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APPEARED IN BULLETIN OF THE

arXiv:math/9204236v1 [math.CA] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 258-263THE AℓAND CℓBAILEY TRANSFORM AND LEMMAStephen C. Milne and Glenn M. LillyAbstract. We announce a higher-dimensional generalization of the Bailey Trans-form, Bailey Lemma, and iterative “Bailey chain” concept in the setting of basichypergeometric series very well-poised on unitary Aℓor symplectic Cℓgroups.

Theclassical case, corresponding to A1 or equivalently U(2), contains an immense amountof the theory and application of one-variable basic hypergeometric series, includingelegant proofs of the Rogers-Ramanujan-Schur identities. In particular, our programextends much of the classical work of Rogers, Bailey, Slater, Andrews, and Bressoud.1.

IntroductionThe purpose of this paper is to announce a higher-dimensional generalization ofthe Bailey Transform [2] and Bailey Lemma [2] in the setting of basic hypergeo-metric series very well-poised on unitary [19] or symplectic [14] groups. Both typesof series are directly related [14, 18] to the corresponding Macdonald identities.The series in [19] were strongly motivated by certain applications of mathematicalphysics and the unitary groups U(n) in [10, 11, 15, 16].

The unitary series use thenotation Aℓ, or equivalently U(ℓ+ 1); the symplectic case, Cℓ. The classical BaileyTransform, Lemma, and very well-poised basic hypergeometric series correspond tothe case A1, or equivalently U(2).The classical Bailey Transform and Bailey Lemma contain an immense amountof the theory and application of one-variable basic hypergeometric series [2, 12,25].

They were ultimately inspired by Rogers’ [24] second proof of the Rogers-Ramanujan-Schur identities [23]. The Bailey Transform was first formulated byBailey [8], utilized by Slater in [25], and then recast by Andrews [4] as a fundamentalmatrix inversion result.

This last version of the Bailey Transform has immediateapplications to connection coefficient theory and “dual” pairs of identities [4], andq-Lagrange inversion and quadratic transformations [13].The most important application of the Bailey Transform is the Bailey Lemma.This result was mentioned by Bailey [8; §4], and he described how the proof wouldwork.However, he never wrote the result down explicitly and thus missed thefull power of iterating it. Andrews first established the Bailey Lemma explicitlyin [5] and realized its numerous possible applications in terms of the iterative“Bailey chain” concept.This iteration mechanism enabled him to derive many1991 Mathematics Subject Classification.

Primary: 33D70, 05A19.S. C. Milne was partially supported by NSF grants DMS 86-04232, DMS 89-04455, and DMS90-96254G.

M. Lilly was fully supported by NSA supplements to the above NSF grants and by NSAgrant MDA 904-88-H-2010Received by the editors April 28, 1991c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2S. C. Milne and G. M. Lillyq-series identities by “reducing” them to more elementary ones.

For example, theRogers-Ramanujan-Schur identities can be reduced to the q-binomial theorem. Fur-thermore, general multiple series Rogers-Ramanujan-Schur identities are a directconsequence of iterating suitable special cases of Bailey’s Lemma.

In addition, An-drews notes that Watson’s q-analog of Whipple’s transformation is an immediateconsequence of the second iteration of one of the simplest cases of Bailey’s Lemma.Continued iteration of this same case yields Andrews’ [3] infinite family of exten-sions of Watson’s q-Whipple transformation. Even Whipple’s original work [26,27] fits into the q = 1 case of this analysis.

Paule [22] independently discoveredimportant special cases of Bailey’s Lemma and how they could be iterated. Essen-tially all the depth of the Rogers-Ramanujan-Schur identities and their iterationsis embedded in Bailey’s Lemma.The process of iterating Bailey’s Lemma has led to a wide range of applica-tions in additive number theory, combinatorics, special functions, and mathematicalphysics.

For example, see [2, 5, 6, 7, 9].The Bailey Transform is a consequence of the terminating 4φ3 summation theo-rem. The Bailey Lemma is derived in [1] directly from the 6φ5 summation and thematrix inversion formulation [4, 13] of the Bailey Transform.

We employ a similarmethod in the Aℓand Cℓcases by starting with a suitable, higher-dimensional, ter-minating 6φ5 summation theorem extracted from [19] and [14], respectively. TheAℓproofs appear in [20, 21], and the Cℓcase is established in [17].

Many otherconsequences of the Aℓand Cℓgeneralizations of Bailey’s Transform and Lemmawill appear in future papers.These include Aℓand Cℓq-Pfaff-Saalsch¨utz sum-mation theorems, q-Whipple transformations, connection coefficient results, andapplications of iterating the Aℓor CℓBailey Lemma.2. ResultsThroughout this article, let i, j, N, and y be vectors of length ℓwith nonnegativeinteger components.

Let q be a complex number such that |q| < 1. Define(2.1a)(α)∞≡(α; q)∞:=Yk≥0(1 −αqk)and, thus,(2.1b)(α)n ≡(α; q)n := (α)∞/ (αqn)∞.Define the Bailey transform matricies, M and M ∗, as follows.Definition (M and M ∗for Aℓ).

Let a, x1, . .

. , xℓbe indeterminate.

Suppose thatnone of the denominators in (2.2a–b) vanishes. Then let(2.2a)M(i; j; Aℓ) :=ℓYr,s=1q xrxsqjr−js−1ir−jrℓYk=1aq xkxℓ−1ik+(j1+···+jℓ);

THE AℓAND CℓBAILEY TRANSFORM AND LEMMA3andM ∗(i; j; Aℓ)(2.2b):=ℓYk=11 −axkxℓqik+(i1+···+iℓ)ℓYk=1aq xkxℓjk+(i1+···+iℓ)−1×ℓYr,s=1q xrxsqjr−js−1ir−jr(−1)(i1+···+iℓ)−(j1+···+jℓ) q((i1+···+iℓ)−(j1+···+jℓ)2).Definition (M and M ∗for Cℓ). Let x1, .

. .

, xℓbe indeterminate. Suppose thatnone of the denominators in (2.3a–b) vanishes.

Then let(2.3a)M(i; j; Cℓ) :=ℓYr,s=1"q xrxsqjr−js−1ir−jrqxrxsqjr+js−1ir−jr#;and(2.3b)M ∗(i; j; Cℓ):=ℓYr,s=1"q xrxsqjr−js−1ir−jrxrxsqjr+is−1ir−jr#Y1≤r

Let M and M ∗be defined as in (2.2) and (2.3), with rows and columns ordered lexicographically.Then M and M ∗are inverse, infinite, lower-triangular matricies. That is,(2.4)ℓYk=1δ(ik, jk) =Xj k≤yk≤ikk=1,2,...,ℓM(i; y; G) M ∗(y; j; G),where δ(r, s) = 1 if r = s, and 0 otherwise.Equations (2.2) and (2.3) motivate the definition of the Aℓand CℓBailey pair.Definition (G-Bailey Pair).

Let G = Aℓor Cℓ.Let Nk ≥0 be integers fork = 1, 2, . .

. , ℓ.

Let A = {A(y;G)} and B = {B(y; G)} be sequences. Let M and M ∗be as above.

Then we say that A and B form a G-Bailey Pair if(2.5)B(N;G) =X0≤yk≤Nkk=1,2,...,ℓM(N; y; G) A(y;G).As a consequence of the Bailey transform, (2.4), and the definition of the G-Bailey pair, (2.5), we have the following result.

4S. C. MILNE AND G. M. LILLYCorollary (Bailey Pair Inversion).

A and B satisfy equation (2.5) if and only if(2.6)A(N;G) =X0≤yk≤Nkk=1,2,...,ℓM ∗(N; y; G) B(y;G).Define the sequences A′ = {A′(y;Aℓ)} and B′ = {B′(y;Aℓ)} by(2.7a)A′(N;Aℓ) :=ℓYk=1aqρxkxℓ−1NkℓYk=1σ xkxℓNk×(ρ)N1+···+Nℓ(aq/σ)N1+···+Nℓ(aq/ρσ)N1+···+NℓA(N; Aℓ)andB′(N;Aℓ) :=X0≤yk≤Nkk=1,2,...,ℓ( ℓYk=1"σ xkxℓykaqρxkxℓ−1Nk#ℓYr,s=1q xrxsqyr−ys−1Nr−yr(2.7b)×(aq/ρσ)(N1+···+Nℓ)−(y1+···+yℓ) (ρ)y1+···+yℓ(aq/σ)N1+···+Nℓ× (aq/ρσ)y1+···+yℓB(y; Aℓ))Define the sequences A′ = {A′(y;Cℓ)} and B′ = {B′(y;Cℓ)} by(2.8a)A′(N;Cℓ) :=ℓYk=1"(αxk)Nkqxkβ−1Nk(βxk)Nk (qxkα−1)Nk# βαN1+···+NℓA(N;Cℓ)and(2.8b)B′(N; Cℓ) :=X0≤yk≤Nkk=1,2,...,ℓ(crℓYk=1" (αxk)ykqxkβ−1yk(βxk)Nk (qxkα−1)Nk#ℓYr,s=1q xrxsqyr−ys−1Nr−yr×Y1≤r

SupposeA = {A(N;G)} and B = {B(N;G)} form a G-Bailey Pair. If A′ = {A′(N;G)} andB′ = {B′(N;G)} are as above, then A′ and B′ also form a G-Bailey Pair.

THE AℓAND CℓBAILEY TRANSFORM AND LEMMA53. Sketches of ProofsProof of (2.4).

In each case, Aℓand Cℓ, we begin with a terminating 4φ3 summationtheorem.In the Cℓcase, it is first necessary to specialize Gustafson’s Cℓ6ψ6summation theorem, see [14], terminate it from below and then from above, andfurther specialize the resulting terminating 6φ5 to yield a terminating 4φ3. In boththe Aℓand Cℓcases, the 4φ3 is modified by multiplying both the sum and productsides by some additional factors.

Finally, that result is transformed term-by-termto yield the sum side of (2.4).□Proof of (2.6). Equation (2.6) follows directly from the definition, (2.5), and thetermwise nature of the calculations in the proof of (2.4).□Proof of Bailey’s Lemma.

The definitions in (2.7) and (2.8) are substituted into(2.5). After an interchange of summation, the inner sum is seen to be a special caseof the appropriate 6φ5.

The 6φ5 is then summed, and the desired result follows.□Detailed proofs of the Cℓcase will appear in [17], as will a discussion of theCℓBailey chain and a connection coefficient result associated with the CℓBaileyTransform.References1. A. K. Agarwal, G. Andrews, and D. Bressoud, The Bailey lattice, J. Indian Math.

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122(1987), 223–256.20., Balanced 3φ2 summation theorems for U(n) basic hypergeometric series, (inpreparation).21., A U(n) generalization of Bailey’s lemma, in preparation.22. P. Paule, Zwei neue Transformationen als elementare Anwendungen der q-VandermondeFormel, Ph.D. thesis, 1982, University of Vienna.23.

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Soc. (2) 25 (1926), 525–544.Department of Mathematics, The Ohio State University, Columbus, Ohio 43210E-mail address: milne@function.mps.ohio-state.edu

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