Some Remarks on Very-Well-Poised ₈∅₇ Series

Nonpolynomial basic hypergeometric eigenfunctions of the Askey-Wilson second order difference operator are known to be expressible as very-well-poised ₈∅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...

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Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2012
Main Author:	Stokman, J.V.
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Language:	English
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Cite this:	Some Remarks on Very-Well-Poised ₈∅₇ Series / J.V. Stokman // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 26 назв. — англ.

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spelling	Stokman, J.V. 2019-02-18T12:38:03Z 2019-02-18T12:38:03Z 2012 Some Remarks on Very-Well-Poised ₈∅₇ Series / J.V. Stokman // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 26 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D15; 33D45 DOI: http://dx.doi.org/10.3842/SIGMA.2012.039 https://nasplib.isofts.kiev.ua/handle/123456789/148446 Nonpolynomial basic hypergeometric eigenfunctions of the Askey-Wilson second order difference operator are known to be expressible as very-well-poised ₈∅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∅₇ series. We also provide a link to Chalykh's theory on (rank one, BC type) Baker-Akhiezer functions. I thank Tom Koornwinder for drawing my attention to the quadratic transformation formula for continuous q-Jacobi polynomials. I thank Mizan Rahman for pointing out to me how the quadratic transformations (5.2) and (5.3) for very-well-poised ₈∅₇ series are related to the known quadratic transformation formula [6, (3.5.10)] (see Reamark 5.3(i)). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Some Remarks on Very-Well-Poised ₈∅₇ Series Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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title	Some Remarks on Very-Well-Poised ₈∅₇ Series
spellingShingle	Some Remarks on Very-Well-Poised ₈∅₇ Series Stokman, J.V.
title_short	Some Remarks on Very-Well-Poised ₈∅₇ Series
title_full	Some Remarks on Very-Well-Poised ₈∅₇ Series
title_fullStr	Some Remarks on Very-Well-Poised ₈∅₇ Series
title_full_unstemmed	Some Remarks on Very-Well-Poised ₈∅₇ Series
title_sort	some remarks on very-well-poised ₈∅₇ series
author	Stokman, J.V.
author_facet	Stokman, J.V.
publishDate	2012
language	English
container_title	Symmetry, Integrability and Geometry: Methods and Applications
publisher	Інститут математики НАН України
format	Article
description	Nonpolynomial basic hypergeometric eigenfunctions of the Askey-Wilson second order difference operator are known to be expressible as very-well-poised ₈∅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∅₇ series. We also provide a link to Chalykh's theory on (rank one, BC type) Baker-Akhiezer functions.
issn	1815-0659
url	https://nasplib.isofts.kiev.ua/handle/123456789/148446
citation_txt	Some Remarks on Very-Well-Poised ₈∅₇ Series / J.V. Stokman // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 26 назв. — англ.
work_keys_str_mv	AT stokmanjv someremarksonverywellpoised87series
first_indexed	2025-11-24T16:26:16Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 039, 17 pages Some Remarks on Very-Well-Poised 8φ7 Series Jasper V. STOKMAN Korteweg–de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands E-mail: j.v.stokman@uva.nl URL: http://staff.science.uva.nl/~jstokman/ Received April 05, 2012, in final form June 18, 2012; Published online June 27, 2012 http://dx.doi.org/10.3842/SIGMA.2012.039 Abstract. Nonpolynomial basic hypergeometric eigenfunctions of the Askey–Wilson second order difference operator are known to be expressible as very-well-poised 8φ7 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 8φ7 series. We also provide a link to Chalykh’s theory on (rank one, BC type) Baker–Akhiezer functions. Key words: very-well-poised basic hypergeometric series; Askey–Wilson functions; quadratic transformation formulas; theta functions 2010 Mathematics Subject Classification: 33D15; 33D45 1 Introduction In this paper we present derivations of various basic hypergeometric and theta function iden- tities using the interpretation of very-well-poised 8φ7 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 8φ7 series [7] as the connection formula for the asymptotically free eigenfunctions of D, we investigate the eigenfunctions of D with trivial quan- tum 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 8φ7 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 va- rious 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 qz + 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∨C1 (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. mailto:j.v.stokman@uva.nl http://staff.science.uva.nl/~jstokman/ http://dx.doi.org/10.3842/SIGMA.2012.039 2 J.V. Stokman 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 co- linear (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 sl2, 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 step- size of D. We use this observation to prove a quadratic transformation formula for a two param- eter 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 poly- nomials 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 8φ7 series. For example, the quadratic transformation formula for Φ(·, z) becomes the known quadratic transfor- mation formula [6, (3.5.10)] for very-well-poised 8φ7 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 ope- rators 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 difference operator We use, besides a deformation parameter 0 < q = e2π √ −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 Some Remarks on Very-Well-Poised 8φ7 Series 3 name comes from their interpretation as multiplicity parameters in the Cherednik–Macdonald theory of the double affine Hecke algebra of type C∨C1, see [16]). We assume that the Hecke parameters κ, λ, υ and ς are real. We set qx := e2π √ −1τx 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 {as, bs, cs, ds} := { qκ+λ,−qκ−λ, q s 2 +υ+ς ,−q s 2 +υ−ς} explicit. We define dual Askey–Wilson parameters by{ ãs, b̃s, c̃s, d̃s } := { qκ+υ,−qκ−υ, q s 2 +λ+ς ,−q s 2 +λ−ς}, i.e. the roles of the Hecke parameters λ and υ are interchanged. The parameters as, bs, ãs and b̃s 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 ã2s = q−sabcd, ãsb̃s = asbs, qsãs b̃s = csds, ãsc̃s = ascs, qsãs c̃s = bsds, ãsd̃s = asds, qsãs d̃s = bscs. We now first recall the asymptotically free eigenfunction of the Askey–Wilson [1] second order qs-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( Df ) (x) := A(x) ( f(x+ s)− f(x) ) +A(−x) ( f(x− s)− f(x) ) , A(x) := (1− asqx)(1− bsqx)(1− csqx)(1− dsqx) ãs(1− q2x)(1− qs+2x) . 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 qs-difference operator when acting on τ−1-translation invariant meromorphic functions on C (which are the meromorphic functions of the form g(qx) with g meromorphic on C∗). We now define the elementary function W (x, z) = W (x, z;κ, λ, υ, ς; q, s) by W (x, z) := q(κ+λ+x)(κ+υ+z)/s. Note that W (x+ s, z) = qκ+υ+zW (x, z) = ãqzW (x, z). For generic bj the r+1φr basic hypergeometric series is defined by the convergent power series r+1φr ( a1, a2, . . . , ar+1 b1, b2, . . . , br ; q, z ) := ∞∑ j=0 ( a1, a2, . . . , ar+1; q ) j( q, b1, . . . , br; q ) j zj , \|z\| < 1, 4 J.V. Stokman where ( a1, . . . , as; q ) j = s∏ r=1 j−1∏ i=0 (1 − arqi) for j ∈ Z≥0 ∪ {∞} (empty products are equal to one by convention). The very-well-poised 8φ7 series is defined by 8W7 ( α0;α1, α2, α3, α4, α5; q, z ) = 8φ7 ( α0, qα 1 2 0 ,−qα 1 2 0 , α1, . . . , α5 α 1 2 0 ,−α − 1 2 0 , qα0/α1, . . . , qα0/α5 ; q, z ) = ∞∑ r=0 1− α0q 2r 1− α0 zr 5∏ j=0 ( αj ; q ) r( qα0/αj ; q ) r . In case z = α2 0q 2/α1α2α3α4α5 it has a meromorphic continuation to ( C∗ )6 as function of (α0, α1, . . . , α5). This follows from the identity [6, (III.36)] expressing such very-well-poised 8φ7 series as a sum of two 4φ3 series. Define the holomorphic function St(x) = St(x;κ, λ, υ, ς; q, s) (“St” is standing for singular term) in x ∈ C by St(x) := ( qs+x/as, q s+x/bs, q s+x/cs, q s+x/ds; q s ) ∞. We write Std(z) := St(z;κ, υ, λ, ς; q, s) for the singular term with respect to dual parameters, Std(z) = ( qs+z/ãs, q s+z/b̃s, q s+z/c̃s, q s+z/d̃s; q s ) ∞. The following proposition combines and refines observations from [9, 10, 25]. Proposition 2.2. There exist unique holomorphic functions Γr on C (r ≥ 0) satisfying the following three conditions, (1) Γ0(z) = ( qs+2z; qs ) ∞. (2) The power series Ψ(x, z) := ∞∑ r=0 Γr(z)q rx is normally convergent on compacta of (x, z) ∈ C× C (consequently Ψ(x, z) is a holomorphic function in (x, z) ∈ C× C). (3) The meromorphic function Φ(x, z) := W (x, z) St(x)Std(z) Ψ(x, z) (2.1) satisfies(( Dx − qz − q−z + ã+ ã−1 ) Φ ) (x, z) = 0, (2.2) where Dx stands for the Askey–Wilson second order difference operator D = D(s) κ,λ,υ,ς;q acting on the x-variable. Furthermore, Γr is τ−1-translation invariant and Ψ(x, z) = ( qs+x+zas ãs , q s+x+zbs ãs , q s+x+zcs ãs , q s+x+z ãs ds , qs+2z, dsq x; qs ) ∞( q2s+x+2z ds ; qs ) ∞ × 8W7 ( qs+x+2z ds ; qs+z ãs , qs+z d̃s , b̃sq z, c̃sq z, qs+x ds ; qs, dsq x ) (2.3) if \|dsqx\| < 1. Some Remarks on Very-Well-Poised 8φ7 Series 5 Proof. The explicit expression (2.3) is Ψ(x, z) = ( qs+x+zas ãs , q s+x+zbs ãs , q s+x+zcs ãs , q s+x+z ãs ds , qs+2z, dsq x; qs ) ∞( qs+x+2z ds ; qs ) ∞ × ∞∑ r=0 ( 1− qs+2sr+x+2z ds ) ( qs+z ãs , q s+z d̃s , b̃sq z, c̃sq z, q s+x ds ; qs ) r drsq rx( qs+x+z ãs ds , q s+x+zas ãs , q s+x+zbs ãs , q s+x+zcs ãs , qs+2z; qs ) r . (2.4) It is a well defined meromorphic function in (x, z) ∈ C × C provided that \|dsqx\| < 1, with possible poles at qs+sr+x+2z = ds (r ∈ Z≥0). It can be expressed as sum of two 4φ3 series using [6, (III.36)] with parameters (a, b, c, d, e, f, q) in [6, (III.36)] specialized to (qs+x+2z/ds, q s+z/ãs, q s+z/d̃s, b̃sq z, c̃sq z, qs+x/ds, q s), leading to the expression Ψ(x, z) = ( qs+x as , dsq x, q s+z b̃s , q s+z c̃s , q s+x+zas ãs , q s+x+z ãs ds ; qs ) ∞( ds as ; qs ) ∞ × 4φ3 ( asq x, q s+x ds , b̃sq z, c̃sq z qs+x+zas ãs , q s+x+z ãs ds , q sas ds ; qs, qs ) + ( asq x, q s+x ds , b̃sq z, c̃sq z, q s+x+z ãs as , q s+x+z d̃s as ; qs ) ∞( as ds ; qs ) ∞ × 4φ3 ( qs+x as , dsq x, q s+z b̃s , q s+z c̃s qs+x+z ãs as , q s+x+z d̃s as , q sds as ; qs, qs ) . (2.5) 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 Φ(x, z) = W (x, z) St(x)Std(z) ∞∑ r≥0 Γr(z)q rx is a formal solution of (2.2) then the Γr(z) (r ≥ 0) satisfy recursion relations of the form ã(qsr − 1)(qz − q−sr−z)Γr(z) = r−1∑ t=0 vrt (q z)Γt(z), r ≥ 1 for some Laurent polynomials vrt . This shows that the Γr(z) (r ≥ 1) are uniquely determined by Γ0(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). Remark 2.3. The characterization of Φ(x, z) as eigenfunction of D is equivalent to a characteri- zation of Ψ(x, z) = ∞∑ r=0 Γr(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 6 J.V. Stokman relevant eigenvalue are symmetric under arbitrary permutations of the Askey–Wilson parame- ters as, bs, cs, ds. Since the normalization Γ0(z) = ( qs+2z; qs ) ∞ 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 as, bs, cs, ds. Let θ(u; q) := ( u, q/u; q ) ∞ be the modified Jacobi theta function and write θ(u1, . . . , ur; q) = r∏ i=1 θ(ui; q) for products of theta functions. Define the (normalized) c-function c(x, z) = c(x, z;κ, λ, υ, ς; q, s) by c(x, z) := θ ( ãsq −z, b̃sq −z, c̃sq −z, dsq x−z ãs ; qs ) W (x, z)θ ( q−2z, dsqx; qs ) . (2.6) Using θ(qu; q) = −u−1θ(u; q) it follows that c(x + s, z) = c(x, z) and c(x, z + s) = c(x, z). We write cd(x, z) = c(x, z;κ, υ, λ, ς; q, s) for the c-function with respect to dual parameters. The Askey–Wilson function E(x, z) = E(x, z;κ, λ, υ, ς; q, s) is defined by E(x, z) := ( ãsqs+z−x ds , ãsq s+z+x ds , asbs, ascs, qsas ds ; qs ) ∞( qs+x ds , q s−x ds , q s+z d̃s , ãsb̃sc̃sqz; qs ) ∞ × 8W7 ( ãsb̃sc̃sq −s+z; asq x, asq −x, ãsq z, b̃sq z, c̃sq z; qs, qs−z d̃s ) (2.7) for \|qs−z/d̃s\| < 1 (see [10]). Using [6, (III.36)] to express E(x, z) as sum of two 4φ3 series it follows that E(·, ·) has a meromorphic extension to C × C. We write Ed and Φd 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]). Proposition 2.4. (i) E(x, z) = Ed(z, x) (selfduality). (ii) E(−x, z) = E(x, z) and E(x,−z) = E(x, z). (iii) Φ(x, z) = Φd(z, x) (selfduality). (iv) E(x, z) = c(x, z)Φ(x, z) + c(x,−z)Φ(x,−z) (c-function expansion). Proof. (i) This follows from the transformation formula [6, (III.23)] for very-well-poised 8φ7 series. (ii) By the explicit expression (2.7) it is clear that E(−x, z) = E(x, z). By (i) it then also follows that E(x,−z) = E(x, z). (iii) This follows again by application of the transformation formula [6, (III.23)] for very- well-poised 8φ7 series. (iv) Use Bailey’s three term transformation formula [6, (III.37)] for very-well-poised 8φ7 series. � Remark 2.5. The selfduality of Φ(x, z) and E(x, z) ensures that Φ(x, ·) and E(x, ·) are eigen- functions 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 7→ −x, hence Φ(−x, z) is again an eigenfunction of Dx with eigenvalue qz + q−z + ã + ã−1. It can be expressed in terms of the eigenfunctions Φ(x, z) and Φ(x,−z) as follows. Some Remarks on Very-Well-Poised 8φ7 Series 7 Corollary 2.6. Φ(−x, z) = ( c(x, z)− cd(z, x) cd(z,−x) ) Φ(x, z) + c(x,−z) cd(z,−x) Φ(x,−z) (2.8) (connection formula). Proof. By the previous proposition we have E(x, z) = Ed(z,−x) = cd(z,−x)Φd(z,−x) + cd(z, x)Φd(z, x) = cd(z, x)Φ(x, z) + cd(z,−x)Φ(−x, z). Compared with the c-function expansion for E(x, z) (see Proposition 2.4(iv)) we get the desired result. � Remark 2.7. The connection formula (2.8) is not a direct consequence of Bailey’s three term transformation formula [6, (III.37)] for very-well-poised 8φ7 series. This is reflected by the fact that the coefficient of Φ(x, z) in (2.8) does not admit an explicit expression as a single product of theta functions. The connection formula (2.8) is though directly related to the three term transformation formula [7, (5.8)]. Corollary 2.8. The c-function satisfies c(x, z)cd(z,−x)cd(−z,−x) + c(−x, z)cd(z, x)cd(−z,−x) = c(x, z)c(−x, z)cd(−z,−x) + c(x, z)c(−x,−z)cd(z,−x). (2.9) Proof. Since E(−x, z) = E(x, z) we have c(x, z)Φ(x, z) + c(x,−z)Φ(x,−z) = c(−x, z)Φ(−x, z) + c(−x,−z)Φ(−x,−z). Applying (2.8) twice to the right hand side of this formula implies α(x, z)Φ(x, z) = −α(x,−z)Φ(x,−z) (2.10) with the function α(x, z) given by α(x, z) = c(−x, z)c(x, z)− c(−x, z)cd(z, x) cd(z,−x) + c(−x,−z)c(x, z) cd(−z,−x) − c(x, z). 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 ã−mq−mzΦ(x+ms, z) = Γ0(z) ( 1 +O(qsm) ) 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 θ ( d ã qx+z, d ã qx−z, aqx, bqx, cqx, dq−x, q2z; q ) − θ ( d ã q−x+z, d ã qx+z, ãqz, b̃qz, c̃qz, d̃q−z, q2x; q ) = q2xθ ( d ã q−x−z, d ã q−x+z, aq−x, bq−x, cq−x, dqx, q2z; q ) − q2zθ ( d ã q−x−z, d ã qx−z, ãq−z, b̃q−z, c̃q−z, d̃qz, q2x; q ) , 8 J.V. Stokman where we have taken s = 1 and have written a = a1, . . . , d = d1 (and similarly for the dual parameters). This is a special case of Slater’s theta function identity [6, Exercise 5.22], with the parameters (a, b, c, d, e, f, g, h) in [6, Exercise 5.22] specialized to( 1, d ã q−x+z, ãq−z, b̃q−z, c̃q−z, q1−z d̃ , d ã qx+z, q2z ) . Note that Slater’s formula [6, Exercise 5.22] is more general since it has, besides q, seven free parameters, while the formula (2.9) only has six. 3 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 di- rectly related to well known three term transformation formulas for the Gauss’ hypergeomet- ric function 2F1. From the above notion of monodromy only its incarnation in terms of con- nection 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 renormaliza- tion Φ̃(x, z) = Φ̃(x, z;κ, λ, υ, ς; q, s) of the asymptotically free eigenfunction Φ(x, z), Φ̃(x, z) := c(x, z)Φ(x, z), see Proposition 3.1(iii) below. Secondly, for an important subclass with trivial quantum mon- odromy, a suitable renormalization of the asymptotically free eigenfunction Φ(x, z) has a termi- nating series expansion, see Remark 3.2(i) for further discussions on this issue. Note that the renormalization Φ̃(x, z) of Φ(x, z) still satisfies the eigenvalue equation D ( Φ̃(·, z) ) = ( qz + q−z − ã− ã−1 ) Φ̃(·, z) since c(x, z) is s-translation invariant in x. Write Φ̃d(z, x) := Φ̃(z, x;κ, υ, λ, ς; q, s). Proposition 3.1. (i) If κ s , λ s , υ s , ς s ∈ 1 2Z with an even number of them being integers, then c(x, z) = cd(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) = cd(z, x). (ii) Φ̃(x, z) = Φ̃d(z, x) (selfduality). (iii) Φ̃(−x, z) = Φ̃(x,−z). (iv) E(x, z) = Φ̃(x, z) + Φ̃(−x, z). Proof. (i) Consider the quotient α := c(x, z)/cd(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 ) , ( 1 2 , 1 2 , 0, 0 ) , ( 1 2 , 0, 1 2 , 0 ) , ( 1 2 , 0, 0, 1 2 ) ,( 0, 1 2 , 1 2 , 0 ) , ( 0, 1 2 , 0, 1 2 ) , ( 0, 0, 1 2 , 1 2 ) , ( 1 2 , 1 2 , 1 2 , 1 2 )} . Some Remarks on Very-Well-Poised 8φ7 Series 9 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 Φ̃(x, z) for Hecke parameters (κ, λ, υ, ς) satisfying, besides the integrality conditions from Proposition 3.1(i), suitable additional (positivity) condi- tions, is up to normalization Chalykh’s BC1 type normalized Baker–Akhiezer function [4, § 6]. An important property of the Baker–Akhiezer function is the fact that it has a ter- minating series expansion. Surprisingly this is not evident from the explicit expressions of Φ̃(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] re- lating 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. (ii) The relation between the A1 type Baker–Akhiezer function (see [4, § 4.1]) and Heine’s basic hypergeometric series 2φ1 was stressed in an informal note of Koornwinder [12]. 4 The factorization of the Askey–Wilson second order difference operator Ruijsenaars [20] analyzed when the square ( D + ã + ã−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 ( D(s/2) κ,λ,0,0;q − q z 2 − q− z 2 + qκ + q−κ )( D(s/2) κ,λ,0,0;q + q z 2 + q− z 2 + qκ + q−κ ) = D(s) κ,λ,κ,λ;q − q z − q−z + q2κ + q−2κ. (4.1) Remark 4.1. Our operator D + ã + ã−1 = D + q2κ + q−2κ for arbitrary Hecke parameters (κ, λ, υ, ς) is essentially the second order difference operator Aδ(c; ·) in Ruijsenaars’ paper [20, (2.1)] with the parameters (c0, c1, c2, c3) in [20] related to our Hecke parameters by κ + λ = − √ −1c0, κ− λ = − √ −1c1, υ + ς = − √ −1c2 and υ − ς = − √ −1c3. A simple derivation of (4.1) is as follows. Consider the second order difference operator L = L(s/2)κ,λ;q defined by ( Lf ) (x) := (1− qκ+λ+x)(1 + qκ−λ+x) qκ(1− q2x) f ( x+ s 2 ) + (1− qκ+λ−x)(1 + qκ−λ−x) qκ(1− q−2x) f ( x− s 2 ) . 10 J.V. Stokman Then (( L(s/2)κ,λ,q − q κ − q−κ ) f ) (x) = (1− qκ+λ+x)(1 + qκ−λ+x) qκ(1− q2x) ( f ( x+ s 2 ) − f(x) ) + (1− qκ+λ−x)(1 + qκ−λ−x) qκ(1− q−2x) ( f ( x− s 2 ) − f(x) ) = ( D(s/2) κ,λ,0,0;qf ) (x). Formula (4.1) thus is equivalent to( L(s/2)κ,λ;q − q z 2 − q− z 2 )( L(s/2)κ,λ;q + q z 2 + q− z 2 ) = D(s) κ,λ,κ,λ;q − q z − q−z + q2κ + q−2κ, (4.2) which is an easy check (use that the Askey–Wilson parameters associated to Hecke parameters (κ, λ, κ, λ) satisfy cs = q s 2as = q s 2 +κ+λ and ds = q s 2 bs = −q s 2 +κ−λ). The eigenfunctions of Dx of the form pn(qx + q−x) (n ∈ Z≥0) with pn 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 cs = q s 2as and ds = q s 2 bs) 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 4φ3 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). Ruijsenaars [20] used the factorization (4.1) to motivate and analyze quadratic transforma- tion formulas for the hyperbolic nonpolynomial generalization of the continuous q-Jacobi poly- nomial, which is Ruijsenaars’ R-function with the continuous q-Jacobi specialization (κ, λ, κ, λ) of the associated Hecke algebra parameters (see Remark 4.1 for the relation with Ruijsenaars’ notations [20]). In the following section we use the factorization (4.1) to prove quadratic trans- formation formulas for the asymptotically free eigenfunction Φ(·, z;κ, λ, κ, λ; q, s). 5 Quadratic transformation formulas For a function Ξ(x, z;κ, λ, υ, ς; q, s) we write ΞR(x, z) := Ξ ( x, z 2 ;κ, λ, 0, 0; q, s 2 ) . 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 ΞdR(x, z) := Ξ(x, z2 ;κ, 0, λ, 0; q, s2) for its dual version. Since L(s/2)κ,λ,q = D(s/2) κ,λ,0,0;q + qκ + q−κ, we know from the previous section that the meromorphic function ΦR(x, z) satisfies( Lx − q z 2 − q− z 2 ) ΦR(·, z) = 0. (5.1) In fact, ΦR(x, z) is the unique solution to (5.1) which is of the form ΦR(x, z) = WR(x, z) StR(x)StdR(z) ∞∑ r=0 ΓR,r(z)q rx with ΓR,r(z) holomorphic in z ∈ C, with ΓR,0(z) = ( q s 2 +z; q s 2 ) ∞ and with the power series ΨR(x, z) := ∞∑ r=0 ΓR,r(z)q rx converging normally on compacta of (x, z) ∈ C× C. Some Remarks on Very-Well-Poised 8φ7 Series 11 For a function Ξ(x, z;κ, λ, υ, ς; q, s) we write ΞJ(x, z) := Ξ(x, z;κ, λ, κ, λ; q, s). This is the parameter specialization which reduces the Askey–Wilson polynomials to Askey’s and Wilson’s version of the continuous q-Jacobi polynomials [1, § 4]. In addition we write ΞdJ(x, z) := Ξ(x, z;κ, κ, λ, λ; q, s) for its dual version. Theorem 5.1. ΦJ(x, z) = ΦR(x, z). 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( qκ+λ,−qκ−λ, q s 4 ,−q s 4 ) . The expression of Ψ(x, z) as a sum of two 4φ3’s, see (2.5), shows that( −q s 4 +x; q s 2 )−1 ∞ ΨR(x, z) is holomorphic in (x, z) ∈ C× C. By Remark 2.3 also( q s 4 +x,−q s 4 +x; q s 2 )−1 ∞ ΨR(x, z) = ( q s 2 +2x; qs )−1 ∞ ΨR(x, z) is holomorphic in (x, z) ∈ C× C. We write its power series expansion in qx as ( q s 2 +2x; qs )−1 ∞ ΨR(x, z) = ∞∑ r=0 H̃r(z)q rx. The H̃r(z) are holomorphic in z ∈ C, the series converges normally on compacta of (x, z) ∈ C×C, and H̃0(z) = ( q s 2 +z; q s 2 ) ∞. Define new holomorphic functions by Hr(z) := ( −q s 2 +z; q s 2 ) ∞H̃r(z) for r ∈ Z≥0. Then H0(z) = ( qs+2z; qs ) ∞ and ΦR(x, z) = WR(x, z) ( q s 2 +2x; qs ) ∞ StR(x)StdR(z) ( −q s 2 +z; q s 2 ) ∞ ∞∑ r=0 Hr(z)q rx with the series converging normally on compacta of (x, z) ∈ C× C. A direct computation now shows that ΦR(x, z) = WJ(x, z) StJ(x)StdJ(z) ∞∑ r=0 Hr(z)q rx. By (5.1) and (4.2) we furthermore have( D(s) κ,λ,κ,λ;q − q z − q−z + q2κ + q−2κ ) ΦR(·, z) = 0. Hence ΦR(x, z) satisfies the characterizing properties of ΦJ(x, z). � By the proof of the theorem we in particular have ΨR(x, z) = ( q s 2 +2x; qs ) ∞( −q s 2 +z; q s 2 ) ∞ ΨJ(x, z). Substituting the explicit expressions of ΨR and ΨJ in this formula gives the following quadratic transformation formula for very-well-poised 8φ7 series. 12 J.V. Stokman Corollary 5.2. ( qβxz,− qxz β , q 3 2 xz α ,−q 1 2αxz; q ) ∞( q 1 2x,−q 3 2xz2, q 2βxz2 α ; q ) ∞ 8W7 ( −q 1 2xz2; qz α ,−q 1 2 z β ,−αz, q 1 2βz,−q 1 2x; q,−q 1 2x ) = ( − q2xz2 αβ ,−qαβxz2,− qαx β ; q2 ) ∞( − q3βxz4 α ; q2 ) ∞ × 8W7 ( −qβxz 4 α ; q2z2 α2 ,−qz2,−z2, qβ2z2,−qβx α ; q2,−qαx β ) (5.2) if both \|q 1 2x\| < 1 and \| qαxβ \| < 1. Remark 5.3. (i) As observed by Mizan Rahman (private communication), the quadratic transformation formula (5.2) for 8W7 is equivalent to the quadratic transformation formula [6, (3.5.10)] by applying to the left hand side of (5.2) the transformation formula [6, (III.23)] with parameters (a, b, c, d, e, f) in [6, (III.23)] specialized to ( −q 1 2xz2,−q 1 2 z β ,−αz,−q 1 2x, qz α , q 1 2βz ) and to the right hand side of (5.2) the transformation formula [6, (III.23)] with parameters (a, b, c, d, e, f, q) in [6, (III.23)] specialized to ( −qβxz 4 α ,−qz2,−z2, qβ2z2, q 2z2 α2 ,−qβx α , q2 ) . (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, qz, qα, qβ) 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 2F1 ( 1 + 2z − 2α, 12 + 2z + 2β 1 + 4z ; 4x (1 + x)2 ) , where 2F1 is Gauss’ hypergeometric series (in contrast to the classical limits of the quadra- tic transformations of very-well-poised 8φ7’s from [18, § 5], which reduce to nontriv- ial quadratic transformations for 2F1). 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 (−qx, qz, qα, qβ). 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 ΦJ(x, z) = ΦR(x, z) before substituting the explicit expression as a 8W7 series. It leads to the following quadratic transformation formula. Some Remarks on Very-Well-Poised 8φ7 Series 13 Corollary 5.4. ( −q 1 2αxz,−q 1 2βz, q 3 2 xz α ; q ) ∞( − q 3 2 x2z β ; q ) ∞ 8W7 ( −q 1 2x2z β ; qx αβ ,−q 1 2x,−αx β , q 1 2x,−q 1 2 z β ; q,−q 1 2βz ) = ( −qαβxz2, q 2αxz2 β ,− q2xz2 αβ , q 3βxz2 α ; q2 ) ∞( −q3x2z2,−q2z2, q2x2z2 β2 ; q2 ) ∞ × 8W7 ( −qx2z2; q 2x αβ ,−qβx α ,−αx β , qαβx,−qβz 2 α ; q2,−qx ) . (5.3) Also (5.3) can be related to [6, (3.5.10)] by applying the transformation formula [6, (III.23)] to both sides, cf. Remark 5.3(i). 6 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, Φ(x,−κ− υ − ns) = a−2ns ( asbs, ascs, asds; q s ) n ( q2(1−n)s asbscsds ; qs ) ∞( q(n−1)sasbscsds; qs ) n Std(−κ− υ − ns) Pn(x) (6.1) with Pn(x) = Pn(x;κ, λ, υ, ς; q, s) the normalized Askey–Wilson polynomial in qx + q−x of deg- ree n, Pn(x) := 4φ3 ( q−ns, q(n−1)sasbscsds, asq x, asq −x asbs, ascs, asds ; qs, qs ) . This can be proved directly from the expression of Φ(x,−κ − υ − ns) as a very-well-poised 8W7 series (see Proposition 2.2) by first applying Watson’s transformation [6, (III.17)] with parameters (a, b, c, d, e, f, q) in [6, (III.17)] specialized to( q2(1−n)s+x asbscsd2s , qs+x ds , q(2−n)s asbscsds , q(1−n)s asds , q(1−n)s bsds , q(1−n)s csds , qs ) , followed by Sear’s transformation [6, (III.16)] with parameters (a, b, c, d, e, f, q) in [6, (III.16)] specialized to( q(1−n)s asds , q(1−n)s bsds , q(1−n)s csds , q2(1−n)s asbscsds , q(1−n)s+x ds , q(1−n)s−x ds , qs ) . Formula (6.1) can also be proved by observing that Pn(x) is an eigenfunction of Dx with eigen- value ã(qn + 1) + ã−1(q−n + 1) (see [1]) of the form Pn(x) = ( q−ns, q(n−1)sasbscsds; q s ) n (−as)nq s 2 n(n+1)( asbs, ascs, asds; qs ) n qnx + ∑ m<n cmq mx, 14 J.V. Stokman 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) = ( asbs, ascs; q s ) ∞( qs asds ; qs ) ∞ Pn(x). (6.2) This can again be proved directly using transformation formulas and the expression of E(x, z) as a 8W7 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 ΦJ(x, z) = ΦR(x, z) and using (6.1) twice gives, after straightforward simplifications, the following result (we take s = 2 without loss of generality), 4φ3 ( q−2n, aqx, aq−x, q2na2b2 ab, qa2, qab ; q2, q2 ) = 4φ3 ( q−n, aqx, aq−x,−qnab ab, q 1 2a,−q 1 2a ; 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)] (−a2q 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)]. 7 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 EJ(x, z) = cJ(x, z)ΦJ(x, z) + cJ(x,−z)ΦJ(x,−z) = cJ(x, z)ΦR(x, z) + cJ(x,−z)ΦR(x,−z) while on the other hand, ER(x, z) = cR(x, z)ΦR(x, z) + cR(x,−z)ΦR(x,−z). The c-functions cJ(x, z) and cR(x, z) are explicitly given by cJ(x, z) = θ ( q2κ−z,−q−z, q s 2 +2λ−z,−q s 2 −κ−λ+x−z; qs ) WJ(x, z)θ ( q−2z,−q s 2 +κ−λ+x; qs ) , cR(x, z) = θ ( q2κ−z; qs ) θ ( q s 4 +λ− z 2 ,−q s 4 −κ+x− z 2 ; q s 2 ) WR(x, z)θ ( q−z,−q s 4 +x; q s 2 ) . (7.1) Note that WJ(x, z) = q(κ+λ+x)(2κ+z)/s = WR(x, z) and that cJ(x, z) cR(x, z) = θ ( −q s 2 −κ−λ+x−z; qs ) θ ( −q s 4 +λ− z 2 ,−q 3 4 +x; q s 2 ) θ ( −q s 2 −z,−q s 2 +κ−λ+x; qs ) θ ( −q s 4 −κ+x− z 2 ; q s 2 ) is not invariant under z → −z for generic Hecke parameters, since it is zero at z = 2λ + s 2 + 2π √ −1τ but nonzero at z = −2λ − s 2 − 2π √ −1τ . Hence EJ(x, z) cannot be of the form α(x, z)ER(x, z) with α(x, z) meromorphic and s-invariant in the variable x. Some Remarks on Very-Well-Poised 8φ7 Series 15 Remark 7.1. Both EJ(·, z) and ER(·, z) are eigenfunctions of D(s) κ,λ,κ,λ;q with eigenvalue qz + q−z − q2κ − q−2κ (for ER(·, z) this follows from (4.1) and the fact that cR(·, z) is s 2 -translation invariant). On the other hand, ER(·, z) is an eigenfunction of L(s/2)κ,λ,q with eigenvalue q z 2 + q− z 2 , but this is not true for EJ(·, z) since cJ(·, z) is not s 2 -translation invariant. Remark 7.2. Ruijsenaars’ R-function R(·, z) [20] is a hyperbolic eigenfunction of the Askey– Wilson second order difference operator (implying in particular that it admits an analytic con- tinuation to the regime \|q\| = 1) satisfying R(−x, z) = R(x, z). For the R-function R a quadratic transformation formula of the form RJ(x, z) = RR(x, z) holds true, see [20, (3.16)]. The dis- crepancy with the fact that EJ(x, z) 6= ER(x, z) is not unexpected since the R-function R in the trigonometric regime \|q\| < 1 has a nontrivial factorization in Askey–Wilson functions: it expands as a sum of two terms, each term being essentially the product of two Askey–Wilson functions, see [3, Theorem 6.5]. The quadratic transformation formula ΦJ(x, z) = ΦR(x, z) implies the following result for the connection coefficients in (2.8). Proposition 7.3. We have cJ(x,−z) cdJ(z,−x) = cR(x,−z) cdR( z2 ,−2x) , cJ(x, z)− cdJ(z, x) cdJ(z,−x) = cR(x, z)− cdR( z2 , 2x) cdR( z2 ,−2x) . (7.2) Proof. Using (2.8) and Theorem 5.1 we have( cJ(x, z)− cdJ(z, x) cdJ(z,−x) ) ΦJ(x, z) + cJ(x,−z) cdJ(z,−x) ΦJ(x,−z) = ( cR(x, z)− cdR( z2 , 2x) cdR( z2 ,−2x) ) ΦJ(x, z) + cR(x,−z) cdR( z2 ,−2x) ΦJ(x,−z), (7.3) since both sides are equal to ΦJ(−x, z) = ΦR(−x, z). By a straightforward asymptotic argument (compare with the proof of Corollary 2.8) it follows that the coefficients of ΦJ(x, z) (resp. of ΦJ(x,−z)) on both sides of this equation should be the same. Alternatively, one verifies by a direct computation that cJ(x,−z) cdJ(z,−x) = q− 4κx s q 2(κ+λ)z s θ ( q2x, q s 2 +2λ+z, q2κ+z; qs ) θ ( qz, qκ+λ+x,−qκ−λ+x; q s 2 ) = cR(x,−z) cdR( z2 ,−2x) , yielding the first equality of (7.2). Combined with (7.3) this implies the second equality of (7.2). � Since c(x, z) is s-translation invariant in both x and z, it follows from the right hand sides of (7.2) that the quotients cJ(x,−z) cdJ(z,−x) , cJ(x, z)− cdJ(z, x) cdJ(z,−x) are s 2 -translation invariant in x (although cJ(x, z) is not, cf. Remark 7.1). Using (7.1) and using the explicit expressions cdJ(z, x) = θ ( qκ+λ−x,−qκ−λ−x, q s 2 +κ+λ−x,−q s 2 −κ−λ+z−x; qs ) W d J (z, x)θ ( q−2x,−q s 2 +z; qs ) , cdR(z, x) = θ ( qκ+λ− x 2 ,−qκ−λ− x 2 , q s 4 −x 2 ,−q s 4 −κ+z−x 2 ; q s 2 ) W d R(z, x)θ ( q−x,−q s 4 +λ+z; q s 2 ) 16 J.V. Stokman with W d J (z, x) = q(κ+λ+x)(2κ+z)/s and W d R(z, x) = q2(κ+z)(κ+λ+ x 2 )/s, we have explicitly, cJ(x, z)− cdJ(z, x) cdJ(z,−x) = q−2x(2κ+z)/s θ ( q2x; qs ) θ ( qκ+λ+x,−qκ−λ+x, q s 2 +κ+λ+x,−q s 2 −κ−λ+x+z; qs ) × { θ ( q2κ−z,−q−z, q s 2 +2λ−z,−q s 2 −κ−λ+x−z,−q s 2 +z; qs ) θ ( q−2z,−q s 2 +κ−λ+x; qs ) − θ ( qκ+λ−x,−qκ−λ−x, q s 2 +κ+λ−x,−q s 2 −κ−λ−x+z; qs ) θ ( q−2x; qs ) } and cR(x, z)− cdR( z2 , 2x) cdR( z2 ,−2x) = q−2x(2κ+z)/s θ ( q2x; q s 2 ) θ ( qκ+λ+x,−qκ−λ+x, q s 4 +x,−q s 4 −κ+x+ z 2 ; q s 2 ) × { θ ( q2κ−z; qs ) θ ( q s 4 +λ− z 2 ,−q s 4 −κ+x− z 2 ,−q s 4 +λ+ z 2 ; q s 2 ) θ ( q−z,−q s 4 +x; q s 2 ) − θ ( qκ+λ−x,−qκ−λ−x, q s 4 −x,−q s 4 −κ−x+ z 2 ; q s 2 ) θ ( q−2x; q s 2 ) } . Hence (7.2) is equivalent to the following theta function identity. Corollary 7.4. θ ( a2, b2c2,−b2, qb2d2,− qb 2 acd ,−qb2; q2 ) − θ ( b4,− qc ad , acd,−ac d , qacd,− qa b2cd ; q2 ) = θ ( −qab2cd; q2 ) θ ( −b2; q ) θ ( qa2; q2 ) θ ( −q 1 2abc; q ) {θ(bc,−bc, a2, q 1 2 bd,−q 1 2 b ac ,−q 1 2d b ; q ) − θ ( b2,−q 1 2a, acd,−ac d , q 1 2a,−q 1 2a bc ; q )} . (7.4) Acknowledgements I thank Tom Koornwinder for drawing my attention to the quadratic transformation formula for continuous q-Jacobi polynomials. I thank Mizan Rahman for pointing out to me how the quadratic transformations (5.2) and (5.3) for very-well-poised 8φ7 series are related to the known quadratic transformation formula [6, (3.5.10)] (see Remark 5.3(i)). References [1] Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319. [2] van de Bult F.J., Ruijsenaars’ hypergeometric function and the modular double of Uq(sl2(C)), Adv. 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[26] Suslov S.K., Some orthogonal very-well-poised 8φ7-functions that generalize Askey–Wilson polynomials, Ramanujan J. 5 (2001), 183–218, math.CA/9707213. http://dx.doi.org/10.1017/CBO9780511526251 http://dx.doi.org/10.1090/S0002-9947-98-01879-0 http://dx.doi.org/10.1090/S0002-9947-98-01879-0 http://arxiv.org/abs/math.CA/9511218 http://dx.doi.org/10.1007/s11139-006-8478-6 http://arxiv.org/abs/math.QA/0203136 http://dx.doi.org/10.2307/2001881 http://dx.doi.org/10.1155/S1073792801000575 http://arxiv.org/abs/math.CA/0004053 http://dx.doi.org/10.2977/prims/1145477332 http://arxiv.org/abs/math.QA/9911163 http://staff.science.uva.nl/~thk/art/comment/ChalykhComment.pdf http://dx.doi.org/10.1112/plms/pdm055 http://dx.doi.org/10.1112/plms/pdm055 http://arxiv.org/abs/math.QA/0701218 http://dx.doi.org/10.1007/s00029-010-0039-6 http://dx.doi.org/10.1007/s00029-010-0039-6 http://arxiv.org/abs/0912.3784 http://dx.doi.org/10.1093/imrn/rnp165 http://arxiv.org/abs/0812.1005 http://arxiv.org/abs/math.QA/0001033 http://dx.doi.org/10.4153/CJM-1981-076-8 http://dx.doi.org/10.2307/2154269 http://dx.doi.org/10.2307/2154269 http://dx.doi.org/10.1007/s002200050840 http://dx.doi.org/10.1007/s11139-006-0257-x http://dx.doi.org/10.1112/jlms/s1-34.1.15 http://dx.doi.org/10.1006/jath.2001.3647 http://arxiv.org/abs/math.CA/0105093 http://arxiv.org/abs/1109.0613 http://dx.doi.org/10.1023/A:1011439924912 http://arxiv.org/abs/math.CA/9707213 1 Introduction 2 Eigenfunctions of the Askey-Wilson second order difference operator 3 The case of trivial quantum monodromy 4 The factorization of the Askey-Wilson second order difference operator 5 Quadratic transformation formulas 6 Polynomial reduction 7 A theta function identity References

Some Remarks on Very-Well-Poised ₈∅₇ Series

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