Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems
We list An, Cn, and Dn extensions of the elliptic WP Bailey transform and lemma, given for n=1 by Andrews and Spiridonov. Our work requires multiple series extensions of Frenkel and Turaev's terminating, balanced, and very-well-poised ₁₀V₉ elliptic hypergeometric summation formula due to Roseng...
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| Цитувати: | Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems / G. Bhatnagar, M.J. Schlosser // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 52 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860206827008950272 |
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| author | Bhatnagar, G. Schlosser, M.J. |
| author_facet | Bhatnagar, G. Schlosser, M.J. |
| citation_txt | Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems / G. Bhatnagar, M.J. Schlosser // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 52 назв. — англ. |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We list An, Cn, and Dn extensions of the elliptic WP Bailey transform and lemma, given for n=1 by Andrews and Spiridonov. Our work requires multiple series extensions of Frenkel and Turaev's terminating, balanced, and very-well-poised ₁₀V₉ elliptic hypergeometric summation formula due to Rosengren and Rosengren and Schlosser. In our study, we discover two new An ₁₂V₁₁ transformation formulas that reduce to two new An extensions of Bailey's 10ϕ9 transformation formulas when the nome p is 0, and two multiple series extensions of Frenkel and Turaev's sum.
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| first_indexed | 2025-12-07T18:12:50Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 025, 44 pages
Elliptic Well-Poised Bailey Transforms
and Lemmas on Root Systems
Gaurav BHATNAGAR and Michael J. SCHLOSSER
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
E-mail: bhatnagarg@gmail.com, michael.schlosser@univie.ac.at
URL: http://www.gbhatnagar.com, http://www.mat.univie.ac.at/~schlosse/
Received September 01, 2017, in final form March 13, 2018; Published online March 22, 2018
https://doi.org/10.3842/SIGMA.2018.025
Abstract. We list An, Cn and Dn extensions of the elliptic WP Bailey transform and
lemma, given for n � 1 by Andrews and Spiridonov. Our work requires multiple series
extensions of Frenkel and Turaev’s terminating, balanced and very-well-poised 10V9 elliptic
hypergeometric summation formula due to Rosengren, and Rosengren and Schlosser. In our
study, we discover two new An 12V11 transformation formulas, that reduce to two new An
extensions of Bailey’s 10φ9 transformation formulas when the nome p is 0, and two multiple
series extensions of Frenkel and Turaev’s sum.
Key words: An elliptic and basic hypergeometric series; elliptic and basic hypergeometric
series on root systems; well-poised Bailey transform and lemma
2010 Mathematics Subject Classification: 33D67
1 Introduction
The many different proofs of the famous Rogers–Ramanujan identities have led to a plethora of
fruitful ideas in mathematics and physics. This paper contains some results ultimately following
a path that began with Watson’s 1929 proof of these identities. Watson [49] proved a very
general transformation formula with many parameters. The Rogers–Ramanujan identities follow
by taking the limit as (most of) these parameters go to infinity, and then invoking the Jacobi
triple product identity. The proof of Watson’s transformation was later simplified by Bailey
during the course of his study of Rogers’ work; and the ensuing ideas used by Slater to prove
more than a hundred Rogers–Ramanujan type identities. Eventually, Bailey’s approach was
perfected by Andrews as a combination of three ideas. According to Andrews’ formulation, the
Bailey transform is a specific (invertible) lower-triangular matrix that transforms a sequence to
another sequence. A Bailey pair is a pair of sequences which satisfies such a relationship, and
the Bailey lemma is a method to generate a new Bailey pair (with additional parameters) from
a given pair. Thus, the Bailey lemma can be used to generate new identities from known results.
In particular, Watson’s transformation follows by two steps of the Bailey lemma, applied to the
unit Bailey pair.
In two important papers, Milne [27] and Milne and Lilly [29] lifted the Bailey transform and
lemma machinery to the context of multiple basic hypergeometric series associated with the root
systems An and Cn.
The primary purpose of this paper is to extend Milne’s and Milne and Lilly’s work to the
setting of the well-poised (WP) Bailey transform and lemma. In the n � 1 case this is again
based on Bailey’s [6] ideas; and again, these ideas have been made accessible by Andrews’ [2]
exposition. In particular, the WP Bailey transform and lemma captures the generalizations of
This paper is a contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications.
The full collection is available at https://www.emis.de/journals/SIGMA/EHF2017.html
mailto:bhatnagarg@gmail.com
mailto:michael.schlosser@univie.ac.at
http://www.gbhatnagar.com
http://www.mat.univie.ac.at/~schlosse/
https://doi.org/10.3842/SIGMA.2018.025
https://www.emis.de/journals/SIGMA/EHF2017.html
2 G. Bhatnagar and M.J. Schlosser
the matrix formulation of the Bailey transform due to Bressoud [11] and Bailey’s [5] famous
10φ9 transformation which further generalizes Watson’s formula.
At the same time, we work in the setting of elliptic hypergeometric series associated with the
root systems An, Cn and Dn. Elliptic hypergeometric series appeared explicitly in the work of
Frenkel and Turaev [16] in 1997, and it was quickly realized that some of the classical methods
used for studying basic hypergeometric series apply as well to this kind of series. In this context,
we find the work of Warnaar [46] very useful and influential, because it introduced a notation for
elliptic hypergeometric series very much like the one used for basic hypergeometric series. The
elliptic series contain a parameter p, called the nome. When the nome p � 0, the formulas reduce
to formulas for basic hypergeometric series. A key ingredient in Warnaar’s paper [46] is an elliptic
matrix inverse which as a special case contains an elliptic extension of the WP Bailey transform.
Spiridonov [39] found an elliptic analogue of the WP Bailey lemma, and Warnaar [47] applied
and further extended Andrews’ ideas in the elliptic setting. A comprehensive survey of elliptic
hypergeometric functions has been given by Spiridonov [42]. Many of the central summation
and transformation formulas concerning multiple elliptic hypergeometric series were given by
Warnaar [46] and Rosengren [31].
In this paper, we provide several extensions of the elliptic WP Bailey transform and lemma.
A feature of the theory of series associated with root systems is that often there are many
extensions on root systems of the same result. Indeed, in this paper we present six extensions
of the elliptic WP Bailey transform (and associated Bressoud matrices), and eight elliptic WP
Bailey lemmas which can be used in multiple ways to generate different identities. For n � 1
all our six elliptic WP Bailey transforms specialize to the same result by Warnaar, and, also, all
our eight elliptic WP Bailey lemmas specialize to the same result by Spiridonov.
When n � 1 and p � 0, the WP Bailey transform and lemma depend in an essential way on
a summation result of Jackson [20], which is contained in Watson’s transformation. Multivariate
extensions of Jackson’s sum on the root systems An, Cn and Dn have been given by Milne [25],
Milne and Lilly [29], Denis and Gustafson [14], the first author [9], and the second author [36, 38].
As the second author showed in [37], Milne’s [25] An Jackson sum can be used to form a bridge
between An basic hypergeometric series and Macdonald polynomials [22, Chapter VI].
Our work in this paper depends on the elliptic An, Cn and Dn generalizations of this sum-
mation due to Rosengren [31] and one such result due to Rosengren and the second author [35].
(Recently, Rosengren [34] gave yet another such result, which we do not include in our study.)
One of the goals of our study is to recover the extensions of Bailey’s transformation formula listed
by Rosengren [31], which were obtained by a straightforward extension of the approach followed
in [10] for the basic hypergeometric case. Indeed, we recover all these results. In addition, we
give two new elliptic generalizations of Bailey’s transformation formula.
Our results are closely related with other work in this area. One of our elliptic generalizations
of Bailey’s 10φ9 sum was found independently by Rosengren [33]. It is motivated by formulas
previously given by the authors [8, 38]. The p � 0 case of one of our matrices was considered by
Milne [27] and a related Bailey lemma (again when p � 0) was previously obtained by Zhang
and Liu [52]. One of our Bailey lemmas is equivalent to a result of Zhang and Huang [51]. Most
of the matrix inversions that appear in our work can be obtained as special cases of very general
matrix inversions due to Rosengren and the second author [35]. Warnaar [48] found four of the
WP Bailey lemmas (including the one due to Zhang and Huang [51]) a few years ago but did
not publish his work.
Like Milne’s [27] important paper, we hope that our work too will open the door for further
work in this area. Further development of Bailey’s ideas in, for example, [2, 3, 21, 23, 39, 41, 44,
45, 47] certainly suggests that there is a lot that can be done. Notably, Andrews, Schilling and
Warnaar [4] established A2 Bailey lemmas and corresponding identities of Rogers–Ramanujan
type. However, the type of series these authors deal with in [4] are quite different from the
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 3
(n � 2 instances of the) An series considered in this paper. Spiridonov and Warnaar [43],
building upon work by Spiridonov [41], obtained integral Bailey transforms relating integrals on
root systems of type An and Cn. Brünner and Spiridonov [12] recently gave applications of the
integral analogue of the An Bailey lemma to supersymmetric linear quiver theories. Coskun [13]
had previously found a BCn WP Bailey transform for a slightly different kind of series than
the one we consider and found multiple Rogers–Ramanujan identities. In this context, we also
mention the work of Griffin, Ono and Warnaar [18].
This paper is organized as follows. In Section 2, we record the notation and terminology of
elliptic and basic hypergeometric series. In Section 3 we give a short introduction to the WP
Bailey transform and lemma, in a format that suits our work. From Section 4 until Section 9
(except Section 6) we systematically extend the analysis of Section 3 by considering in each
section a particular multivariable extension of the elliptic Jackson summation. In Section 6, we
record some basic hypergeometric results which follow as special cases of our work. In Section 10
we summarize our work and motivate our final set of results in Section 11, which contain an
extra parameter. The extra parameter disappears when n � 1.
2 The notation and terminology
In this section, we record the notation and terminology used in this paper, following, for the
most part, Gasper and Rahman [17]. We recommend Rosengren’s lectures [32] for a friendly
introduction to elliptic hypergeometric series.
The series considered in this paper are all of the form¸
kr¥0
r�1,2,...,n
Sk,
where the k � pk1, . . . , knq is an n-tuple of non-negative integers k1, k2, . . . , kn. The positive
integer n is called the dimension of the sum. We use the notation |k| :� k1 � � � � � kn for the
sum of components of the n-tuple; k� j for pk1 � j1, . . . , kn � jnq, obtained by component-wise
addition; and k � j for pk1 � j1, . . . , kn � jnq.
The summand Sk itself is a product of various q, p-shifted factorials, which we define shortly.
First, for arbitrary integers k and m, we define products as follows.
m¹
j�k
Aj :�
$'&
'%
AkAk�1 � � �Am if m ¥ k,
1 if m � k � 1,�
Am�1Am�2 � � �Ak�1
��1
if m ¤ k � 2.
(2.1)
The primary reason why this definition is useful is because the relation
m�1¹
j�k
Aj �
m¹
j�k
Aj �Am�1,
applies for all integers k and m.
Next, we define the q-shifted factorials, for k any integer, as
pa; qqk :�
k�1¹
j�0
�
1� aqj
�
,
and for |q| 1,
pa; qq8 :�
8¹
j�0
�
1� aqj
�
.
4 G. Bhatnagar and M.J. Schlosser
The parameter q is called the base. With this definition, we have the modified Jacobi theta
function defined as
θ pa; pq :� pa; pq8pp{a; pq8,
where a � 0 and |p| 1. We define the q, p-shifted factorials (or theta shifted factorials), for k
an integer, as
pa; q, pqk :�
k�1¹
j�0
θ
�
aqj ; p
�
.
The parameter p is called the nome. When the nome p � 0, the modified theta function θ pa; pq
reduces to p1� aq; and pa; q, pqk reduces to pa; qqk.
We use the short-hand notations
θ pa1, a2, . . . , ar; pq :� θ pa1; pq θ pa2; pq � � � θ par; pq ,
pa1, a2, . . . , ar; q, pqk :� pa1; q, pqkpa2; q, pqk � � � par; q, pqk,
pa1, a2, . . . , ar; qqk :� pa1; qqkpa2; qqk � � � par; qqk.
Observe that in view of (2.1), we have
pa; q, pq0 � 1,
and, for k an arbitrary integer
pa; q, pq�k �
1
paq�k; q, pqk
.
Two important properties of the modified theta function are [17, equation (11.2.42)]
θ pa; pq � θ pp{a; pq � �aθ p1{a; pq , (2.2a)
and [50, p. 451, Example 5]
θ pxy, x{y, uv, u{v; pq � θ pxv, x{v, uy, u{y; pq �
u
y
θ pyv, y{v, xu, x{u; pq . (2.2b)
Using (2.2a), we can “reverse the products”, and obtain the identity [17, equation (11.2.53)]
pa; q, pq�k �
p�q{aqk
pq{a; q, pqk
qp
k
2q. (2.3)
We can combine the two identities above to obtain�
aq�k; q, p
�
k
� pq{a; q, pqk p�a{qq
k q�p
k
2q. (2.4)
More generally, we have [17, equation (11.2.49) rewritten]
�
q1�n{a; q, p
�
k
�
pa; q, pqn
pa; q, pqn�k
p�q{aqk qp
k
2q�nk. (2.5)
Two other useful identities we use are [17, equation (11.2.47)]
pa; q, pqn�k � pa; q, pqnpaq
n; q, pqk, (2.6)
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 5
and a special case of [17, equation (11.2.55)],
ppa; q, pqk � p�1qka�kq�p
k
2qpa; q, pqk. (2.7)
We will use the notation of basic hypergeometric series (or rφs series). This series is of the
form
rφs
�
a1, a2, . . . , ar
b1, b2, . . . , bs
; q, z
�
:�
8̧
k�0
pa1, a2, . . . , ar; qqk
pq, b1, b2, . . . , bs; qqk
�
p�1qkqp
k
2q
�1�s�r
zk.
See Gasper and Rahman [17] for the convergence conditions for these series.
A series is terminating if it is not an infinite series. Usually, this happens due to a factor�
q�N ; q
�
k
in the numerator of the summand. When k ¡ N , this factor is 0, and so the series
terminates. Observe that in view of (2.1)
1
pq; qqk
� 0, when k 0. (2.8)
This ensures that the series terminates naturally from below too. That is, the terms with
negative index in the series are 0.
An r�1φr series is called well-poised if a1q � a2b1 � a3b2 � � � � � ar�1br. In addition, if
a2 � qa
1{2
1 , a3 � �qa
1{2
1 , then the series is called very-well-poised. Note that in such a case,
taking the special parameter a1 � a, the summand contains the term�
qa1{2,�qa1{2; q
�
k�
a1{2,�a1{2; q
�
k
�
1� aq2k
1� a
,
which we call the very-well-poised part. This suggests the compact notation
r�1Wrpa; a4, a5, . . . , ar�1; q, zq
for very-well-poised series.
An r�1φr series is called balanced if b1 � � � br � qa1 � � � ar�1 and z � q.
Recall that for an ordinary (resp. basic) hypergeometric series
°
ck the quotient gpkq �
ck�1{ck is a rational function in k (resp. qk). Now, a series
°
ck is called an elliptic hypergeometric
series if gpkq � ck�1{ck is an elliptic function of k with k considered as a complex variable, i.e.,
the function gpxq is a doubly periodic meromorphic function of the complex variable x. Without
loss of generality, by the theory of theta functions, we may assume that
gpxq �
θ pa1q
x, a2q
x, . . . , ar�1q
x; pq
θ pq1�x, b1qx, . . . , brqx; pq
z,
where the elliptic balancing condition,
a1a2 � � � ar�1 � qb1b2 � � � br,
holds. If we write q � e2πiσ, p � e2πiτ , with complex σ, τ , then gpxq is indeed doubly periodic
in x with periods σ�1 and τσ�1.
One usually requires ar�1 � q�N (N being a nonnegative integer), so that the sum of an
elliptic hypergeometric series is terminating, and hence convergent.
Very-well-poised elliptic hypergeometric series are defined as
r�1Vrpa1; a6, . . . , ar�1; q, pq :�
8̧
k�0
θ
�
a1q
2k; p
�
θ pa1; pq
pa1, a6, . . . , ar�1; q, pqk
pq, a1q{a6, . . . , a1q{ar�1; q, pqk
qk,
6 G. Bhatnagar and M.J. Schlosser
where
q2a26a
2
7 � � � a
2
r�1 � pa1qq
r�5.
Note that in the elliptic case the number of pairs of numerator and denominator parameters
involved in the construction of the very-well-poised term θpa1q
2k; pq{θpa1; pq is four (whereas in
the basic case this number is two, in the ordinary case only one). See Gasper and Rahman [17,
Chapter 11] for details. The notions of balancing, well-poisedness and very-well-poisedness were
explained from the point of view of elliptic functions for the first time in Spiridonov’s paper [40].
This justifies the notations 10V9 and 12V11 (corresponding to 8φ7 and 10φ9 in the p � 0 case) for
the series below.
Frenkel and Turaev [16] discovered the following 10V9 summation formula (as a result of
a more general 12V11 transformation), see [17, equation (11.4.1)]:
10V9
�
a; b, c, d, e, q�N ; q, p
�
�
paq, aq{bc, aq{bd, aq{cd; q, pqN
paq{b, aq{c, aq{d, aq{bcd; q, pqN
, (2.9)
where a2qN�1 � bcde.
It should be clear that for p � 0 elliptic hypergeometric series reduce to basic hypergeometric
series. When the reference point is a basic hypergeometric result, we refer to the result by
adding the word “elliptic” to it. For example, we refer to Frenkel and Turaev’s [16] elliptic
extension of Bailey’s 10φ9 transformation formula [17, equation (11.5.1)] as the elliptic Bailey
10φ9 transformation. This formula transforms a 12V11 series which is terminating, balanced and
very-well-poised to a multiple of another series of the same kind. Similarly, we refer to elliptic
extensions of Bressoud’s matrix.
Next we note some identities useful in our calculations with multiple series.
The summand of the series we consider typically has a factor of the form
¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
.
This is referred to as the elliptic Vandermonde product of type A. When p � 0, it reduces to
¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
.
Next we note two useful lemmas which in the p � 0 case were given by Milne [27]. The first
lemma shows an alternate way of writing the elliptic Vandermonde product.
Lemma 2.1. We have
n¹
r,s�1
pqxr{xs; q, pqkr�ks �
¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
� p�1qpn�1q|k|q
n°
r�1
pr�1qkr�pn�1q
n°
r�1
pkr2 q�
°
r s
krks
n¹
r�1
xnkr�|k|r .
The p � 0 case of this identity appeared as [27, Lemma 3.12]. Its proof in the elliptic case
proceeds along the same lines.
Next, we have a lemma that extends the elementary identity [17, equation (I.12)]
1
pq; qqN�k
�
�
q�N ; q
�
k
pq; qqN
p�1qkqNk�p
k
2q.
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 7
Lemma 2.2. We have
n¹
r,s�1
1
pq1�kr�ksxr{xs; q, pqNr�kr
�
¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
�
q�Nsxr{xs; q, p
�
kr
pqxr{xs; q, pqNr
� p�1q|k|q
|N ||k|�p|k|2 q�
n°
r�1
pr�1qkr
.
The special case Nr � kr appears in Rosengren [31, equation (3.8)]. When p � 0, this reduces
to Milne [27, Lemma 4.3]. The proof uses Lemma 2.1 but otherwise just proceeds as in the p � 0
case.
Next, as motivation, we outline the basic hypergeometric case of our work, for n � 1.
3 The WP Bailey transform and lemma:
a very short introduction
Here we outline an approach to the theory of basic hypergeometric series beginning with Jack-
son’s sum. From this result one can extract all the components of the well-poised Bailey trans-
form and lemma. Our purpose here is to explain and motivate the steps of our approach to find
analogous results on root systems. This should help the reader get a bird’s eye view of our work
in later sections.
The idea for the WP Bailey transform and lemma was given by Bailey [6] and explained (and
then extended) by Andrews [2]. It was further studied by various authors (see [3, 21, 23, 39,
44, 47, 52] for a small selection of references). The matrix inverse due to Bressoud [11] is an
important ingredient. The ideas presented below are primarily based on Andrews [2], but they
have been sequenced in a way to explain our work in later sections.
The essential idea is as follows. A pair of sequences α � pαkq and β � pβkq is given which
follows a relationship of the form
βN �
Ņ
j�0
BNjαj , (3.1)
where B � pBkjq is an infinite lower-triangular matrix called the Bressoud matrix (to be defined
shortly), with entries indexed by k and j. The entries Bkj � Bkjpa, bq of the matrix depend on
two parameters a and b (in addition to the parameter q). This relationship is called the Bailey
transform, since it transforms a sequence into another sequence. Given B, the pair of sequences
pαk, βkq is called a WP Bailey pair. The WP Bailey lemma is a method to construct sequences
pα1k, β
1
kq that also form a WP Bailey pair. The αk and βk are also dependent on a and b. B is
a lower triangular matrix, so the equation (3.1) corresponds to the matrix equation β � Bα.
For our purposes, it is useful to note that one can extract the Bressoud matrix and a key
WP Bailey pair from Jackson’s sum, the p � 0 case of (2.9) (given in Gasper and Rahman [17,
equation (2.6.2)]),
Ņ
k�0
p1�aq2kq
�
a, b, c, d, a2q1�N{bcd, q�N ; q
�
k
p1�aqpq, aq{b, aq{c, aq{d, bcdq�N{a, aqN�1; qqk
qk�
paq, aq{bc, aq{bd, aq{cd; qqN
paq{b, aq{c, aq{d, aq{bcd; qqN
. (3.2)
The b ÞÑ qa2{bcd case of (3.2) may be written in the form (3.1), where the Bressoud matrix
B � pBkjpa, bqq is defined as
Bkjpa, bq :�
pb; qqj�kpb{a; qqk�j
paq; qqj�kpq; qqk�j
, (3.3)
8 G. Bhatnagar and M.J. Schlosser
and the sequences pakq and pbkq are defined as
αkpa, bq :�
1� aq2k
1� a
�
a, c, d, a2q{bcd; q
�
k
pq, aq{c, aq{d, bcd{a; qqk
�
b
a
k
, (3.4a)
βkpa, bq :�
pb, bc{a, bd{a, aq{cd; qqk
pq, aq{c, aq{d, bcd{a; qqk
. (3.4b)
Note that, in view of (2.8), Bkjpa, bq � 0 unless k ¥ j, so B is indeed a lower-triangular matrix.
If B, pαkq and pβkq satisfy (3.1), we say pαkq and pβkq form a WP Bailey pair. Observe that
if we set d � aq{c, we obtain the unit WP Bailey pair
αkpa, bq :�
1� aq2k
1� a
pa, a{b; qqk
pq, bq; qqk
�
b
a
k
,
βkpa, bq :� δk,0 �
#
1 if k � 0,
0 otherwise.
The fact that this is a WP Bailey pair translates into an expression of the form
Ņ
j�0
BNjpa, bqαjpa, bq � δN,0. (3.5)
One can view this as a matrix inversion and from here obtain an explicit formula for the inverse
of B. To do that, we replace N by N �K, shift the index, and after a change of variables (see
remarks below), write this sum in the form
Ņ
j�K
BNjB
�1
jK � δN,K .
From here, one can read off the formula for the inverse B�1 of the matrix B. The entries of the
(uniquely determined) inverse are given by
pBpa, bqq�1
kj �
1� aq2k
1� a
1� bq2j
1� b
pa; qqj�kpa{b; qqk�j
pbq; qqj�kpq; qqk�j
�
b
a
k�j
. (3.6)
Remarks.
1. The entries of the inverse of the Bressoud matrix in (3.6) can be computed as follows:
In (3.5), replace N by N �K, a by aq2K , b by bq2K , and shift the index j ÞÑ j�K. Then
by
BN�K,j�K
�
aq2K , bq2K
�
�
paq; qq2K
pb; qq2K
BN,jpa, bq
it follows that
paq; qq2K
pb; qq2K
αj�K
�
aq2K , bq2K
�
can be identified as the pj,Kq entry of the inverse Bressoud matrix B�1. The details of
the analogous computation in our work, for example, in the proof of Corollary 4.5 below,
vary slightly from those given here, but the essential idea is the same.
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 9
2. By comparing the entries of B and B�1, one sees that they are almost the same. In fact,
Bressoud [11] expressed them more symmetrically. Let Mpa, bq � pMkjpa, bqq with
Mkjpa, bq :�
1� aq2j
1� a
pb; qqj�kpb{a; qqk�j
paq; qqj�kpq; qqk�j
ak�j .
Then Bressoud showed that Mpb, cqMpa, bq � Mpa, cq. In particular, this implies Mpb, aq
is the inverse of Mpa, bq. However, in our work, we find it beneficial to follow the exposi-
tion of Andrews [2] rather than Bressoud’s symmetric formulation. Both approaches are
equivalent.
Given B�1, and the WP Bailey pair pαk, βkq as in (3.4), one has the inverse relation
αN �
Ņ
j�0
pBpa, bqq�1
Njβj .
This is again equivalent to Jackson’s sum (3.2).
Given a Bailey pair, the WP Bailey lemma (see Andrews [2, Theorem 7]) gives a method to
construct a new WP Bailey pair, with two additional parameters ρ1 and ρ2, given by
α1N pa, bq :�
pρ1, ρ2; qqN
paq{ρ1, aq{ρ2; qqN
�
aq
ρ1ρ2
N
αN pa, bρ1ρ2{aqq,
β1N pa, bq :�
pbρ1{a, bρ2{a; qqN
paq{ρ1, aq{ρ2; qqN
Ņ
k�0
�
pρ1, ρ2; qqk
pbρ1{a, bρ2{a; qqk
pb; qqk�N paq{ρ1ρ2; qqN�k
pbρ1ρ2{a; qqk�N pq; qqN�k
�
1� bρ1ρ2q
2k{aq
1� bρ1ρ2{aq
�
aq
ρ1ρ2
k
βkpa, bρ1ρ2{aqq
.
The WP Bailey lemma is the assertion that α1kpa, bq and β1kpa, bq also form a WP Bailey pair.
This step too depends on Jackson’s sum (3.2).
If one begins with the Bailey pair pαkq and pβkq given by (3.4), then substituting the WP pair
α1kpa, bq and β1kpa, bq into the definition of a WP Bailey pair gives a transformation equivalent
to Bailey’s 10φ9 transformation [17, equation (2.9.1)]:
Ņ
k�0
p1� aq2kq
�
a, b, c, d, e, f, λaq1�N{ef, q�N ; q
�
k
p1� aqpq, aq{b, aq{c, aq{d, aq{e, aq{f, efq�N{λ, aqN�1; qqk
qk
�
paq, aq{ef, λq{e, λq{f ; qqN
paq{e, aq{f, λq, λq{ef ; qqN
�
Ņ
k�0
p1� λq2kq
�
λ, λb{a, λc{a, λd{a, e, f, λaq1�N{ef, q�N ; q
�
k
p1� λqpq, aq{b, aq{c, aq{d, λq{e, λq{f, efq�N{a, λqN�1; qqk
qk, (3.7)
where λ � qa2{bcd.
The main summation and transformation formulas of basic hypergeometric series now follow
from (3.7). For example, Watson’s q-analog of Whipple’s transformation formula follows by
taking the limit as d Ñ 8, and relabeling the parameters. Other key identities such as the
terminating, very well-poised 6φ5 summation and the terminating, balanced 3φ2 summation are
immediate consequences of Watson’s transformation formula. (Note that in the case of elliptic
hypergeometric series, one cannot let parameters go to 0 or 8.)
To summarize, a special case of the Jackson summation yields a Bressoud matrix as well
as a WP Bailey pair. A further special case allows us to compute the inverse of the matrix.
10 G. Bhatnagar and M.J. Schlosser
Another application of Jackson’s sum is used to find the WP Bailey lemma. And finally, an
application of this yields Bailey’s 10φ9 transformation formula.
The ideas outlined above extend immediately to elliptic hypergeometric series. This was
shown by Spiridonov [39]. Here we begin with Frenkel and Turaev’s [16] 10V9 summation [17,
equation (11.4.1)], equation (2.9) above, the elliptic extension of Jackson’s sum. Our goal is to
extend this analysis to elliptic extensions of multiple basic hypergeometric series associated with
root systems. We take a step in this direction in the next section.
4 Consequences of an An elliptic Jackson summation
of Rosengren
When the dimension n � 1, the WP Bailey transform and lemma are consequences of Jackson’s
sum. In this section, we consider one of Rosengren’s [31] An elliptic Jackson sums, and investigate
whether the ideas of Section 3 can be extended to this setting.
A multiple series extension of (3.1) is as follows
βN �
¸
0¤jr¤Nr
r�1,2,...,n
BNjαj . (4.1)
Here the sequences α � pαkq and β � pβkq are indexed by n-tuples k with non-negative integer
components. The rows and columns of the matrix B � pBkjq are indexed by n-tuples k and j of
non-negative integers. Following Milne [27], one can consider these n-tuples to be ordered lexi-
cographically. With this ordering the matrix operations can be carried out in the usual manner.
Moreover, the matrix is lower-triangular, so BNj � 0 if j ¡ N . The entries Bkj � Bkjpa, bq,
and the sequences αk � αkpa, bq and βk � βkpa, bq depend on two parameters a and b (in
addition to p and q and perhaps other parameters).
The An elliptic Jackson summation theorem that we use is due to Rosengren [31, Corol-
lary 5.3], its p � 0 case being due to Milne [25]. Rosengren’s result is
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
�
q�Nsxr{xs; q, p
�
kr
pqxr{xs; q, pqkr
�
n¹
r�1
θ
�
axrq
kr�|k|; p
�
θ paxr; pq
n¹
r�1
paxr; q, pq|k|
�
dxr, a
2xrq
1�|N |{bcd; q, p
�
kr
paxrq1�Nr ; q, pq|k|paxrq{b, axrq{c; q, pqkr
�
pb, c; q, pq|k|�
aq{d, bcdq�|N |{a; q, p
�
|k|
q
n°
r�1
rkr
�
�
paq{bd, aq{cd; q, pq|N |
paq{d, aq{bcd; q, pq|N |
n¹
r�1
paxrq, axrq{bc; q, pqNr
paxrq{b, axrq{c; q, pqNr
. (4.2)
To extract the An extension of Bressoud’s matrix and the definition of a WP Bailey pair, we
wish to write the case b ÞÑ qa2{bcd of (4.2) in the form of (4.1). After multiplying both sides by
n±
r�1
pbxr; q, pq|N |
n±
r,s�1
pqxr{xs; q, pqNr
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 11
and rearranging factors, we obtain
pb{a; q, pq|N |
n±
r,s�1
pqxr{xs; q, pqNr
n¹
r�1
pbxr; q, pq|N |
paxrq; q, pqNr
�
¸
0¤jr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
θ
�
qjr�jsxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
�
q�Nsxr{xs; q, p
�
jr
pqxr{xs; q, pqjr
�
n¹
r�1
θ
�
axrq
jr�|j|; p
�
θ paxr; pq
paxr; q, pq|j|
�
dxr, bxrq
|N |; q, p
�
jr
paxrq1�Nr ; q, pq|j|pbcdxr{a, axrq{c; q, pqjr
�
�
qa2{bcd, c; q, p
�
|j|�
aq{d, aq1�|N |{b; q, p
�
|j|
q
n°
r�1
rjr
�
�
pbc{a, aq{cd; q, pq|N |
paq{d; q, pq|N |
n±
r,s�1
pqxr{xs; q, pqNr
n¹
r�1
pbxr; q, pq|N |pbdxr{a; q, pqNr
pbcdxr{a, axrq{c; q, pqNr
. (4.3)
For a moment, ignore the product in front of the sum and compare the rest with the form (4.1).
It is easy to separate the terms in the summand that depend on both N and j (and so appear
as a part of BNj) and the others that depend on j (and thus comprise αj). The product on the
right-hand side depends only on N , of course.
We use the elementary identity (2.6) to combine terms, for example:
n¹
r�1
pbxr; q, pq|N |
�
bxrq
|N |; q, p
�
jr
�
n¹
r�1
pbxr; q, pq|N |�jr
.
Further, we use (2.5) to reverse the products in
�
aq1�|N |{b; q, p
�
|j|
. We also require Lemma 2.2.
In this manner, we obtain an equation of the form¸
0¤jr¤Nr
r�1,2,...,n
B
p1q
Njpa, bqαjpa, bq � βN pa, bq,
where the matrix Bp1q � pB
p1q
kj pa, bqq is defined in (4.4), and the sequences αkpa, bq and βkpa, bq
are as defined by (4.7). These considerations motivate the following definition and Theorem 4.3
below.
Definition 4.1 (an An elliptic Bressoud matrix). Let Bp1q �
�
B
p1q
kj pa, bq
�
with entries indexed
by pk, jq be defined as
B
p1q
kj pa, bq :�
pb{a; q, pq|k|�|j|
n±
r,s�1
pq1�jr�jsxr{xs; q, pqkr�jr
n¹
r�1
pbxr; q, pqjr�|k|
paxrq; q, pqkr�|j|
. (4.4)
We call Bp1q an elliptic Bressoud matrix, because it reduces to a form equivalent to (3.3)
when n � 1 and p � 0. The label An is placed to indicate that it is associated with An series.
An equivalent form of the p � 0 case of this matrix appeared in Milne [27, Theorem 3.41].
For some applications, it helps to use Lemma 2.2 to rewrite the terms of the matrix Bp1q as
follows
B
p1q
kj pa, bq �
pb{a; q, pq|k|
n±
r,s�1
pqxr{xs; q, pqkr
n¹
r�1
pbxr; q, pq|k|
paxrq; q, pqkr
12 G. Bhatnagar and M.J. Schlosser
�
¹
1¤r s¤n
θ
�
qjr�jsxr{xs; p
�
θ pxr{xs; pq
n¹
r�1
�
bxrq
|k|; q, p
�
jr
paxrq1�kr ; q, pq|j|
�
n±
r,s�1
�
q�ksxr{xs; q, p
�
jr�
aq1�|k|{b; q, p
�
|j|
�a
b
|j|
q
n°
r�1
rjr
. (4.5)
Definition 4.2 (WP Bailey pair with respect to a Bressoud matrix). Two sequences αN pa, bq
and βN pa, bq are said to form a WP Bailey pair with respect to a Bressoud matrix B if
βN pa, bq �
¸
0¤jr¤Nr
r�1,2,...,n
BNjpa, bqαjpa, bq. (4.6)
As we shall see, there are many multivariable Bressoud matrices. That is why we find it
necessary to mention the matrix B with respect to which the sequences form a WP Bailey pair.
Theorem 4.3 (an elliptic WP Bailey pair with respect to Bp1q). The following sequences
αkpa, bq :�
n¹
r�1
θ
�
axrq
kr�|k|; p
�
θ paxr; pq
paxr; q, pq|k|pdxr; q, pqkr
paxrq{c, bcdxr{a; q, pqkr
�
�
c, a2q{bcd; q, p
�
|k|
paq{d; q, pq|k|
n±
r,s�1
pqxr{xs; q, pqkr
�
b
a
|k|
, (4.7a)
and
βkpa, bq :�
pbc{a, aq{cd; q, pq|k|
paq{d; q, pq|k|
n±
r,s�1
pqxr{xs; q, pqkr
n¹
r�1
pbxr; q, pq|k|pbdxr{a; q, pqkr
paxrq{c, bcdxr{a; q, pqkr
, (4.7b)
form a WP Bailey pair with respect to Bp1q.
Proof. We have already indicated how to discover this theorem. Alternatively, we can verify
the theorem as follows. With αjpa, bq as above, we compute the sum¸
0¤jr¤Nr
r�1,2,...,n
BNjpa, bqαjpa, bq, (4.8)
with B replaced by Bp1q to calculate βN pa, bq. (It is helpful to take the form (4.5) for B
p1q
Njpa, bq.)
The sum can be summed using the b ÞÑ qa2{bcd case of (4.2). After canceling some factors, we
immediately obtain the expression in (4.7b) (with k replaced by N). �
As a corollary, we obtain a unit WP Bailey pair.
Corollary 4.4. The two sequences
αkpa, bq :�
n¹
r�1
θ
�
axrq
kr�|k|; p
�
θ paxr; pq
paxr; q, pq|k|
pbxrq; q, pqkr
�
pa{b; q, pq|k|
n±
r,s�1
pqxr{xs; q, pqkr
�
b
a
|k|
,
and
βkpa, bq :�
n¹
r�1
δkr,0,
form a WP Bailey pair with respect to Bp1q.
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 13
Proof. Take d � aq{c in (4.7) to obtain this simpler WP Bailey pair. �
Take d � aq{c in (4.3) to obtain an equivalent form of Corollary 4.4. This is an expression
of the form
¸
0¤jr¤Nr
r�1,2,...,n
FNjαj �
n¹
r�1
δNr,0. (4.9)
One can view this as a matrix inversion and from here obtain an explicit formula for the inverse
of Bp1q.
Corollary 4.5 (inverse of Bp1q). Let Bp1q �
�
B
p1q
kj pa, bq
�
be defined by (4.4). Then the entries
of the inverse are given by
�
Bp1qpa, bq
��1
kj
�
n¹
r�1
θ
�
axrq
kr�|k|; p
�
θ paxr; pq
θ
�
bxrq
jr�|j|; p
�
θ pbxr; pq
�
�
b
a
|k|�|j|
�
pa{b; q, pq|k|�|j|
n±
r,s�1
pq1�jr�jsxr{xs; q, pqkr�jr
n¹
r�1
paxr; q, pqjr�|k|
pbxrq; q, pqkr�|j|
. (4.10)
Remark. When p � 0, this matrix inversion is equivalent to a result of Milne [27, Theorem 3.41].
Rosengren and the second author [35] have proved more general elliptic matrix inversions, which
contain the inverses of most of the matrices in this paper.
Proof. Take d � aq{c in (4.3) to obtain an expression of the form (4.9). We replace Nr by
Nr �Kr, for r � 1, 2, . . . , n and obtain an expression of the form
¸
0¤jr¤Nr�Kr
r�1,2,...,n
FpN�Kq,jαj �
n¹
r�1
δNr,Kr .
We shift the indices to write it as
¸
Kr¤jr¤Nr
r�1,2,...,n
FpN�Kq,pj�Kqαj�K �
n¹
r�1
δNr,Kr .
Next, we substitute xr ÞÑ xrq
Kr , for r � 1, 2, . . . , n, a ÞÑ aq|K|, b ÞÑ bq|K|, and simplify terms
using (2.6) to obtain
¸
Kr¤jr¤Nr
r�1,2,...,n
�
pb{a; q, pq|N |�|j|
n±
r,s�1
pq1�jr�jsxr{xs; q, pqNr�jr
n¹
r�1
pbxr; q, pqjr�|N |
paxrq; q, pqNr�|j|
�
n¹
r�1
θ
�
axrq
jr�|j|; p
�
θ paxr; pq
θ
�
bxrq
Kr�|K|; p
�
θ pbxr; pq
�
�
b
a
|j|�|K|
�
pa{b; q, pq|j|�|K|
n±
r,s�1
pq1�Kr�Ksxr{xs; q, pqjr�Kr
n¹
r�1
paxr; q, pqKr�|j|
pbxrq; q, pqjr�|K|
�
�
n¹
r�1
δNr,Kr .
From here it is easy to read off the entries of the inverse matrix. �
14 G. Bhatnagar and M.J. Schlosser
Consider the inverse relation
αN pa, bq �
¸
0¤jr¤Nr
r�1,2,...,n
pBpa, bqq�1
Njβjpa, bq, (4.11)
where B � Bp1q, and αk and βk are defined as in Theorem 4.3 and pBp1qpa, bqq�1
kj is given
by (4.10). Next, simplify using (2.4), (2.6) and Lemma 2.2. If we now make the substitutions
a ÞÑ qa2{bcd, b ÞÑ c, c ÞÑ aq{bd and d ÞÑ aq{bc, we again obtain (4.2). This is an interesting
symmetry of Rosengren’s result.
Theorem 4.6 (an elliptic
�
Bp1q Ñ Bp1q
�
WP Bailey lemma). Suppose αN pa, bq and βN pa, bq
form a WP Bailey pair with respect to the matrix Bp1q. Let α1N pa, bq and β1N pa, bq be defined as
follows
α1N pa, bq :�
pρ1; q, pq|N |
paq{ρ2; q, pq|N |
n¹
r�1
pρ2xr; q, pqNr
paxrq{ρ1; q, pqNr
�
�
aq
ρ1ρ2
|N |
αN pa, bρ1ρ2{aqq, (4.12a)
β1N pa, bq :�
pbρ1{a; q, pq|N |
paq{ρ2; q, pq|N |
n¹
r�1
pbρ2xr{a; q, pqNr
paxrq{ρ1; q, pqNr
�
¸
0¤kr¤Nr
r�1,2,...,n
�
pρ1; q, pq|k|
pbρ1{a; q, pq|k|
n¹
r�1
pρ2xr; q, pqkr
pbρ2xr{a; q, pqkr
�
n¹
r�1
θ
�
bρ1ρ2xrq
kr�|k|{aq; p
�
θ pbρ1ρ2xr{aq; pq
pbxr; q, pqkr�|N |
pbρ1ρ2xr{a; q, pq|k|�Nr
�
paq{ρ1ρ2; q, pq|N |�|k|
n±
r,s�1
pq1�kr�ksxr{xs; q, pqNr�kr
�
aq
ρ1ρ2
|k|
βkpa, bρ1ρ2{aqq
�
. (4.12b)
Then α1N pa, bq and β1N pa, bq also form a WP Bailey pair with respect to Bp1q.
Remark. When p � 0, n � 1 and x1 � 1, then Theorem 4.6 reduces to Theorem 7 of An-
drews [2]. When n � 1 and x1 � 1, then Theorem 4.6 reduces to the elliptic WP Bailey lemma
by Spiridonov [39, Theorem 4.3]. When p � 0, Theorem 4.6 reduces to an An WP Bailey
lemma given by Zhang and Liu [52]. A slightly different formulation of Theorem 4.6 appears in
unpublished notes of Warnaar [48].
Proof. Our proof is an extension of Andrews’ proof [2, Theorem 7]. We begin with the expres-
sion (4.12b) for β1N pa, bq. Substitute for βkpa, bρ1ρ2{aqq from (4.6) written in the form:
βkpa, bρ1ρ2{aqq �
¸
0¤jr¤kr
r�1,2,...,n
Bkjpa, bρ1ρ2{aqqαjpa, bρ1ρ2{aqq, (4.13)
with B replaced by Bp1q, to obtain a double sum. After interchanging the sums, the inner sum is
summed using (4.2) and the result can be recognized as the defining condition for a WP Bailey
pair with respect to the matrix Bp1q. The details are as follows.
We interchange the sums and shift the index using the following:¸
0¤kr¤Nr
r�1,2,...,n
¸
0¤jr¤kr
r�1,2,...,n
Aj,k �
¸
0¤jr¤Nr
r�1,2,...,n
¸
0¤kr¤Nr�jr
r�1,2,...,n
Aj,k�j .
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 15
We now need the following simplification which follows from Lemma 2.2, by replacing Nr ÞÑ
Nr � jr and xr ÞÑ xrq
jr , for r � 1, 2, . . . , n,
n¹
r,s�1
1
pq1�kr�ks�jr�jsxr{xs; q, pqNr�kr�jr
�
¹
1¤r s¤n
θ
�
qkr�ks�jr�jsxr{xs; p
�
θ pqjr�jsxr{xs; pq
�
n¹
r,s�1
�
q�Ns�jrxr{xs; q, p
�
kr
pq1�jr�jsxr{xs; q, pqNr�jr
� p�1q|k|q
|N ||k|�|k||j|�p|k|2 q�
n°
r�1
pr�1qkr
.
We also need the elementary identities (2.4) and (2.6). In this manner, we obtain the following
expression for β1N pa, bq
pbρ1{a; q, pq|N |
paq{ρ2; q, pq|N |
n¹
r�1
pbρ2xr{a; q, pqNr
paxrq{ρ1; q, pqNr
�
¸
0¤jr¤Nr
r�1,2,...,n
�
� pρ1; q, pq|j|paq{ρ1ρ2; q, pq|N |�|j|
pbρ1{a; q, pq|j|
n±
r,s�1
pq1�jr�jsxr{xs; q, pqNr�jr
�
n¹
r�1
pρ2xr; q, pqjrpbxr; q, pqjr�|N |pbρ1ρ2xr{aq; qqjr�|j|
pbρ2xr{a; q, pqjrpaxrq; q, pqjr�|j|pbρ1ρ2xr{a; q, pq|j|�Nr
�
n¹
r�1
θ
�
bρ1ρ2xrq
jr�|j|{aq; p
�
θ pbρ1ρ2xr{aq; pq
�
�
aq
ρ1ρ2
|j|
αjpa, bρ1ρ2{aqq
�
¸
0¤kr¤Nr�jr
r�1,2,...,n
� ¹
1¤r s¤n
θ
�
qjr�js�kr�ksxr{xs; p
�
θ pqjr�jsxr{xs; pq
n¹
r,s�1
�
q�Ns�jrxr{xs; q, p
�
kr
pq1�jr�jsxr{xs; q, pqkr
�
n¹
r�1
θ
�
bρ1ρ2xrq
jr�|j|�kr�|k|{aq; p
�
θ
�
bρ1ρ2xrqjr�|j|{aq; p
� n¹
r�1
�
bρ1ρ2xrq
jr�|j|{aq; q, p
�
|k|�
bρ1ρ2xrqNr�|j|{a; q, p
�
|k|
�
n¹
r�1
�
ρ2xrq
jr , bxrq
jr�|N |; q, p
�
kr�
bρ2xrqjr{a, axrq1�jr�|j|; q, p
�
kr
�
�
ρ1q
|j|, bρ1ρ2{a
2q; q, p
�
|k|�
bρ1q|j|{a, ρ1ρ2q|j|�|N |{a; q, p
�
|k|
q
n°
r�1
r kr
��
. (4.14)
The inner sum can be summed using (4.2). Take the equivalent formulation of (4.2) obtained by
replacing c by a2q1�|N |{bcd and use the following substitutions: xr ÞÑ xrq
jr and Nr ÞÑ Nr � jr
for r � 1, 2, . . . , n, a ÞÑ bρ1ρ2q
|j|{aq, b ÞÑ ρ1q
|j|, c ÞÑ ρ2, d ÞÑ bq|N |. In this manner, we find that
the inner sum in (4.14) equals�
b{a, ρ2q
�|N |{a; q, p
�
|N |�|j|�
ρ1ρ2q|j|�|N |{a, bρ1q|j|{a; q, p
�
|N |�|j|
n¹
r�1
�
bρ1ρ2xrq
jr�|j|{a, ρ1q
�Nr{axr; q, p
�
Nr�jr�
bρ2xrqjr{a, q�Nr�|j|{axr; q, p
�
Nr�jr
.
Now we use (2.4) and (2.6) to write the sum (and therefore, β1N pa, bq) in the form¸
0¤jr¤Nr
r�1,2,...,n
BNjα
1
jpa, bq,
where B � Bp1q and α1jpa, bq is defined by (4.12a). This shows that α1N pa, bq and β1N pa, bq form
a WP Bailey pair with respect to Bp1q. �
16 G. Bhatnagar and M.J. Schlosser
Remark 4.7. An elliptic An Bailey 10φ9 transformation formula follows immediately by apply-
ing the Bp1q Ñ Bp1q elliptic Bailey lemma in Theorem 4.6 to the WP Bailey pair in Theorem 4.3.
This An elliptic Bailey transformation is due to Rosengren [31, Corollary 8.1]. When p � 0,
this reduces to an An Bailey 10φ9 transformation formula found by Denis and Gustafson [14]
and independently, by Milne and Newcomb [30, Theorem 3.1]. When p � 0, this was noted
previously by Zhang and Liu [52].
We used an equivalent, altered formulation of the elliptic Jackson sum in (4.2) in our proof
of Theorem 4.6. More precisely, we altered (4.2) by a specific substitution of variables with the
effect that the occurrence of the nonnegative integer sequence N in the respective factors got
changed. By using the altered formulation of (4.2), we can extract another Bressoud matrix
and corresponding WP Bailey pair. The matrix Bp2q � pB
p2q
kj pa, bqq is defined as follows.
Definition 4.8 (an An elliptic Bressoud matrix). We define the matrix Bp2q with entries indexed
by pk, jq as
B
p2q
kj pa, bq :�
pb; q, pq|k|�|j|
n±
r�1
�
bq|k|�kr{axr; q, p
�
kr�jr
n±
r,s�1
pq1�jr�jsxr{xs; q, pqkr�jr
n±
r�1
paxrq; q, pqkr�|j|
. (4.15)
Theorem 4.9 (an elliptic WP Bailey pair with respect to Bp2q). The two sequences
αkpa, bq :�
n¹
r�1
θ
�
axrq
kr�|k|; p
�
θ paxr; pq
paxr; q, pq|k|pcxr, dxr; q, pqkr
pbcdxr{a; q, pqkr
�
�
a2q{bcd; q, p
�
|k|
paq{c, aq{d; q, pq|k|
n±
r,s�1
pqxr{xs; q, pqkr
� b
a
|k|
q
°
r s
krks
n¹
r�1
x�krr , (4.16a)
and
βkpa, bq :�
pb, bc{a, bd{a; q, pq|k|
paq{c, aq{d; q, pq|k|
n±
r,s�1
pqxr{xs; q, pqkr
n¹
r�1
�
aq1�|k|�kr{cdxr; q, p
�
kr
pbcdxr{a; q, pqkr
, (4.16b)
form a WP Bailey pair with respect to Bp2q.
Proof. The proof is analogous to that of Theorem 4.3. The proof requires (4.2). With αjpa, bq
as above, we compute the sum (4.8), with B � Bp2q, to calculate βN pa, bq. We take the altered
formulation of (4.2) obtained by replacing c by a2q1�|N |{bcd. The sum can be summed using
the b ÞÑ qa2{bcd case of this altered form of (4.2). After canceling some factors, we immediately
obtain the expression (4.16b) (with k replaced by N). �
Theorem 4.10 (an elliptic (Bp1q Ñ Bp2q) WP Bailey lemma). Suppose αN pa, bq and βN pa, bq
form a WP Bailey pair with respect to Bp1q. Let α1N pa, bq and β1N pa, bq be defined as follows
α1N pa, bq :�
n±
r�1
pρ1xr, ρ2xr; q, pqNr
paq{ρ1, aq{ρ2; q, pq|N |
�
aq
ρ1ρ2
|N | n¹
r�1
x�Nr
r � q
°
r s
NrNs
αN pa, bρ1ρ2{aqq, (4.17a)
β1N pa, bq :�
pbρ1{a, bρ2{a; q, pq|N |
paq{ρ1, aq{ρ2; q, pq|N |
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 17
�
¸
0¤kr¤Nr
r�1,2,...,n
� n±
r�1
pρ1xr, ρ2xr; q, pqkr
pbρ1{a, bρ2{a; q, pq|k|
pb; q, pq|N |�|k|
n±
r,s�1
pq1�kr�ksxr{xs; q, pqNr�kr
�
n¹
r�1
θ
�
bρ1ρ2xrq
kr�|k|{aq; p
�
θ pbρ1ρ2xr{aq; pq
n¹
r�1
�
aq1�|N |�Nr{ρ1ρ2xr; q, p
�
Nr�kr
pbρ1ρ2xr{a; q, pq|k|�Nr
�
�
aq
ρ1ρ2
|k| n¹
r�1
x�krr � q
°
r s
krks
βkpa, bρ1ρ2{aqq
�
. (4.17b)
Then α1N pa, bq and β1N pa, bq form a WP Bailey pair with respect to Bp2q.
Proof. The proof is analogous to the proof of Theorem 4.6. We begin with the expres-
sion (4.17b) for β1N pa, bq. Substitute for βkpa, bρ1ρ2{aqq in (4.13) with B � Bp1q. The inner
sum can be summed using (4.2). We take the altered formulation of (4.2) obtained by replac-
ing c by a2q1�|N |{bcd. We use the following substitutions: xr ÞÑ xrq
jr and Nr ÞÑ Nr � jr for
r � 1, 2, . . . , n, a ÞÑ bρ1ρ2q
|j|{aq, b ÞÑ bq|j|�|N |, c ÞÑ ρ1, d ÞÑ ρ2. Now after some simplification,
we find that α1N pa, bq and β1N pa, bq form a WP Bailey pair with respect to Bp2q. �
An elliptic An Bailey 10φ9 transformation formula due to Rosengren [31, Corollary 8.1] follows
immediately by applying the Bp1q Ñ Bp2q elliptic Bailey lemma in Theorem 4.10 to the WP
Bailey pair in Theorem 4.3. This An elliptic Bailey transformation formula is the same as
obtained in Remark 4.7.
We have seen two An Bressoud matrices which follow from the same An elliptic Jackson sum.
We also obtained a WP Bailey lemma that transforms a WP Bailey pair with respect to a mat-
rix Bp1q into a more complicated WP Bailey pair (with 2 additional parameters). However, this
time the WP Bailey pair is with respect to the matrix Bp2q. In the next section, we explore the
consequences of another An elliptic Jackson sum.
5 An An elliptic Bailey transformation
In the previous section, we examined the consequences of one of Rosengren’s elliptic Jackson
summation over An. In this section, we consider another An elliptic Jackson summation. We
will find that we can obtain some WP Bailey lemmas relating to the Bressoud matrices obtained
in Section 4. In addition, we find another useful Bressoud matrix closely related to Bp2q.
The An elliptic Jackson summation we use in this section is due to Rosengren and the second
author [35]. The p � 0 case is due to the second author [38]. We have
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
�
q�Nsxr{xs; q, p
�
kr
pqxr{xs; q, pqkr
�
n¹
r�1
pd{xr; q, pq|k|
�
a2xrq
|N |�1{bcd; q, p
�
kr
pbcd{axr; q, pq|k|�kr
pd{xr; q, pq|k|�krpaxrq{d; q, pqkrpbcdq
�Nr{axr; q, pq|k|
�
θ
�
aq2|k|; p
�
θ pa; pq
pa, b, c; q, pq|k|�
aq|N |�1, aq{b, aq{c; q, p
�
|k|
q
n°
r�1
rkr
�
�
paq, aq{bc; q, pq|N |
paq{b, aq{c; q, pq|N |
n¹
r�1
paxrq{bd, axrq{cd; q, pqNr
paxrq{d, axrq{bcd; q, pqNr
. (5.1)
18 G. Bhatnagar and M.J. Schlosser
Observe that (5.1) has a simpler very-well-poised part, namely
θ
�
aq2|k|; p
�
θ pa; pq
compared to the one in (4.2), given by
n¹
r�1
θ
�
axrq
kr�|k|; p
�
θ paxr; pq
.
We begin with a WP Bailey lemma which follows from (5.1).
Theorem 5.1 (an elliptic
�
Bp2q Ñ Bp1q
�
WP Bailey lemma). Suppose αN pa, bq and βN pa, bq
form a WP Bailey pair with respect to the matrix Bp2q. Let α1N pa, bq and β1N pa, bq be defined as
follows
α1N pa, bq :�
pρ1, ρ2; q, pq|N |
n±
r�1
paxrq{ρ1, axrq{ρ2; q, pqNr
�
aq
ρ1ρ2
|N | n¹
r�1
xNr
r � q
�
°
r s
NrNs
αN pa, bρ1ρ2{aqq,
β1N pa, bq :�
pbρ1{a, bρ2{a; q, pq|N |
n±
r�1
paxrq{ρ1, axrq{ρ2; q, pqNr
�
¸
0¤kr¤Nr
r�1,2,...,n
�
pρ1, ρ2; q, pq|k|
pbρ1{a, bρ2{a; q, pq|k|
θ
�
bρ1ρ2q
2|k|{aq; p
�
θ pbρ1ρ2{aq; pq
�
n±
r�1
pbxr; q, pq|N |�kr
�
axrq
1�kr�|k|{ρ1ρ2; q, p
�
Nr�kr
pbρ1ρ2{a; q, pq|N |�|k|
n±
r,s�1
pq1�kr�ksxr{xs; q, pqNr�kr
�
�
aq
ρ1ρ2
|k| n¹
r�1
xkrr � q
�
°
r s
krks
βkpa, bρ1ρ2{aqq
�
.
Then α1N pa, bq and β1N pa, bq form a WP Bailey pair with respect to Bp1q.
Proof. The proof is analogous to the proof of Theorem 4.6. The only difference is that this
time we sum the inner sum using (5.1), with the substitutions: xr ÞÑ xrq
jr and Nr ÞÑ Nr � jr
for r � 1, 2, . . . , n, a ÞÑ bρ1ρ2q
2|j|{aq, b ÞÑ ρ1q
|j|, c ÞÑ ρ2q
|j|, d ÞÑ bρ1ρ2q
|j|{a2q. The rest of the
computations are very similar. �
A new elliptic An Bailey 10φ9 transformation formula follows immediately by applying the
Bp2q Ñ Bp1q elliptic Bailey lemma in Theorem 5.1 to the WP Bailey pair in Theorem 4.9.
Theorem 5.2 (an An elliptic Bailey 10φ9 transformation). Let λ � qa2{bcd. Then
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
�
q�Nsxr{xs; q, p
�
kr
pqxr{xs; q, pqkr
�
n¹
r�1
θ
�
axrq
kr�|k|; p
�
θ paxr; pq
n¹
r�1
paxr; q, pq|k|
paxrq1�Nr ; q, pq|k|
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 19
�
n¹
r�1
�
cxr, dxr, λaxrq
|N |�1{ef ; q, p
�
kr
paxrq{b, axrq{e, axrq{f ; q, pqkr
�
pb, e, f ; q, pq|k|�
aq{c, aq{d, efq�|N |{λ; q, p
�
|k|
q
n°
r�1
rkr
�
�
pλq{e, λq{f ; q, pq|N |
pλq, λq{ef ; q, pq|N |
n¹
r�1
paxrq, axrq{ef ; q, pqNr
paxrq{e, axrq{f ; q, pqNr
�
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
�
q�Nsxr{xs; q, p
�
kr
pqxr{xs; q, pqkr
�
θ
�
λq2|k|; p
�
θ pλ; pq
pλ, λc{a, λd{a, e, f ; q, pq|k|�
λq|N |�1, aq{c, aq{d, λq{e, λq{f ; q, p
�
|k|
�
n¹
r�1
pλb{axr; q, pq|k|
�
λaxrq
|N |�1{ef ; q, p
�
kr
pefq�Nr{axr; q, pq|k|paxrq{b; q, pqkr
�
n¹
r�1
pef{axr; q, pq|k|�kr
pλb{axr; q, pq|k|�kr
� q
n°
r�1
rkr
�
. (5.2)
Remark. When p � 0 and n � 1, this reduces to an equivalent form of Bailey’s 10φ9 transfor-
mation formula (3.7), given in [17, equation (2.9.1)]. After discovering this result, the authors
were informed by Rosengren [33] that he obtained the same result by following the approach
used in [10, 31]. Other elliptic Bailey transformations on root systems were given previously by
Rosengren [31, 34].
Proof. This elliptic An Bailey 10φ9 transformation formula is obtained by applying the Bp2q Ñ
Bp1q elliptic Bailey lemma in Theorem 5.1 to the WP Bailey pair with respect to Bp2q in Theo-
rem 4.9. The result is a WP Bailey pair with respect to the matrix Bp1q. Written explicitly, this
is an equivalent form of (5.2). The details are as follows.
First write the relation (4.6) explicitly
β1N pa, bq �
¸
0¤jr¤Nr
r�1,2,...,n
B
p1q
Njpa, bqα
1
jpa, bq,
with B
p1q
Njpa, bq written in the form (4.5), and α1k and β1k given by Theorem 5.1. Here, replace
αkpa, bρ1ρ2{aqq, βkpa, bρ1ρ2{aqq by the corresponding expressions from Theorem 4.9 (with b
replaced by bρ1ρ2{aq). After some algebraic calculations involving the use of (2.3), (2.6), and
Lemma 2.2, we obtain a formula resembling (5.2). Next, set ρ1 � e, ρ2 � f and b ÞÑ bq�|N |.
Finally, replace b by a3q2�|N |{bcdef � λaq1�|N |{ef , where λ � qa2{bcd, to obtain (5.2). �
Using an analytic continuation argument, we can write (5.2) with sums over an n-simplex.
Theorem 5.3 (an An elliptic Bailey 10φ9 transformation). Let λ � qa2{bcd. Then
¸
0¤|k|¤N
k1,k2,...,kn¥0
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
pfsxr{xs; q, pqkr
pqxr{xs; q, pqkr
�
n¹
r�1
θ
�
axrq
kr�|k|; p
�
θ paxr; pq
n¹
r�1
paxr; q, pq|k|
paxrq{fr; q, pq|k|
20 G. Bhatnagar and M.J. Schlosser
�
n¹
r�1
�
cxr, dxr, λaxrq
N�1{ef1 � � � fn; q, p
�
kr
paxrq{b, axrq{e, axrq1�N ; q, pqkr
�
�
b, e, q�N ; q, p
�
|k|
paq{c, aq{d, ef1 � � � fnq�N{λ; q, pq|k|
q
n°
r�1
rkr
�
�
pλq{e, λq{f1 � � � fn; q, pqN
pλq, λq{ef1 � � � fn; q, pqN
n¹
r�1
paxrq, axrq{efr; q, pqN
paxrq{e, axrq{fr; q, pqN
�
¸
0¤|k|¤N
k1,k2,...,kn¥0
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
pfsxr{xs; q, pqkr
pqxr{xs; q, pqkr
�
θ
�
λq2|k|; p
�
θ pλ; pq
�
λ, λc{a, λd{a, e, q�N ; q, p
�
|k|
pλq{f1 � � � fn, aq{c, aq{d, λq{e, λq1�N ; q, pq|k|
�
n¹
r�1
pλb{axr; q, pq|k|
�
eq�N{axr; q, p
�
|k|�kr
pλb{axr; q, pq|k|�krpefrq
�N{axr; q, pq|k|
�
n¹
r�1
�
λaxrq
N�1{ef1 � � � fn; q, p
�
kr
paxrq{b; q, pqkr
� q
n°
r�1
rkr
�
. (5.3)
Proof. Denote the left-hand side minus the right-hand side of the transformation in (5.3)
by F pf1, . . . , fnq. Now F is meromorphic in each of the variables f1, . . . , fn in the domain
0 |fs| 8, for 1 ¤ s ¤ n. Using (2.7), it can be easily checked that F is periodic in each fs,
i.e.,
F pf1, . . . , pfs, . . . , fnq � F pf1, . . . , fnq,
for s � 1, . . . , n. For technical reasons we shall first assume that p, q are chosen such that pmqn
are distinct for all integers m and n.
We will first demonstrate the existence of a convergent sequence of distinct points, such
that F is zero when f1 is equal to any term of that sequence. This will imply that F is zero
for any f1 where F is defined. It follows from the f � q�N case of Theorem 5.2 that F is
zero for pf1, . . . , fnq �
�
q�N1 , . . . , q�Nn
�
, where N1, . . . , Nn are nonnegative integers. Now for
every r � 0, 1, 2, . . . , there is an integer mr such that if zr � pmrq�r, then p ¤ |zr| ¤ 1. By
periodicity of F , F must be 0 when f1 � zr, for every r. Since there is an infinite number of zr
in the annulus p ¤ |z| ¤ 1, the zr must have a limit point. Thus there is a convergent infinite
subsequence pzrkq of distinct points, such that F is 0 on this subsequence, and therefore F must
be zero for any f1, as required.
By iterating this argument for f2, . . . , fn, we conclude that F is identically zero for f1, . . . , fn.
By analytic continuation this is extended to the degenerate cases where the pmqn are not all
different (for various m, n), as long as those choices for p, q leave F well-defined. �
Some consequences of Theorem 5.2 are noted in Section 6. Next we have an An elliptic
well-poised Bp2q Ñ Bp2q Bailey lemma.
Theorem 5.4 (an elliptic (Bp2q Ñ Bp2q) WP Bailey lemma). Suppose αN pa, bq and βN pa, bq
form a WP Bailey pair with respect to the matrix Bp2q. Let α1N pa, bq and β1N pa, bq be defined as
follows
α1N pa, bq :�
pρ1; q, pq|N |
paq{ρ2; q, pq|N |
n¹
r�1
pρ2xr; q, pqNr
paxrq{ρ1; q, pqNr
�
�
aq
ρ1ρ2
|N |
αN pa, bρ1ρ2{aqq,
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 21
β1N pa, bq :�
pbρ2{a; q, pq|N |
paq{ρ2; q, pq|N |
n¹
r�1
�
bρ1q
|N |�Nr{axr; q, p
�
Nr
paxrq{ρ1; q, pqNr
�
¸
0¤kr¤Nr
r�1,2,...,n
�
pρ1; q, pq|k|
pbρ2{a; q, pq|k|
n¹
r�1
pρ2xr; q, pqkr
�
bρ1q
|N |{axr; q, p
�
|k|�kr�
bρ1q|N |�Nr{axr; q, p
�
|k|
�
θ
�
bρ1ρ2q
2|k|{aq; p
�
θ pbρ1ρ2{aq; pq
pb; q, pq|N |�|k|
pbρ1ρ2{a; q, pq|N |�|k|
�
paq{ρ1ρ2; q, pq|N |�|k|
n±
r,s�1
pq1�kr�ksxr{xs; q, pqNr�kr
�
aq
ρ1ρ2
|k|
βkpa, bρ1ρ2{aqq
�
.
Then α1N pa, bq and β1N pa, bq also form a WP Bailey pair with respect to Bp2q.
Proof. The proof is analogous to the proof of Theorem 4.6. However, this time we sum the
inner sum using (5.1). We use the following substitutions: xr ÞÑ xrq
jr and Nr ÞÑ Nr � jr for
r � 1, 2, . . . , n, a ÞÑ bρ1ρ2q
2|j|{aq, b ÞÑ ρ1q
|j|, c ÞÑ bq|N |�|j|, d ÞÑ bρ1ρ2q
|j|{a2q. The rest of the
proof is similar to the earlier one. �
On applying the Bp2q Ñ Bp2q elliptic Bailey lemma in Theorem 5.4 to the WP Bailey pair in
Theorem 4.9, we again obtain Theorem 5.2.
Next, we present another Bressoud matrix related to (5.1).
Definition 5.5 (an An Elliptic Bressoud matrix). We define the matrix Bp3q with entries
B
p3q
kj pa, bq as
B
p3q
kj pa, bq :�
n±
r�1
pbxr; q, pqjr�|k|
�
bxrq
jr�|j|{a; q, p
�
kr�jr
paq; q, pq|k|�|j|
n±
r,s�1
pq1�jr�jsxr{xs; q, pqkr�jr
. (5.4)
Remark. When p � 0 and further b � 0, this reduces to a multivariable Bailey transform
matrix given by Milne [27, Definition 8.24]. This matrix and its inverse were used in [8] to
derive a result related to (6.3) below.
Theorem 5.6 (an elliptic WP Bailey pair with respect to Bp3q). The two sequences
αkpa, bq :�
θ
�
aq2|k|; p
�
θ pa; pq
pa, c, d; q, pq|k|
paq{c, aq{d; q, pq|k|
n±
r,s�1
pqxr{xs; q, pqkr
�
n¹
r�1
�
a2q{bcdxr; q, p
�
|k|
pa2q{bcdxr; q, pq|k|�krpbcdxr{a; q, pqkr
�
�
b
a
|k|
q
�
°
r s
krks
n¹
r�1
xkrr ,
and
βkpa, bq :�
paq{cd; q, pq|k|
paq{c, aq{d; q, pq|k|
n¹
r,s�1
1
pqxr{xs; q, pqkr
n¹
r�1
pbxr; q, pq|k|pbcxr{a, bdxr{a; q, pqkr
pbcdxr{a; q, pqkr
,
form a WP-Bailey pair with respect to Bp3q.
22 G. Bhatnagar and M.J. Schlosser
Proof. The proof is analogous to that of Theorem 4.3. The proof requires the d ÞÑ qa2{bcd case
of (5.1), after interchanging b and d. We verify that αjpa, bq and βN pa, bq satisfy the defining
condition of a WP Bailey pair with respect to Bp3q. �
Corollary 5.7. The two sequences
αkpa, bq :�
θ
�
aq2|k|; p
�
θ pa; pq
pa; q, pq|k|
n¹
r,s�1
1
pqxr{xs; q, pqkr
�
n¹
r�1
pa{bxr; q, pq|k|
pa{bxr; q, pq|k|�krpbxrq; q, pqkr
�
�
b
a
|k|
q
�
°
r s
krks
n¹
r�1
xkrr ,
and
βkpa, bq :�
n¹
r�1
δkr,0,
form a WP-Bailey pair with respect to Bp3q.
Proof. Take d � aq{c in Theorem 5.6 to obtain this unit Bailey pair. �
As before, we can derive a formula for the inverse of Bp3q using this unit Bailey pair.
Corollary 5.8 (inverse of Bp3q). Let Bp3q �
�
B
p3q
kj pa, bq
�
be defined by (5.4). Then the entries
of its inverse are given by
�
Bp3qpa, bq
��1
kj
�
θ
�
aq2|k|; p
�
θ pa; pq
n¹
r�1
θ
�
bxrq
jr�|j|; p
�
θ pbxr; pq
�
�
b
a
|k|�|j|
q
°
r s
pjrjs�krksq
n¹
r�1
xkr�jrr
�
pa; q, pq|k|�|j|
n±
r�1
�
aq|k|�kr{bxr; q, p
�
kr�jr
n±
r,s�1
pq1�jr�jsxr{xs; q, pqkr�jr
n±
r�1
pbxrq; q, pqkr�|j|
. (5.5)
Proof. We first write Corollary 5.7 in the form
¸
0¤jr¤Nr
r�1,2,...,n
B
p3q
Njpa, bqαj �
n¹
r�1
δNr,0,
where αj is as in Corollary 5.7, and the entries of Bp3q is given by (5.4). We now replace N
by N �K, and shift the index by replacing j by j �K. The index of the sum now runs from
Kr ¤ jr ¤ Nr, for r � 1, 2, . . . , n. Next we take xr ÞÑ xrq
Kr , a ÞÑ aq2|K| and b ÞÑ bq|K| to
obtain, after some simplification, the sum
¸
Kr¤jr¤Nr
r�1,2,...,n
B
p3q
Njpa, bq
�
Bp3qpa, bq
��1
jK
pa, bq �
n¹
r�1
δNr,Kr , (5.6)
where
�
Bp3qpa, bq
��1
jK
is given by (5.5) with indices relabeled as pk, jq ÞÑ pj,Kq. �
Observe that the entries of pBp3qpa, bqq�1 consist of the entries of Bp2qpb, aq multiplied by some
additional factors, which can be separated into factors containing either terms with index j or
with index k. This can help us find the inverse of Bp2q.
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 23
Corollary 5.9 (inverse of Bp2q). Let Bp2q � pB
p2q
kj pa, bqq be defined by (4.15). Then the entries
of its inverse are given by
�
Bp2qpa, bq
��1
kj
�
n¹
r�1
θ
�
axrq
kr�|k|; p
�
θ paxr; pq
�
θ
�
bq2|j|; p
�
θ pb; pq
�
b
a
|k|�|j|
q
°
r s
pkrks�jrjsq
n¹
r�1
xjr�krr
�
n±
r�1
paxr; q, pqjr�|k|
�
axrq
jr�|j|{b; q, p
�
kr�jr
pbq; q, pq|k|�|j|
n±
r,s�1
pq1�jr�jsxr{xs; q, pqkr�jr
.
Proof. We can write the sum (5.6) in the form
¸
Kr¤jr¤Nr
r�1,2,...,n
�
B
p3q
Njpa, bqB
p2q
jKpb, aq
θ
�
aq2|j|; p
�
θ pa; pq
n¹
r�1
θ
�
bxrq
Kr�|K|; p
�
θ pbxr; pq
�
�
b
a
|j|�|K|
q
°
r s
pKrKs�jrjsq
n¹
r�1
xjr�Kr
r
�
�
n¹
r�1
δNr,Kr .
Interchanging a and b, this expression can be written as
n¹
r�1
θ
�
axrq
Kr�|K|; p
�
θ
�
axrqNr�|N |; p
� � � b
a
|K|�|N |
q
°
r s
pKrKs�NrNsq
n¹
r�1
xNr�Kr
r
�
¸
Kr¤jr¤Nr
r�1,2,...,n
�
B
p3q
Njpb, aqB
p2q
jKpa, bq
θ
�
bq2|j|; p
�
θ pb; pq
n¹
r�1
θ
�
axrq
Nr�|N |; p
�
θ paxr; pq
�
�
b
a
|N |�|j|
q
°
r s
pNrNs�jrjsq
n¹
r�1
xjr�Nr
r
�
�
n¹
r�1
δNr,Kr .
Note that when N � K, the (non-zero) factors outside the sum on the left-hand side reduce
to 1, and thus cancel. We can now read off the entries of the inverse of Bp2qpa, bq from this
expression. �
Consider the inverse relation (4.11) where B � Bp3q, and αk and βk are defined as in Theo-
rem 5.6 and pBp3qpa, bqq�1
kj is given by (5.5). After using Lemma 2.2 and canceling some products,
we make the substitutions a ÞÑ qa2{bcd, b ÞÑ a, c ÞÑ aq{bd and d ÞÑ aq{bc. Finally, we replace c
by a2q1�|N |{bcd to again obtain (4.2). Thus (4.2) and (5.1) are inverse relations. Indeed, Rosen-
gren and the second author originally obtained (5.1) by taking the inverse relation of (4.2) using
a matrix inversion equivalent to Bp2q.
Theorem 5.10 (an elliptic (Bp3q Ñ Bp3q) WP Bailey lemma). Suppose αN pa, bq and βN pa, bq
form a WP Bailey pair with respect to the matrix Bp3q. Let α1N pa, bq and β1N pa, bq be defined as
follows
α1N pa, bq :�
pρ1, ρ2; q, pq|N |
paq{ρ1, aq{ρ2; q, pq|N |
�
aq
ρ1ρ2
|N |
αN pa, bρ1ρ2{aqq,
β1N pa, bq :�
n±
r�1
pbρ1xr{a, bρ2xr{a; q, pqNr
paq{ρ1, aq{ρ2; q, pq|N |
24 G. Bhatnagar and M.J. Schlosser
�
¸
0¤kr¤Nr
r�1,2,...,n
�
pρ1, ρ2; q, pq|k|
n±
r�1
pbρ1xr{a, bρ2xr{a; q, pqkr
n¹
r�1
θ
�
bρ1ρ2xrq
kr�|k|{aq; p
�
θ pbρ1ρ2xr{aq; pq
�
paq{ρ1ρ2; q, pq|N |�|k|
n±
r,s�1
pq1�kr�ksxr{xs; q, pqNr�kr
n¹
r�1
pbxr; q, pqkr�|N |
pbρ1ρ2xr{a; q, pq|k|�Nr
�
�
aq
ρ1ρ2
|k|
βkpa, bρ1ρ2{aqq
�
.
Then α1N pa, bq and β1N pa, bq also form a WP Bailey pair with respect to Bp3q.
Proof. The proof is analogous to that of Theorem 4.6. Again we use (4.2), but this time with
the following substitutions: xr ÞÑ xrq
jr and Nr ÞÑ Nr � jr for r � 1, 2, . . . , n, a ÞÑ bρ1ρ2q
|j|{aq,
b ÞÑ ρ1q
|j|, c ÞÑ ρ2q
|j|, d ÞÑ bρ1ρ2q
�|j|{a2q. The rest of the proof is similar. �
On applying the Bp3q Ñ Bp3q elliptic Bailey lemma in Theorem 5.10 to the WP Bailey pair
in Theorem 5.6, we again obtain Theorem 5.2.
We have now seen three multivariable Bressoud matrices, and various WP Bailey lemmas
connecting WP Bailey pairs with respect to them. As a result we obtained a new 10φ9 elliptic
Bailey transformation. In the next section, we suspend our study of WP Bailey pairs and
lemmas, to record special cases of this new transformation formula.
6 Special cases: new An Watson transformations
and related identities
We now consider extensions of Watson’s transformation that follow from the An elliptic Bailey
10φ9 transformation formula (5.2). Previously, multiple series extensions of Watson’s transfor-
mations have been obtained by, for example, Milne [25, 26, 28], Milne and Lilly [29], Milne
and Newcomb [30], Coskun [13] and by the authors, see [9] and [10]. Some interesting appli-
cations of one of these transformations to the theory of affine Lie algebras appear in Bartlett
and Warnaar [7] and Griffin, Ono and Warnaar [18]. Below we present some new An Watson
transformation formulas and some further special cases.
Multiple series extensions of Watson’s transformation formula [17, equation (2.5.1)] can be
obtained from the p � 0 case of (5.2) in multiple ways. We can take the limit as b, c, or d goes
to infinity. Alternatively, we can consider the equivalent formulation obtained by replacing e
or f by λaq1�|N |{ef and then take the limits as one of b, c, or d go to infinity. We can also
interchange the role of λ and a and then take limits as above. Many of these limits give rise to
the same formula, depending on the symmetry of the various parameters.
Theorem 6.1 (an An Watson transformation). We have
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
n¹
r,s�1
�
q�Nsxr{xs; q
�
kr
pqxr{xs; qqkr
�
n¹
r�1
1� axrq
kr�|k|
1� axr
paxr; qq|k|pbxr, cxr, exr; qqkr
paxrq1�Nr ; qq|k|paxrq{d; qqkr
n¹
r�1
x�krr
�
pd; qq|k|
paq{b, aq{c, aq{e; qq|k|
�
a2q2�|N |
bcde
�|k|
q
°
r s
krks�
n°
r�1
pr�1qkr
�
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 25
�
paq{de; qq|N |
paq{e; qq|N |
n¹
r�1
paxrq; qqNr
paxrq{d; qqNr
�
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
n¹
r,s�1
�
q�Nsxr{xs; q
�
kr
pqxr{xs; qqkr
�
pd; qq|k|
n±
r�1
�
exr, aq
1�|k|�kr{bcxr; q
�
kr�
aq{b, aq{c, deq�|N |{a; q
�
|k|
q
n°
r�1
rkr
�
.
Proof. We take p � 0 in (5.2), and replace e by λaq1�|N |{ef . Now we take the limit b Ñ 8
and replace d by b and f by d. �
Remark. By applying a standard polynomial argument to Theorem 6.1, we can obtain an
equivalent transformation formula, where both sums are summed over an n-simplex. (This result
could alternatively be obtained from Theorem 5.3 by applying a similar limit and substitution
as that used in the proof of Theorem 6.1.) That is, the summation indices on both sides range in
the region 0 ¤ |k| ¤ N , where N is a non-negative integer, and kr ¥ 0, for r � 1, 2, . . . , n. This
remark applies to all the results of this section. We do not write down these results explicitly.
For an example of such a calculation, see the proof of [27, Theorem 2.4] or [10, Theorem 3.7].
If we take p � 0 in (5.2), and take the limit bÑ8, we obtain the An Watson transformation
formula [10, Theorem 4.3]. If we take p � 0 in (5.2), take the limit d Ñ 8, we obtain an An
Watson transformation formula due to Milne, see [28, Theorem 2.1]. Finally, we take p � 0
in (5.2), and replace e by λaq1�|N |{ef , and take the limit dÑ8 to obtain another An Watson
transformation due to Milne, see [30, Theorem A.3].
Next, we consider the formula obtained from (5.2) by first replacing λ by qa2{bcd, and
then taking (simultaneously) a ÞÑ qa2{bcd, b ÞÑ aq{cd, c ÞÑ aq{bd and d ÞÑ aq{bc. In the
resulting formula, we take λ � qa2{bcd and use the relations λb{a � aq{cd, λc{a � aq{bd and
λd{a � aq{bc. In this manner we can write the right-hand side of the series (5.2) with special
parameter a and the left-hand side with special parameter λ. Now we take p � 0 to obtain the
following An Bailey 10φ9 transformation formula. This is equivalent to the p � 0 case of (5.2),
but with λ and a interchanged. Let λ � qa2{bcd. Then
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
n¹
r,s�1
�
q�Nsxr{xs; q
�
kr
pqxr{xs; qqkr
�
1� aq2|k|
1� a
pa, c, d, e, f ; qq|k|�
aq|N |�1, aq{c, aq{d, aq{e, aq{f ; q
�
|k|
�
n¹
r�1
pb{xr; qq|k|
�
λaxrq
|N |�1{ef ; q
�
kr
pef{λxr; qq|k|�kr
pb{xr; qq|k|�krpaxrq{b; qqkrpefq
�Nr{λxr; qq|k|
� q
n°
r�1
rkr
�
�
paq, aq{ef ; qq|N |
paq{e, aq{f ; qq|N |
n¹
r�1
pλxrq{e, λxrq{f ; qqNr
pλxrq, λxrq{ef ; qqNr
�
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
n¹
r,s�1
�
q�Nsxr{xs; q
�
kr
pqxr{xs; qqkr
n¹
r�1
1� λxrq
kr�|k|
1� λxr
�
n¹
r�1
pλxr; qq|k|
�
λcxr{a, λdxr{a, λaxrq
|N |�1{ef ; q
�
kr
pλxrq1�Nr ; qq|k|pλxrq{e, λxrq{f, axrq{b; qqkr
26 G. Bhatnagar and M.J. Schlosser
�
pe, f, λb{a; qq|k|�
aq{c, aq{d, efq�|N |{a; q
�
|k|
q
n°
r�1
rkr
�
. (6.1)
Three new An Watson transformations follow from this An Bailey transformation formula.
Theorem 6.2 (an An Watson transformation). We have
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
n¹
r,s�1
�
q�Nsxr{xs; q
�
kr
pqxr{xs; qqkr
�
1� aq2|k|
1� a
pa, c, d, e; qq|k|�
aq|N |�1, aq{c, aq{d, aq{e; q
�
|k|
n¹
r�1
xkrr
�
n¹
r�1
pb{xr; qq|k|
pb{xr; qq|k|�krpaxrq{b; qqkr
�
�
a2q2�|N |
bcde
�|k|
q
�
°
r s
krks�
n°
r�1
pr�1qkr
�
�
paq, aq{de; qq|N |
paq{d, aq{e; qq|N |
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
n¹
r,s�1
�
q�Nsxr{xs; q
�
kr
pqxr{xs; qqkr
�
n¹
r�1
paxrq{bc; qqkr
paxrq{b; qqkr
�
pd, e; qq|k|�
aq{c, deq�|N |{a; q
�
|k|
q
n°
r�1
rkr
�
.
Proof. In (6.1), we take the limit as dÑ8 and replace f by d in the result. �
Theorem 6.3 (an An Watson transformation). We have
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
n¹
r,s�1
�
q�Nsxr{xs; q
�
kr
pqxr{xs; qqkr
�
n¹
r�1
pexr; qqkr
�
axrq
1�|N |{e; q
�
|k|�kr�
axrq1�|N |�Nr{e; q
�
|k|
� q
°
r s
krks�
n°
r�1
pr�1qkr
n¹
r�1
x�krr
�
1� aq2|k|
1� a
pa, b, c, d; qq|k|�
aq|N |�1, aq{b, aq{c, aq{d; q
�
|k|
�
a2q2�|N |
bcde
�|k|�
�
paq; qq|N |
paq{d; qq|N |
n¹
r�1
�
aq1�|N |�Nr{dexr; q
�
Nr�
aq1�|N |�Nr{exr; q
�
Nr
�
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
n¹
r,s�1
�
q�Nsxr{xs; q
�
kr
pqxr{xs; qqkr
�
n¹
r�1
pexr; qqkr�
dexrq�|N |{a; q
�
kr
�
pd, aq{bc; qq|k|
paq{b, aq{c; qq|k|
q
n°
r�1
rkr
�
. (6.2)
Remark. If we take c � 1 in (6.2), the sum on the left-hand side reduces to 1, and we obtain
an equivalent formulation of Milne’s balanced 3φ2 sum [27, Theorem 4.1].
Proof. In (6.1), we replace e ÞÑ λaq1�|N |{ef and take b Ñ 8, and take f ÞÑ d and d ÞÑ b in
the resulting identity to obtain (6.2). �
A summation theorem follows immediately from Theorem 6.3.
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 27
Theorem 6.4 (an An very-well-poised 6φ5 summation). We have
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
n¹
r,s�1
�
q�Nsxr{xs; q
�
kr
pqxr{xs; qqkr
�
n¹
r�1
pcxr; qqkr
�
axrq
1�|N |{c; q
�
|k|�kr�
axrq1�|N |�Nr{c; q
�
|k|
n¹
r�1
x�krr
�
1� aq2|k|
1� a
pa, b; qq|k|�
aq|N |�1, aq{b; q
�
|k|
�
aq1�|N |
bc
�|k|
q
°
r s
krks�
n°
r�1
pr�1qkr
�
�
paq; qq|N |
paq{b; qq|N |
n¹
r�1
�
aq1�|N |�Nr{bcxr; q
�
Nr�
aq1�|N |�Nr{cxr; q
�
Nr
. (6.3)
Remark. When n�1, this formula reduces to the very-well-poised 6φ5 sum [17, equation (2.4.2)].
Several other extensions of this formula on root systems have appeared previously, see, for
example, [8, 9, 15, 19, 27, 29, 37].
Proof. We take c � aq{b in (6.2). The sum on the right-hand side becomes 1. In the resulting
identity, we replace d by b and e by c to obtain (6.3). �
The identity (6.3) is related to the An 6φ5 summation due to the first author [8, Theorem 3.6].
It follows from this result by inverting the base or reversing the sum. It can also be obtained
from the An Jackson sum [38, Theorem 4.1], given by the p � 0 case of (5.1), by replacing c by
a2q|N |�1{bcd and letting dÑ8.
Theorem 6.5 (an An Watson transformation). We have
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
n¹
r,s�1
�
q�Nsxr{xs; q
�
kr
pqxr{xs; qqkr
�
n¹
r�1
pexr; qqkrpb{xr; qq|k|
�
axrq
1�|N |{e; q
�
|k|�kr
paxrq{b; qqkr
�
axrq1�|N |�Nr{e; q
�
|k|
pb{xr; qq|k|�kr
�
1� aq2|k|
1� a
pa, c, d; qq|k|�
aq|N |�1, aq{c, aq{d; q
�
|k|
�
a2q2�|N |
bcde
�|k|
q
n°
r�1
pr�1qkr
�
�
paq; qq|N |
paq{d; qq|N |
n¹
r�1
�
aq1�|N |�Nr{dexr; q
�
Nr�
aq1�|N |�Nr{exr; q
�
Nr
�
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
n¹
r,s�1
�
q�Nsxr{xs; q
�
kr
pqxr{xs; qqkr
�
n¹
r�1
paxrq{bc; qqkrpexr; qqkr
paxrq{b; qqkr
�
dexrq�|N |{a; q
�
kr
�
pd; qq|k|
paq{c; qq|k|
q
n°
r�1
rkr
�
. (6.4)
Proof. In (6.1), we replace e ÞÑ λaq1�|N |{ef and take dÑ8. In the result, we take f ÞÑ d to
obtain (6.4). �
When c � 1 in (6.4) the sum on the left-hand side reduces to 1 and we obtain an equivalent
formulation of Milne’s balanced 3φ2 sum [27, Theorem 4.1].
28 G. Bhatnagar and M.J. Schlosser
If we set e � a2q1�|N |{bcd in (6.4), we obtain the second author’s An Jackson’s 8φ7 sum
(the p � 0 case of (5.1)). After replacing e as specified, the sum on the right-hand side can
be evaluated by setting a ÞÑ a2q1�|N |{bcd, b � d, c ÞÑ aq{b, xr ÞÑ xrxn, for r � 1, 2, . . . , n in
Milne’s balanced 3φ2 sum [27, Theorem 4.1].
Before proceeding to the next section, we remark on the motivation for our search for a new
Bailey 10φ9 transformation given in (5.2). Most results of this section contain a very-well-poised
part
1� aq2|k|
1� a
instead of the usual
n¹
r�1
1� axrq
kr�|k|
1� axr
,
where the summation index is k. The first result of this type was given by the first author [8],
followed by several related results by the second author [38]. It was natural to search for an An
10φ9 transformation involving a series with this kind of very-well-poised part that would contain
all those results as special cases.
7 Another WP Bailey pair for Bp1q
The next elliptic Jackson summation we consider does not give rise to another Bressoud matrix.
Rather surprisingly, it provides yet another WP Bailey pair with respect to the matrix Bp1q.
The Dn elliptic Jackson sum we apply in this section is due to Rosengren [31, Corollary 6.4].
Its p � 0 case is due to the first author [9]. Rosengren’s result is
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
�
q�Nsxr{xs; q, p
�
kr
pqxr{xs; q, pqkr
�
±
1¤r s¤n
paxrxsq{d; q, pqkr�ks
n±
r,s�1
paxrxsq{d; q, pqkr
n¹
r�1
θ
�
axrq
kr�|k|; p
�
paxr, d{xr; q, pq|k|
θ paxr; pq paxrq1�Nr ; q, pq|k|pd{xr; q, pq|k|�kr
�
n±
r�1
�
bxr, cxr, a
2xrq
1�|N |{bcd; q, p
�
kr�
aq{b, aq{c, bcdq�|N |{a; q, p
�
|k|
q
n°
r�1
rkr
�
�
±
1¤r s¤n
paxrxsq{d; q, pqNr�Ns
n±
r,s�1
paxrxsq{d; q, pqNr
�
n±
r�1
�
axrq, axrq{bd, axrq{cd, aq
1�|N |�Nr{bcxr; q, p
�
Nr
paq{b, aq{c, aq{bcd; q, pq|N |
. (7.1)
Remark. The Dn series (with summation index k) typically contain the elliptic Vandermonde
product
¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
¹
1¤r s¤n
θ
�
axrxsq
kr�ks ; p
�
θ paxrxs; pq
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 29
as a factor. (This is not followed very strictly. Sometimes the series is labelled as a Dn series
when this factor appears on reversing the sum or inverting the base, as in (8.1), below.)
We use (7.1) to obtain a WP Bailey pair with respect to the matrix Bp1q.
Theorem 7.1 (second elliptic WP Bailey pair with respect to Bp1q). The two sequences
αkpa, bq :�
n¹
r�1
θ
�
axrq
kr�|k|; p
�
θ pa; pq
±
1¤r s¤n
paxrxsq{d; q, pqkr�ks
n±
r,s�1
pqxr{xs; q, pqkrpaxrxsq{d; q, pqkr
�
n±
r�1
�
cxr, a
2xrq{bcd; q, p
�
kr
paxr, d{xr; q, pq|k|
paq{c, bcd{a; q, pq|k|
n±
r�1
pd{xr; q, pq|k|�kr
�
b
a
|k|
,
and
βkpa, bq :�
±
1¤r s¤n
paxrxsq{d; q, pqkr�ks
n±
r,s�1
paxrxsq{d; q, pqkrpqxr{xs; q, pqkr
�
n¹
r�1
pbxr, bd{axr; q, pq|k|
pbd{axr; q, pq|k|�kr
�
n±
r�1
paxrq{cd, bcxr{a; q, pqkr
paq{c, bcd{a; q, pq|k|
,
form a WP-Bailey pair with respect to Bp1q.
Proof. The proof is analogous to that of Theorem 4.3, except that we use the b ÞÑ qa2{bcd case
of (7.1). �
Consider the inverse relation of (7.1), in the form (4.11) where B � Bp1q, and αk and βk are
defined as in Theorem 7.1 and pBp1qpa, bqq�1
kj is given by (4.10). After canceling some products,
we take a ÞÑ bq�|N |, b ÞÑ a, c ÞÑ aq{cd and d ÞÑ bd{aq|N |, to again obtain (7.1). Thus we do not
obtain a new result by taking the inverse relation.
Since we have another WP Bailey pair with respect to the matrix Bp1q, we can apply the
Bp1q Ñ Bp1q WP Bailey lemma or the Bp1q Ñ Bp2q WP Bailey lemma, to obtain a WP Bailey
pair with respect to the matrices Bp1q or Bp2q, respectively.
An elliptic Dn Bailey 10φ9 transformation formula due to Rosengren (which follows by re-
versing the sum in [31, Corollary 8.5]) follows immediately by applying the Bp1q Ñ Bp1q elliptic
Bailey lemma in Theorem 4.6 to the WP Bailey pair in Theorem 7.1. When p � 0 this reduces
to the authors’ formula [10, Theorem 3.9].
The same elliptic Dn Bailey 10φ9 transformation also follows by applying the Bp1q Ñ Bp2q
elliptic Bailey lemma in Theorem 4.10 to the WP Bailey pair in Theorem 7.1.
Next, we find another Bressoud matrix and another WP Bailey lemma, which will allow us
to use the WP Bailey pair of this section again.
8 The matrix Bp4q
In this section we examine some results which are related to multiple series attached to a mix of
root systems, such as An, Cn and Dn. These results are a consequence of a Dn elliptic Jackson
30 G. Bhatnagar and M.J. Schlosser
sum due to Rosengren [31, Corollary 6.3]. The p � 0 case is due to the second author [36].
Rosengren’s result is
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
�
q�Nsxr{xs; q, p
�
kr
pqxr{xs; q, pqkr
�
n¹
r�1
θ
�
axrq
kr�|k|; p
�
θ paxr; pq
paxr; q, pq|k|pbcd{axr; q, pq|k|�kr
paxrq1�Nr ; q, pq|k|pbcdq
�Nr{axr; q, pq|k|
�
pb, c, d; q, pq|k|
n±
r,s�1
�
a2xrxsq
1�Ns{bcd; q, p
�
kr
n±
r�1
paxrq{b, axrq{c, axrq{d; q, pqkr
±
1¤r s¤n
pa2xrxsq{bcd; q, pqkr�ks
q
n°
r�1
rkr
�
�
n¹
r�1
paxrq, axrq{bc, axrq{bd, axrq{cd; q, pqNr
paxrq{b, axrq{c, axrq{d, axrq{bcd; q, pqNr
. (8.1)
The above summation implies another Bressoud matrix and a WP Bailey pair. The matrix
Bp4q � pB
p4q
kj pa, bqq is defined as follows:
Definition 8.1 (a Dn elliptic Bressoud matrix). We define the matrix Bp4q with entries indexed
by pk, jq as
B
p4q
kj pa, bq :�
n±
r,s�1
pbxrxs; q, pqkr�js
±
1¤r s¤n
pbxrxs; q, pqjr�js
n±
r,s�1
pq1�jr�jsxr{xs; q, pqkr�jr
�
n¹
r�1
pbxr{a; q, pqkr�|j|
pbxr{a; q, pqjr�|j|paxrq; q, pqkr�|j|
. (8.2)
Theorem 8.2 (an elliptic WP Bailey pair with respect to Bp4q). The two sequences
αkpa, bq :�
n¹
r�1
θ
�
axrq
kr�|k|; p
�
θ paxr; pq
paxr; q, pq|k|
paxrq{c, axrq{d, bcdxr{a; q, pqkr
�
�
c, d, qa2{bcd; q, p
�
|k|
n±
r,s�1
pqxr{xs; q, pqkr
�
b
a
|k|
q
�
°
r s
krks
n¹
r�1
xkrr , (8.3a)
and
βkpa, bq :�
n¹
r,s�1
pbxrxs; q, pqks
pqxr{xs; q, pqkr
n¹
r�1
paxrq{cd, bcxr{a, bdxr{a; q, pqkr
pbcdxr{a, axrq{c, axrq{d; q, pqkr
, (8.3b)
form a WP-Bailey pair with respect to Bp4q.
Proof. The proof is similar to that of Theorem 4.3. We use the b ÞÑ qa2{bcd case of (8.1) to
verify that αkpa, bq and βkpa, bq form a WP-Bailey pair with respect to Bp4q. �
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 31
Theorem 8.3 (an elliptic
�
Bp1q Ñ Bp4q
�
WP Bailey lemma). Suppose αN pa, bq and βN pa, bq
form a WP Bailey pair with respect to the matrix Bp1q. Let α1N pa, bq and β1N pa, bq be defined as
follows:
α1N pa, bq :�
pρ1, ρ2; q, pq|N |
n±
r�1
paxrq{ρ1, axrq{ρ2; q, pqNr
�
aq
ρ1ρ2
|N | n¹
r�1
xNr
r � q
�
°
r s
NrNs
αN pa, bρ1ρ2{aqq,
β1N pa, bq :�
n¹
r�1
pbρ1xr{a, bρ2xr{a; q, pqNr
paxrq{ρ1, axrq{ρ2; q, pqNr
¸
0¤kr¤Nr
r�1,2,...,n
�
pρ1, ρ2; q, pq|k|
n±
r�1
pbρ1xr{a, bρ2xr{a; q, pqkr
�
n¹
r�1
θ
�
bρ1ρ2xrq
kr�|k|{aq; p
�
θ pbρ1ρ2xr{aq; pq
�
axrq
1�kr�|k|{ρ1ρ2; q, p
�
Nr�kr
pbρ1ρ2xr{a; q, pqNr�|k|
�
n±
r,s�1
pbxrxs; q, pqNs�kr
±
1¤r s¤n
pbxrxs; q, pqkr�ks
n±
r,s�1
pq1�kr�ksxr{xs; q, pqNr�kr
�
�
aq
ρ1ρ2
|k| n¹
r�1
xkrr � q
�
°
r s
krks
βkpa, bρ1ρ2{aqq
�
.
Then α1N pa, bq and β1N pa, bq form a WP Bailey pair with respect to Bp4q, defined by (8.2).
Remark. A matrix reformulation of Theorem 8.3 appears in unpublished notes of Warnaar [48].
Proof. The proof is similar to that of Theorem 4.6. We need to use (8.1), with the substitutions:
xr ÞÑ xrq
jr and Nr ÞÑ Nr � jr for r � 1, 2, . . . , n, a ÞÑ bρ1ρ2q
2|j|{aq, b ÞÑ ρ1q
|j|, c ÞÑ ρ2q
|j|,
d ÞÑ bρ1ρ2q
|j|{a2q. �
An elliptic Dn Bailey 10φ9 transformation formula due to Rosengren [31, Corollary 8.5] follows
immediately by applying the Bp1q Ñ Bp4q elliptic Bailey lemma in Theorem 8.3 to the WP Bailey
pair in Theorem 4.3. When p � 0 this reduces to the authors’ formula in [10, Theorem 3.13].
If instead we use the second WP Bailey pair in Theorem 7.1, we obtain a different elliptic Dn
Bailey 10φ9 transformation formula, again due to Rosengren [31, Corollary 8.4]. When p � 0
this reduces to [10, Theorem 3.1].
We will compute the inverse of Bp4q in the next section.
9 Consequences of a Cn elliptic Jackson sum due to Rosengren
In this section we consider a Cn elliptic Jackson summation theorem due to Rosengren [31,
Theorem 7.1]. The p � 0 case was found independently by Denis and Gustafson [14, Theorem 4.1]
and Milne and Lilly [29, Theorem 6.13]. We will find that the Bressoud matrix following from
this result is closely related to the one in Section 8. Again, the results are closely related to
both Dn and An series. The summation theorem we consider is
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
¹
1¤r s¤n
θ
�
axrxsq
kr�ks ; p
�
θ paxrxs; pq
�
n¹
r�1
θ
�
ax2rq
2kr ; p
�
θ pax2r ; pq
n¹
r,s�1
�
q�Nsxr{xs, axrxs; q, p
�
kr
pqxr{xs, axrxsq1�Ns ; q, pqkr
32 G. Bhatnagar and M.J. Schlosser
�
n¹
r�1
�
bxr, cxr, dxr, a
2xrq
1�|N |{bcd; q, p
�
kr�
axrq{b, axrq{c, axrq{d, bcdxrq�|N |{a; q, p
�
kr
� q
n°
r�1
rkr
�
�
n±
r,s�1
paxrxs; q, pqNr±
1¤r s¤n
paxrxs; q, pqNr�Ns
�
paq{bc, aq{bd, aq{cd; q, pq|N |
n±
r�1
�
axrq{b, axrq{c, axrq{d, aq1�|N |�Nr{bcdxr; q, p
�
Nr
. (9.1)
Remark. The Cn series (with summation index k) contain the elliptic Vandermonde product
¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
¹
1¤r s¤n
θ
�
axrxsq
kr�ks ; p
�
θ paxrxs; pq
n¹
r�1
θ
�
ax2rq
2kr ; p
�
θ pax2r ; pq
as a factor.
We use this result to obtain an elliptic WP Bailey lemma.
Theorem 9.1 (an elliptic
�
Bp4q Ñ Bp1q
�
WP Bailey lemma). Suppose αN pa, bq and βN pa, bq
form a WP Bailey pair with respect to the matrix Bp4q defined in (8.2). Let α1N pa, bq and β1N pa, bq
be defined as follows
α1N pa, bq :�
n±
r�1
pρ1xr, ρ2xr; q, pqNr
paq{ρ1, aq{ρ2; q, pq|N |
�
aq
ρ1ρ2
|N | n¹
r�1
x�Nr
r � q
°
r s
NrNs
αN pa, bρ1ρ2{aqq, (9.2a)
β1N pa, bq :�
n±
r�1
pbρ1xr{a, bρ2xr{a; q, pqNr
paq{ρ1, aq{ρ2; q, pq|N |
�
¸
0¤kr¤Nr
r�1,2,...,n
�
n¹
r�1
pρ1xr, ρ2xr; q, pqkr
�
aq1�|N |�Nr{ρ1ρ2xr; q, p
�
Nr�kr
pbρ1xr{a, bρ2xr{a; q, pqkr
�
¹
1¤r s¤n
θ
�
bρ1ρ2xrxsq
kr�ks{aq; p
�
θ pbρ1ρ2xrxs{aq; pq
n¹
r�1
θ
�
bρ1ρ2x
2
rq
2kr{aq; p
�
θ pbρ1ρ2x2r{aq; pq
�
±
1¤r s¤n
pbρ1ρ2xrxs{a; q, pqNr�Ns
n±
r�1
pbxr; q, pq|N |�kr
n±
r,s�1
pbρ1ρ2xrxs{a; q, pqNs�kr
pq1�kr�ksxr{xs; q, pqNr�kr
�
�
aq
ρ1ρ2
|k| n¹
r�1
x�krr � q
°
r s krksβkpa, bρ1ρ2{aqq
�
. (9.2b)
Then α1N pa, bq and β1N pa, bq form a WP Bailey pair with respect to Bp1q, defined by (4.4).
Remark. A matrix reformulation of Theorem 9.1 appears in unpublished notes of Warnaar [48].
Proof. The proof is analogous to that of Theorem 4.6. The only difference is that we use (9.1),
with the substitutions: xr ÞÑ xrq
jr and Nr ÞÑ Nr � jr for r � 1, 2, . . . , n, a ÞÑ bρ1ρ2{aq, b ÞÑ ρ1,
c ÞÑ ρ2, d ÞÑ bρ1ρ2q
�|j|{a2q. The remaining calculations are very similar. �
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 33
Remark 9.2. An elliptic Cn Ñ An Bailey 10φ9 transformation formula due to Rosengren [31,
Corollary 8.3] follows immediately by applying the Bp4q Ñ Bp1q elliptic Bailey lemma in Theo-
rem 9.1 to the WP Bailey pair in Theorem 8.2. When p � 0 this reduces to [10, Theorem 2.1].
Note that the β1N defined in (9.2b) has the very-well-poised part usually present in Cn series.
However the αN in (9.2a) (which comes from the definition (8.3a) of the WP Bailey pair with
respect to Bp4q) contains the usual An very-well-poised part.
Next we have another elliptic Bressoud matrix and a WP Bailey pair from (9.1). The matrix
Bp5q �
�
B
p5q
kj pa, bq
�
is defined as follows.
Definition 9.3 (a Cn elliptic Bressoud matrix). We define the matrix Bp5q with entries indexed
by pk, jq as
B
p5q
kj pa, bq :�
±
1¤r s¤n
paxrxsq; q, pqkr�ks
n±
r�1
�
bq|k|�kr{axr; q, p
�
kr�jr
pbxr; q, pq|k|�jr
n±
r,s�1
pq1�jr�jsxr{xs; q, pqkr�jrpaxrxsq; q, pqks�jr
. (9.3)
Theorem 9.4 (an elliptic WP Bailey pair with respect to Bp5q). The two sequences
αkpa, bq :�
¹
1¤r s¤n
θ
�
axrxsq
kr�ks ; p
�
θ paxrxs; pq
n¹
r�1
θ
�
ax2rq
2kr ; p
�
θ pax2r ; pq
�
�
b
a
|k|
q
°
r s
krks
�
n¹
r,s�1
paxrxs; q, pqkr
pqxr{xs; q, pqkr
n¹
r�1
�
a2xrq{bcd, cxr, dxr; q, p
�
kr
paxrq{c, axrq{d, bcdxr{a; q, pqkr
n¹
r�1
x�krr ,
and
βkpa, bq :�
pbc{a, bd{a, aq{cd; q, pq|k|
n±
r�1
pbxr; q, pq|k|
n±
r,s�1
pqxr{xs; q, pqkr
n±
r�1
pbcdxr{a, axrq{c, axrq{d; q, pqkr
,
form a WP-Bailey pair with respect to Bp5q.
Proof. The proof requires the Cn elliptic Jackson sum given in (9.1). We verify that αkpa, bq
and βkpa, bq form a WP-Bailey pair with respect to Bp5q using the b ÞÑ qa2{bcd case of (9.1). �
As a corollary, we obtain a unit WP Bailey pair.
Corollary 9.5. The two sequences
αkpa, bq :�
¹
1¤r s¤n
θ
�
axrxsq
kr�ks ; p
�
θ paxrxs; pq
n¹
r�1
θ
�
ax2rq
2kr ; p
�
θ pax2r ; pq
�
�
b
a
|k|
q
°
r s
krks
�
n¹
r,s�1
paxrxs; q, pqkr
pqxr{xs; q, pqkr
n¹
r�1
paxr{b; q, pqkr
pbxrq; q, pqkr
n¹
r�1
x�krr ,
and
βkpa, bq :�
n¹
r�1
δkr,0,
form a WP-Bailey pair with respect to Bp5q.
34 G. Bhatnagar and M.J. Schlosser
Proof. Take d � aq{c in Theorem 9.4. �
We can find a formula for the inverse of Bp5q using the unit Bailey pair.
Corollary 9.6 (inverse of Bp5q). Let Bp5q �
�
B
p5q
kj pa, bq
�
be defined by (9.3). Then the entries
of its inverse are given by
�
Bp5qpa, bq
��1
kj
�
¹
1¤r s¤n
θ
�
axrxsq
kr�ks ; p
�
θ paxrxs; pq
n¹
r�1
θ
�
ax2rq
2kr ; p
�
θ pax2r ; pq
θ
�
bxrq
jr�|j|; p
�
θ pbxr; pq
�
n¹
r�1
�
axrq
jr�|j|{b; q, p
�
kr�jr
pbxrq; q, pqkr�|j|
�
�
b
a
|k|�|j|
q
°
r s
pkrks�jrjsq
n¹
r�1
xjr�krr
�
n±
r,s�1
paxrxs; q, pqks�jr
±
1¤r s¤n
paxrxs; q, pqjr�js
n±
r,s�1
pq1�jr�jsxr{xs; q, pqkr�jr
. (9.4)
Proof. The derivation is analogous to that of Corollary 5.8 and left to the reader. �
Corollary 9.7 (inverse of Bp4q). Let Bp4q �
�
B
p4q
kj pa, bq
�
be defined by (8.2). Then the entries
of its inverse is given by
�
Bp4qpa, bq
��1
kj
�
n¹
r�1
θ
�
axrq
kr�|k|; p
�
θ paxr; pq
�
�
b
a
|k|�|j|
q
°
r s
pjrjs�krksq
n¹
r�1
xkr�jrr
�
¹
1¤r s¤n
θ
�
bxrxsq
jr�js ; p
�
θ pbxrxs; pq
n¹
r�1
θ
�
bx2rq
2jr ; p
�
θ pbx2r ; pq
�
±
1¤r s¤n
pbxrxsq; q, pqkr�ks
n±
r,s�1
pq1�jr�jsxr{xs; q, pqkr�jrpbxrxsq; q, pqks�jr
�
n¹
r�1
�
aq|k|�kr{bxr; q, p
kr�jr
paxr; q, pq|k|�jr .
Proof. Observe that the entries of Bp5qpa, bq�1 consist of the entries of Bp4qpb, aq multiplied by
some additional factors, which can be separated into factors containing either terms with index j
or with index k. This can help us find the inverse of Bp4q as in the proof of Corollary 5.9. We
leave the details to the reader. �
Consider the inverse relation (4.11) where B � Bp5q, and αk and βk are defined as in Theo-
rem 9.4 and
�
Bp5qpa, bq
��1
kj
is given by (9.4). After canceling some products, make the substitu-
tions a ÞÑ qa2{bcd, b ÞÑ a, c ÞÑ aq{bd and d ÞÑ aq{bc to obtain (8.1). Thus (9.1) and (8.1) are
inverse relations. This approach provides an alternate derivation of (8.1) beginning with (9.1).
Theorem 9.8 (Zhang and Huang [51]; an elliptic
�
Bp5q Ñ Bp5q
�
WP Bailey lemma). Suppose
αN pa, bq and βN pa, bq form a WP Bailey pair with respect to the matrix Bp5q. Let α1N pa, bq and
β1N pa, bq be defined as follows
α1N pa, bq :�
n¹
r�1
pρ1xr, ρ2xr; q, pqNr
paxrq{ρ1, axrq{ρ2; q, pqNr
�
�
aq
ρ1ρ2
|N |
αN pa, bρ1ρ2{aqq,
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 35
β1N pa, bq :�
pbρ1{a, bρ2{a; q, pq|N |
n±
r�1
paxrq{ρ1, axrq{ρ2; q, pqNr
�
¸
0¤kr¤Nr
r�1,2,...,n
� n±
r�1
pρ1xr, ρ2xr; q, pqkr
pbρ1{a, bρ2{a; q, pq|k|
paq{ρ1ρ2; q, pq|N |�|k|
n±
r,s�1
pq1�kr�ksxr{xs; q, pqNr�kr
�
n¹
r�1
θ
�
bρ1ρ2xrq
kr�|k|{aq; p
�
θ pbρ1ρ2xr{aq; pq
pbxr; q, pq|N |�kr
pbρ1ρ2xr{a; q, pqNr�|k|
�
�
aq
ρ1ρ2
|k|
βkpa, bρ1ρ2{aqq
�
.
Then α1N pa, bq and β1N pa, bq form a WP Bailey pair with respect to Bp5q.
Remark. Theorem 9.4 is equivalent to a theorem of Zhang and Huang [51, Theorem 5.3]. The
matrix they consider is equivalent to (9.3), with slightly different notation. This result appears
in unpublished notes of Warnaar [48] too.
Proof. The proof follows the model of Theorem 4.6, except that we use (7.1), with the following
substitutions: xr ÞÑ xrq
jr and Nr ÞÑ Nr�jr for r � 1, 2, . . . , n, a ÞÑ bρ1ρ2q
|j|{aq, b ÞÑ ρ1, c ÞÑ ρ2,
d ÞÑ bρ1ρ2q
|j|{a2q. �
An elliptic Cn Ñ An Bailey 10φ9 transformation formula due to Rosengren [31, Corollary 8.3]
follows immediately by applying the Bp5q Ñ Bp5q elliptic Bailey lemma in Theorem 9.8 to the
WP Bailey pair in Theorem 9.4. This Cn Ñ An elliptic transformation formula is the same as
obtained in Remark 9.2.
We have completed our study of the existing elliptic Jackson theorems on root systems that
are relevant in this theory. In the next section, we summarize our results, and examine our
results from another perspective to see whether we have missed anything that fits our approach.
10 Summary of results
In Sections 4–9, we have systematically considered the consequences of five elliptic Jackson
summation theorems. In this section, we provide a summary of our findings so far and examine
our results. Our examination suggests one more idea to follow up before closing this study.
Here is a list of our findings.
1. We have considered five elliptic Jackson summations on root systems.
2. We have defined five elliptic Bressoud matrices, denoted Bp1q–Bp5q, so far.
3. We found six WP Bailey pairs. There were two WP Bailey pairs with respect to Bp1q.
(We are not counting the unit WP Bailey pairs.)
4. In case there is a unit WP Bailey pair, we are able to find a formula for the inverse of the
matrix.
5. Up to normalization, the inverse of Bp1qpa, bq is given by Bp1qpb, aq. Similarly, the inverse
of Bp2qpa, bq is (up to normalization) Bp3qpb, aq. The matrices Bp4q and Bp5q are similarly
related.
6. We computed the inverse relations arising out of the elliptic Jackson summations and the
related Bressoud matrix inverses. We found the following relationships between the elliptic
Jackson summations.
36 G. Bhatnagar and M.J. Schlosser
Matrix: Bp1q Bp2q Bp3q Bp4q Bp5q
Bp1q Theorem 4.6 Theorem 4.10 Theorem 8.3
Bp2q Theorem 5.1 Theorem 5.4
Bp3q Theorem 5.10
Bp4q Theorem 9.1
Bp5q Theorem 9.8
Table 1. The WP Bailey lemmas.
• If we write the inverse relation of (4.2) with respect to Bp1q, we obtain an equivalent
form of (4.2).
• If we write the inverse relation of (5.1) with respect to Bp3q, we obtain an equivalent
form of (4.2). To go in the other direction, we use Bp2q.
• If we write the inverse relation of (7.1) with respect to Bp1q, we obtain an equivalent
form of (7.1).
• If we write the inverse relation of (9.1) with respect to Bp5q, we obtain an equivalent
form of (8.1). In the other direction, we use Bp4q.
7. We have found eight WP Bailey lemmas (see Table 1 for the list).
8. We recovered the elliptic Bailey transformations in Rosengren [31] by the WP Bailey
lemma approach. In addition, we found one new An elliptic Bailey transformation, so far.
9. As basic hypergeometric special cases of the new An Bailey 10φ9 transformation, we found
four new An Watson transformations and one An terminating, very-well-poised, 6φ5 sum-
mation.
As we have seen, we can apply the WP Bailey lemmas to obtain elliptic Bailey transformation
formulas on root systems. As in the dimension 1 case, we can iterate the WP Bailey lemmas
and obtain results with more parameters. For example, one can begin with a WP Bailey pair
with respect to the matrix Bp1q and apply a Bp1q Ñ Bp4q Bailey lemma to obtain a WP Bailey
pair with respect to Bp4q. One can now use the Bp4q Ñ Bp1q WP Bailey lemma to obtain a WP
Bailey pair with respect to Bp1q. In the next step, one can apply the Bp1q Ñ Bp2q WP Bailey
lemma.
If we look closely at the definitions of β1N in any of the WP Bailey lemmas, we can observe
an interesting pattern. For example, consider the definition of β1N given in the Bp4q Ñ Bp1q
WP Bailey lemma in (9.2b). Observe that this expression contains the expression B
p5q
Nkpa, bq,
where a is replaced by bρ1ρ2{aq. One can say something similar for all the WP Bailey lemmas
presented in this paper, except for one.
The one exception is the Bp2q Ñ Bp2q WP Bailey lemma given by Theorem 5.4. This suggests
that we may have missed the Bressoud matrix (stated here for p � 0):
Bkjpa, bq :�
pb; qq|k|�|j|pb{a; qq|k|�|j|
paq; qq|k|�|j|
n±
r,s�1
pq1�jr�jsxr{xs; qqkr�jr
.
Indeed, we have the An Jackson sum
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
n¹
r,s�1
�
q�Nsxr{xs; q
�
kr
pqxr{xs; qqkr
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 37
�
p1� aq2|k|q
�
a, b, c, d, a2q1�|N |{bcd; q
�
|k|
p1� aq
�
aq|N |�1, aq{b, aq{c, aq{d, bcdq�|N |{a; q
�
|k|
q
n°
r�1
rkr
�
�
paq, aq{bc, aq{bd, aq{cd; qq|N |
paq{b, aq{c, aq{d, aq{bcd; qq|N |
, (10.1)
which follows from the following result due to Milne [27], which the first author [8] dubbed the
“Fundamental Theorem of Upnq series”, namely
¸
|k|�K
k1,k2,...,kn¥0
¹
1¤r s¤n
1� qkr�ksxr{xs
1� xr{xs
n¹
r,s�1
pasxr{xs; qqkr
pqxr{xs; qqkr
� q
n°
r�1
pr�1qkr
�
pa1 � � � an; qqK
pq; qqK
.(10.2)
Unfortunately, if we formally replace each term of (10.1) by its elliptic analogue, the resulting
summation is false. However, in the next section we find an elliptic Jackson summation which
contains (10.1) as a special case.
Before heading to the next section, we note that the observation above can be explained by
the matrix approach to the WP Bailey lemma, given by Agarwal, Andrews and Bressoud [1]
and Warnaar [47]. Indeed, Warnaar [48] extended this matrix formulation for his (unpublished)
work on multivariable WP Bailey lemmas.
11 Other elliptic Jackson summations, with an extra parameter
The objective of this section is to give an elliptic extension of (10.1) by adding another parameter.
To do that, we present a nice trick that is useful in many contexts. Essentially, this trick is an
elliptic extension of one of Milne’s lemmas [27, Lemma 7.3] that he used [24] to prove one of the
Macdonald identities.
We will use the following theorem of Rosengren, which can be shown to be equivalent to (4.2),
by replacing n by n� 1 relabeling parameters, and using an analytic continuation argument
¸
|k|�K
k1,k2,...,kn¥0
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
pasxr{xs; q, pqkr
pqxr{xs; q, pqkr
�
n¹
r�1
pbxr{a1 � � � an; q, pqkr
pbxr; q, pqkr
� q
n°
r�1
pr�1qkr
�
�
pa1 � � � an; q, pqK
pq; q, pqK
n¹
r�1
pbxr{ar; q, pqK
pbxr; q, pqK
. (11.1)
This is equivalent to Rosengren’s result [31, Theorem 5.1], where we take N � K, zk ÞÑ xk,
ak ÞÑ ak{xk (for k � 1, 2, . . . , n) and replace an�1 by b{a1a2 � � � an. Note that when p � 0 and
b � 0, (11.1) reduces to (10.2).
Theorem 11.1. Given the sequence fk, k � 0, 1, 2, . . . , and N ¥ 0, we have
¸
0¤|k|¤N
k1,k2,...,kn¥0
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
pasxr{xs; q, pqkr
pqxr{xs; q, pqkr
�
n¹
r�1
pbxr{a1 � � � an; q, pqkrpbxr; q, pq|k|
pbxr; q, pqkrpbxr{ar; q, pq|k|
� q
n°
r�1
pr�1qkr
f|k|
�
38 G. Bhatnagar and M.J. Schlosser
�
Ņ
K�0
pa1 � � � an; q, pqK
pq; q, pqK
fK . (11.2)
Remark. When p � 0 and b � 0 in (11.2), we obtain Milne’s lemma [27, Lemma 7.3]. Observe
that the right-hand side of (11.2) is not dependent on x1, x2, . . . , xn and b. Further, note that
the sums are indefinite.
Proof. The theorem follows by taking the products with parameter b from the right-hand side
of (11.1) to the left, multiplying both sides by fK and then summing over K from 0 to N . �
Remark. We can obtain a similar result from Rosengren [34, Theorem 3.1].
Theorem 11.1 allows us to choose the fk appropriately and use a dimension 1 identity to
obtain its multiple series extension. In particular, we now obtain an An elliptic Jackson sum in
this manner.
Theorem 11.2 (an An elliptic Jackson summation). We have
¸
0¤|k|¤N
k1,k2,...,kn¥0
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
pbsxr{xs; q, pqkr
pqxr{xs; q, pqkr
�
θ
�
aq2|k|; p
�
θ pa; pq
�
�
a, c, d, a2q1�N{b1 � � � bncd, q
�N ; q, p
�
|k|
paq{b1 � � � bn, aq{c, aq{d, b1 � � � bncdq�N{a, aqN�1; q, pq|k|
�
n¹
r�1
pexr{b1 � � � bn; q, pqkrpexr; q, pq|k|
pexr; q, pqkrpexr{br; q, pq|k|
� q
n°
r�1
rkr
�
�
paq, aq{b1 � � � bnc, aq{b1 � � � bnd, aq{cd; q, pqN
paq{b1 � � � bn, aq{c, aq{d, aq{b1 � � � bncd; q, pqN
. (11.3)
Proof. We take ak � bk for k � 1, 2, . . . , n, b � e in (11.2), and take
fK �
θ
�
aq2K ; p
�
θ pa; pq
�
a, c, d, a2q1�N{b1 � � � bncd, q
�N ; q, p
�
K
paq{b1 � � � bn, aq{c, aq{d, b1 � � � bncdq�N{a, aqN�1; q, pqK
qK .
The left-hand side of (11.2) then becomes the left-hand side of (11.3). Now in the right-hand
side of (11.2), we use the Frenkel–Turaev summation (2.9), with b ÞÑ b1 � � � bn. In this manner,
we obtain the right-hand side of (11.3). �
We can rewrite this identity so that the sum is over an n-dimensional rectangle.
Theorem 11.3 (an An elliptic Jackson summation). We have
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
�
q�Nsxr{xs; q, p
�
kr
pqxr{xs; q, pqkr
�
θ
�
aq2|k|; p
�
θ pa; pq
�
a, b, c, d, a2q1�|N |{bcd; q, p
�
|k|�
aq|N |�1, aq{b, aq{c, aq{d, bcdq�|N |{a; q, p
�
|k|
�
n¹
r�1
�
exrq
|N |; q, p
�
kr
pexr; q, pq|k|
pexr; q, pqkrpexrq
Nr ; q, pq|k|
q
n°
r�1
rkr
�
�
paq, aq{bc, aq{bd, aq{cd; q, pq|N |
paq{b, aq{c, aq{d, aq{bcd; q, pq|N |
. (11.4)
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 39
Proof. This can be obtained by an analytic continuation argument, or directly from Theo-
rem 11.1, by observing that when ar � q�Nr , then the indices k satisfy the additional condition
kr ¤ Nr, for r � 1, . . . , n. Then taking N � |N |, and choosing fK appropriately, we ob-
tain (11.4). �
Next, we obtain another An Bressoud matrix, from the b ÞÑ qa2{bcd case of (11.4).
Definition 11.4 (an An elliptic Bressoud matrix). We define the matrix Bp6q with entries
indexed by pk, jq as
B
p6q
kj pa, bq :�
pb; q, pq|k|�|j|pb{a; q, pq|k|�|j|
paq; q, pq|k|�|j|
n±
r,s�1
pq1�jr�jsxr{xs; q, pqkr�jr
n¹
r�1
pexr; q, pq|k|�jr
pexr; q, pqkr�|j|
. (11.5)
The inverse of this matrix is as follows.
Corollary 11.5 (inverse of Bp6q). Let Bp6q �
�
B
p6q
kj pa, bq
�
be defined by (11.5). Then its inverse
is given by
�
Bp6qpa, bq
��1
kj
�
θ
�
aq2|k|; p
�
θ pa; pq
θ
�
bq2|j|; p
�
θ pb; pq
n¹
r�1
pexr; q, pq|k|�jr
pexr; q, pqkr�|j|
�
�
b
a
|k|�|j|
�
pa; q, pq|k|�|j|pa{b; q, pq|k|�|j|
pbq; q, pq|k|�|j|
n±
r,s�1
pq1�jr�jsxr{xs; q, pqkr�jr
.
Proof. We begin with the equivalent formulation of Theorem 11.3 which is obtained by taking
b ÞÑ qa2{bcd. Now take d � aq{c to obtain a Kronecker delta function on the product side. The
rest of the calculation is analogous to the proof of Corollary 5.8. �
We have not stated the WP Bailey pair with respect to Bp6q and the corresponding Bp6q Ñ
Bp6q WP Bailey lemma explicitly. While it is not difficult to find a corresponding WP Bailey
lemma, it appears that in this case the WP Bailey lemma is not so useful to derive further
identities. (We therefore have decided to omit it.) For this purpose it is actually better to apply
Theorem 11.1 instead. We illustrate this by writing down an elliptic Bailey 10φ9 transformation
formula, which transforms an n dimensional sum to a multiple of an m-dimensional sum. (This
result cannot be obtained from the Bp6q Ñ Bp6q WP Bailey lemma.)
Theorem 11.6 (an An Ñ Am elliptic Bailey 10φ9 transformation theorem). Let λ � qa2{bcd.
Then
¸
0¤|k|¤N
k1,k2,...,kn¥0
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
pesxr{xs; q, pqkr
pqxr{xs; q, pqkr
�
θ
�
aq2|k|; p
�
θ pa; pq
pa, b, c, d; q, pq|k|
paqN�1, aq{b, aq{c, aq{d; q, pq|k|
�
�
f1 � � � fm, λaq
1�N{e1 � � � enf1 � � � fm, q
�N ; q, p
�
|k|
paq{e1 � � � en, aq{f1 � � � fm, e1 � � � enf1 � � � fmq�N{λ; q, pq|k|
�
n¹
r�1
pgxr{e1 � � � en; q, pqkrpgxr; q, pq|k|
pgxr; q, pqkrpgxr{er; q, pq|k|
� q
n°
r�1
rkr
�
�
paq, aq{e1 � � � enf1 � � � fm, λq{e1 � � � en, λq{f1 � � � fm; q, pqN
paq{e1 � � � en, aq{f1 � � � fm, λq, λq{e1 � � � enf1 � � � fm; q, pqN
40 G. Bhatnagar and M.J. Schlosser
�
¸
0¤|j|¤N
j1,j2,...,jm¥0
� ¹
1¤r s¤m
θ
�
qjr�jsyr{ys; p
�
θ pyr{ys; pq
m¹
r,s�1
pfsyr{ys; q, pqjr
pqyr{ys; q, pqjr
�
θ
�
λq2|j|; p
�
θ pλ; pq
pλ, λb{a, λc{a, λd{a; q, pq|j|
pλqN�1, aq{b, aq{c, aq{d; q, pq|j|
�
�
e1 � � � en, λaq
1�N{e1 � � � enf1 � � � fm, q
�N ; q, p
�
|j|
pλq{e1 � � � en, λq{f1 � � � fm, e1 � � � enf1 � � � fmq�N{a; q, pq|j|
�
m¹
r�1
phyr{f1 � � � fm; q, pqjrphyr; q, pq|j|
phyr; q, pqjrphyr{fr; q, pq|j|
� q
m°
r�1
rjr
�
. (11.6)
Note that the series on the right-hand side is of the same type as that on the left-hand-side.
Proof. We begin with the left-hand side, and write it in the form
Ņ
K�0
fK
¸
|k|�K
Ak.
We now use Theorem 11.1 (with ak ÞÑ ek and b ÞÑ g) to obtain a single sum. We transform
this sum using the n � 1 case of the elliptic Bailey transformation formula given in (5.2). Once
again, we use Theorem 11.1, this time with n � m, xk ÞÑ yk, b ÞÑ h, and aj ÞÑ fj . In this
manner, we obtain the right-hand side of (11.6). �
An analytic continuation argument similar to the one used in the proof of Theorem 5.3 can be
applied to Theorem 11.6 to obtain a transformation for a sum over an n-dimensional rectangle
into a multiple of a sum over an n-simplex.
Theorem 11.7 (an An elliptic Bailey 10φ9 transformation theorem). Let λ � qa2{bcd. Then
¸
0¤kr¤Nr
r�1,2,...,n
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θ pxr{xs; pq
n¹
r,s�1
�
q�Nsxr{xs; q, p
�
kr
pqxr{xs; q, pqkr
�
θ
�
aq2|k|; p
�
θ pa; pq
pa, b, c, d; q, pq|k|�
aq|N |�1, aq{b, aq{c, aq{d; q, p
�
|k|
�
�
e, f1 � � � fm, λaq
1�|N |{ef1 � � � fm; q, p
�
|k|�
aq{e, aq{f1 � � � fm, ef1 . . . fmq�|N |{λ; q, p
�
|k|
�
n¹
r�1
�
gxrq
|N |; q, p
�
kr
pgxr; q, pq|k|
pgxr; q, pqkrpgxrq
Nr ; q, pq|k|
� q
n°
r�1
rkr
�
�
paq, aq{ef1 � � � fm, λq{e, λq{f1 � � � fm; q, pq|N |
paq{e, aq{f1 � � � fm, λq, λq{ef1 � � � fm; q, pq|N |
�
¸
0¤|j|¤|N |
j1,j2,...,jm¥0
� ¹
1¤r s¤m
θ
�
qjr�jsyr{ys; p
�
θ pyr{ys; pq
m¹
r,s�1
pfsyr{ys; q, pqjr
pqyr{ys; q, pqjr
�
θ
�
λq2|j|; p
�
θ pλ; pq
pλ, λb{a, λc{a, λd{a; q, pq|j|�
λq|N |�1, aq{b, aq{c, aq{d; q, p
�
|j|
Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems 41
�
�
e, λaq1�|N |{ef1 � � � fm, q
�|N |; q, p
�
|j|�
λq{e, λq{f1 � � � fm, ef1 � � � fmq�|N |{a; q, p
�
|j|
�
m¹
r�1
phyr{f1 � � � fm; q, pqjrphyr; q, pq|j|
phyr; q, pqjrphyr{fr; q, pq|j|
� q
m°
r�1
rjr
�
.
As another example, we obtain a result similar to Theorem 11.1. For this we start with the
following Dn elliptic Jackson sum from Rosengren [31, Theorem 6.1] (rewritten, using elementary
manipulations of q, p-shifted factorials) which is equivalent to (7.1) upon replacing n by n � 1
and using an analytic continuation argument. In the p � 0 case the corresponding Dn Jackson
sum was first given in [36, Theorem 5.17]
¸
|k|�K
k1,k2,...,kn¥0
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θpxr{xs; pqpxrxs; q, pqkr�ks
�
n¹
r�1
pb{xr; q, pq|k|�kr
n�1±
s�1
pxras, xr{as; q, pqkr
pbxr; q, pqkr
n±
s�1
pqxr{xs; q, pqkr
� q
n°
r�1
pr�1qkr
�
�
n�1±
s�1
pbas, b{as; q, pqK
pq; q, pqK
n±
r�1
pbxr; q, pqK
.
We use this identity to obtain a Dn version of Theorem 11.1.
Theorem 11.8. Given the sequence fk, k � 0, 1, 2, . . . , and N ¥ 0, we have
¸
0¤|k|¤N
k1,k2,...,kn¥0
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θpxr{xs; pqpxrxs; q, pqkr�ks
n�1¹
s�1
n±
r�1
pxras, xr{as; q, pqkr
pbas, b{as; q, pq|k|
�
n¹
r�1
pbxr; q, pq|k|pb{xr; q, pq|k|�kr
pbxr; q, pqkr
n±
s�1
pqxr{xs; q, pqkr
� q
n°
r�1
pr�1qkr
f|k|
�
�
Ņ
K�0
fK
pq; q, pqK
. (11.7)
As in Theorem 11.1 we can choose fK in Theorem 11.8 appropriately and use a single series
identity to obtain a multiple series extension. In particular, we now obtain a new Dn elliptic
Jackson sum in this manner which involves two independent bases q and q̃, as well as two
independent nomes p and p̃.
Theorem 11.9 (a Dn Jackson sum). We have
¸
0¤|k|¤N
k1,k2,...,kn¥0
� ¹
1¤r s¤n
θ
�
qkr�ksxr{xs; p
�
θpxr{xs; pqpxrxs; q, pqkr�ks
n�1¹
s�1
n±
r�1
pxrys, xr{ys; q, pqkr
pgys, g{ys; q, pq|k|
�
n¹
r�1
pgxr; q, pq|k|pg{xr; q, pq|k|�kr
pgxr; q, pqkr
n±
s�1
pqxr{xs; q, pqkr
� q
n°
r�1
pr�1qkr
pq; q, pq|k|
42 G. Bhatnagar and M.J. Schlosser
�
θ
�
aq̃2|k|; p̃
�
θ
�
a; p̃
� pa, b, c, d, a2q̃1�N{bcd, q̃�N ; q̃, p̃q|k|
pq̃, aq̃{b, aq̃{c, aq̃{d, bcdq̃�N{a, aq̃N�1; q̃, p̃q|k|
q̃|k|
�
�
paq̃, aq̃{bc, aq̃{bd, aq̃{cd; q̃, p̃qN
paq̃{b, aq̃{c, aq̃{d, aq̃{bcd; q̃, p̃qN
. (11.8)
Proof. We replace b by g, and as by ys, for s � 1, . . . , n� 1 in (11.7), and take
fK � pq; q, pqK
θ
�
aq̃2K ; p̃
�
θ
�
a; p̃
� pa, b, c, d, a2q̃1�N{bcd, q̃�N ; q̃, p̃qK
pq̃, aq̃{b, aq̃{c, aq̃{d, bcdq̃�N{a, aq̃N�1; q̃, p̃qK
q̃K .
The rest of the proof is similar to the proof of Theorem 11.2. �
The last example shows that to obtain an elliptic extension of a terminating basic hypergeo-
metric series identity is not just a matter of replacing q-shifted factorials by q, p-shifted factorials.
While in the elliptic case, the factors depending on g in (11.8) are essential, they are not required
in the basic case (where one could let g Ñ 0).
This brings us to the end of our study.
Acknowledgements
The first author thanks Hjalmar Rosengren and the organizers of OPSF-S6 for the series of
lectures [32] on this subject. We thank Ole Warnaar for showing his notes [48] and much en-
couragement, Slava Spiridonov for some comments, and Zhizheng Zhang and Junli Huang for
sending us their preprint [51]. We thank the anonymous referees for many insightful suggestions.
We thank the Erwin Schrödinger Institute for its workshop on Elliptic hypergeometric functions
in combinatorics, integrable systems and physics held in Vienna in March 2017, where we bene-
fited from discussions with other participants. Finally, research of both authors was supported
by a grant of the Austrian Science Fund (FWF): F 50-N15.
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1 Introduction
2 The notation and terminology
3 The WP Bailey transform and lemma: a very short introduction
4 Consequences of an An elliptic Jackson summation of Rosengren
5 An An elliptic Bailey transformation
6 Special cases: new An Watson transformations and related identities
7 Another WP Bailey pair for B(1)
8 The matrix B(4)
9 Consequences of a Cn elliptic Jackson sum due to Rosengren
10 Summary of results
11 Other elliptic Jackson summations, with an extra parameter
References
|
| id | nasplib_isofts_kiev_ua-123456789-209439 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:12:50Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bhatnagar, G. Schlosser, M.J. 2025-11-21T18:52:47Z 2018 Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems / G. Bhatnagar, M.J. Schlosser // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 52 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D67 arXiv: 1704.00020 https://nasplib.isofts.kiev.ua/handle/123456789/209439 https://doi.org/10.3842/SIGMA.2018.025 We list An, Cn, and Dn extensions of the elliptic WP Bailey transform and lemma, given for n=1 by Andrews and Spiridonov. Our work requires multiple series extensions of Frenkel and Turaev's terminating, balanced, and very-well-poised ₁₀V₉ elliptic hypergeometric summation formula due to Rosengren and Rosengren and Schlosser. In our study, we discover two new An ₁₂V₁₁ transformation formulas that reduce to two new An extensions of Bailey's 10ϕ9 transformation formulas when the nome p is 0, and two multiple series extensions of Frenkel and Turaev's sum. The first author thanks Hjalmar Rosengren and the organizers of OPSF-S6 for the series of lectures [32] on this subject. We thank Ole Warnaar for showing his notes [48] and much encouragement, Slava Spiridonov for some comments, and Zhizheng Zhang and Junli Huang for sending us their preprint [51]. We thank the anonymous referees for many insightful suggestions. We thank the Erwin Schrödinger Institute for its workshop on Elliptic hypergeometric functions in combinatorics, integrable systems, and physics held in Vienna in March 2017, where we benefited from discussions with other participants. Finally, the research of both authors was supported by a grant of the Austrian Science Fund (FWF): F 50-N15. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems Article published earlier |
| spellingShingle | Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems Bhatnagar, G. Schlosser, M.J. |
| title | Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems |
| title_full | Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems |
| title_fullStr | Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems |
| title_full_unstemmed | Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems |
| title_short | Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems |
| title_sort | elliptic well-poised bailey transforms and lemmas on root systems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209439 |
| work_keys_str_mv | AT bhatnagarg ellipticwellpoisedbaileytransformsandlemmasonrootsystems AT schlossermj ellipticwellpoisedbaileytransformsandlemmasonrootsystems |