also [6, § 3.10] for further discussions. It was also obtained in [20, (3.20)] (−a 2 q in [20, (3.20)] should be −abq) by specialization of the quadratic transformation formula [20, (3.16)] for the R-function. A similar polynomial reduction can be done with the dual version (5.3) of the quadratic transformation formula, in which case it reduces to the quadratic transformation formula [1, (3.1)].

A theta function identity
By the c-function expansion of the Askey-Wilson function (see Proposition 2.4(iv)) and by Theorem 5.1 we have while on the other hand, The c-functions c J (x, z) and c R (x, z) are explicitly given by c J (x, z) = θ q 2κ−z , −q −z , q