Gustafson-Rakha-Type Elliptic Hypergeometric Series
We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey...
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irk-123456789-1486362019-02-19T01:31:22Z Gustafson-Rakha-Type Elliptic Hypergeometric Series Rosengren, H. We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case. 2017 Article Gustafson-Rakha-Type Elliptic Hypergeometric Series / H. Rosengren // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 21 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D67 DOI:10.3842/SIGMA.2017.037 http://dspace.nbuv.gov.ua/handle/123456789/148636 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case. |
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Rosengren, H. |
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Rosengren, H. Gustafson-Rakha-Type Elliptic Hypergeometric Series Symmetry, Integrability and Geometry: Methods and Applications |
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Rosengren, H. |
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Rosengren, H. |
| title |
Gustafson-Rakha-Type Elliptic Hypergeometric Series |
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Gustafson-Rakha-Type Elliptic Hypergeometric Series |
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Gustafson-Rakha-Type Elliptic Hypergeometric Series |
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Gustafson-Rakha-Type Elliptic Hypergeometric Series |
| title_full_unstemmed |
Gustafson-Rakha-Type Elliptic Hypergeometric Series |
| title_sort |
gustafson-rakha-type elliptic hypergeometric series |
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Інститут математики НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/148636 |
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Gustafson-Rakha-Type Elliptic Hypergeometric Series / H. Rosengren // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 21 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT rosengrenh gustafsonrakhatypeelliptichypergeometricseries |
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2025-07-12T19:50:33Z |
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2025-07-12T19:50:33Z |
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1837471986541920256 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 037, 11 pages
Gustafson–Rakha-Type Elliptic Hypergeometric
Series
Hjalmar ROSENGREN
Department of Mathematical Sciences, Chalmers University of Technology and
University of Gothenburg, SE-412 96 Gothenburg, Sweden
E-mail: hjalmar@chalmers.se
URL: http://www.math.chalmers.se/~hjalmar/
Received February 02, 2017, in final form May 29, 2017; Published online June 01, 2017
https://doi.org/10.3842/SIGMA.2017.037
Abstract. We prove a multivariable elliptic extension of Jackson’s summation formula
conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson
and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations
and two multivariable elliptic Bailey transformations. The latter four results are all new
even in the trigonometric case.
Key words: elliptic hypergeometric series; multivariable hypergeometric series; Jackson sum-
mation; Bailey transformation
2010 Mathematics Subject Classification: 33D67
1 Introduction
By combining two integral evaluations previously obtained by Gustafson [6], Gustafson and
Rakha [7] evaluated the basic hypergeometric integral
∫ ∏
1≤i<j≤n
(zi/zj)∞(zj/zi)∞
n∏
j=1
(S/zj)∞
∏
1≤i<j≤n
(tzizj)∞
n∏
i=1
(
3∏
j=1
(cjzi)∞
n∏
j=1
(dj/zi)∞
) dz1
z1
· · · dzn−1
zn−1
, (1.1)
where the integration is over |z1| = · · · = |zn−1| = 1, (z)∞ =
∞∏
j=0
(1− qjz), the parameters satisfy
|q|, |t|, |c1|, |c2|, |c3|, |d1|, . . . , |dn| < 1,
zn is determined from the integration variables through z1 · · · zn = 1 and S = tn−2c1c2c3d1 · · · dn.
By applying residue calculus to (1.1), they could evaluate a certain multivariable basic hyper-
geometric finite sum, equivalent to the case p = 0 of Theorem 3.1 below.
Since the seminal work of Date et al. [4] and Frenkel and Turaev [5], it has been recognized
that basic hypergeometric functions appear as the trigonometric limit of more general elliptic
hypergeometric functions. Elliptic extensions of Gustafson’s two integral evaluations mentioned
above were conjectured in [16, 19] and proved in [10]. Spiridonov [16] used these (at the time
conjectural) evaluations to obtain an elliptic extension of (1.1). He also stated the corresponding
summation formula as a conjecture. Although it seems likely that this conjecture can be deduced
from Spiridonov’s integral, such a derivation is still missing from the literature. The purpose 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:hjalmar@chalmers.se
http://www.math.chalmers.se/~hjalmar/
https://doi.org/10.3842/SIGMA.2017.037
https://www.emis.de/journals/SIGMA/EHF2017.html
2 H. Rosengren
the present paper is to give a direct proof of Spiridonov’s conjectured summation and to apply
it to derive some further summation and transformation formulas.
It is worth mentioning that Spiridonov’s elliptic extension of (1.1) can be interpreted as the
identity between superconformal indices of two dual quantum field theories [17, Sections 12.1.2–
12.1.3]. This indicates that (1.1) and related results are not mere curiosities and that it is not
unreasonable to expect further applications.
The plan of the paper is as follows. In Section 3, we prove Spiridonov’s conjecture. The proof
is elementary and provides in particular a significant simplification of the trigonometric case.
The only previously known proof of the Gustafson–Rakha summation is the original one, which
as we recall is based on first proving two auxiliary multiple integral evaluations, combining them
to obtain (1.1) and finally on a technical computation to pass from integrals to finite residue
sums. In Section 4, we give some applications of our result. Namely, combining the elliptic
Gustafson–Rakha sum with a summation from [14], we obtain two transformation formulas and
two further summation formulas for multivariable elliptic hypergeometric series. These four
results are all new even in the trigonometric case.
Note added in proof: After completing this work, I learned from Masahiko Ito and
Masatoshi Noumi that they have independently proved Theorem 3.1, using a different method.
2 Preliminaries
When z = (z1, . . . , zn) is a vector we will write |z| = z1 + · · ·+ zn and Z = z1 · · · zn.
Throughout, p and q will be fixed parameters with |p| < 1. We employ the standard notation
θ(z) =
∞∏
j=0
(
1− pjz
)(
1− pj+1/z
)
,
(z)k =
{
θ(z)θ(qz) · · · θ
(
qk−1z
)
, k ∈ Z≥0,
1/θ
(
qkz
)
θ
(
qk+1z
)
· · · θ
(
q−1z
)
, k ∈ Z<0
as well as
θ(z1, . . . , zm) = θ(z1) · · · θ(zm), (z1, . . . , zm)k = (z1)k · · · (zm)k.
Most of our computations are based on the elementary identities
θ(1/z) = θ(pz) = −θ(z)/z,
(a)n+k = (a)n(aqn)k, (a)n−k = (−1)kq(
k
2)(q1−n/a)k
(a)n
(q1−n/a)k
,
which will be used without comment. All our sums contain the A-type factor
∆(zqx)
∆(z)
=
∏
1≤i<j≤n
qxiθ(qxj−xizj/zi)
θ(zj/zi)
,
where z = (z1, . . . , zn) and x = (x1, . . . , xn) is the summation index. We mention the useful
identity [12, equation (3.8)]
∆(zqx)
∆(z)
= (−1)|x|q−(|x|2 )−|x|
n∏
i,j=1
(qzi/zj)xi
(q−xjzi/zj)xi
(2.1)
Gustafson–Rakha-Type Elliptic Hypergeometric Series 3
and [21, Example 20.53.3]
n∑
k=1
n+1∏
j=1
θ(zk/bj)
θ(zk/t)
n∏
j=1, j 6=k
θ(zk/zj)
=
n+1∏
j=1
θ(bj/t)
n∏
j=1
θ(zj/t)
, (2.2)
valid for tz1 · · · zn = b1 · · · bn+1.
In the one-variable case, the most fundamental results for elliptic hypergeometric series are
the elliptic Jackson (or Frenkel–Turaev) summation
N∑
x=0
θ
(
aq2x
)
θ(a)
(
a, b, c, d, e, q−N
)
x
qx(
q, aq/b, aq/c, aq/d, aq/e, aqN+1
)
x
=
(aq, aq/bc, aq/bd, aq/cd)N
(aq/b, aq/c, aq/d, aq/bcd)N
, (2.3)
valid for a2qN+1 = bcde, and the elliptic Bailey transformation
N∑
x=0
θ
(
aq2x
)
θ(a)
(
a, b, c, d, e, f, g, q−N
)
x
qx(
q, aq/b, aq/c, aq/d, aq/e, aq/f, aq/g, aqN+1
)
x
(2.4)
=
(aq, aq/ef, λq/e, λq/f)N
(λq, λq/ef, aq/e, aq/f)N
N∑
x=0
θ
(
λq2x
)
θ(λ)
(
λ, λb/a, λc/a, λd/a, e, f, g, q−N
)
x
qx(
q, aq/b, aq/c, aq/d, λq/e, λq/f, λq/g, λqN+1
)
x
valid for a3qN+2 = bcdefg, λ = a2q/bcd [5]. Numerous multivariable extensions of (2.3) and (2.4)
are known, see, e.g., [3, 9, 11, 12, 13, 14, 18, 20]; further examples are obtained in the present
paper. We will need one such result, a multivariable extension of (2.3) obtained in [14] (see [15]
for the case p = 0). Namely, for a2q|N |+1 = bcde,
N1,...,Nn∑
x1,...,xn=0
∆(zqx)
∆(z)
θ
(
aq2|x|
)
θ(a)
(a, b, c)|x|
n∏
i=1
(d/zi)|x|(
aq/b, aq/c, aq|N |+1
)
|x|
n∏
i=1
(
aq|N |+1−Ni/ezi
)
|x|
q|x|
×
n∏
i=1
(
aq|N |+1/ezi
)
|x|−xi
(ezi)xi
n∏
j=1
(
q−Njzi/zj
)
xi
(d/zi)|x|−xi(aqzi/d)xi
n∏
j=1
(qzi/zj)xi
=
(aq, aq/bc)|N |
(aq/b, aq/c)|N |
n∏
i=1
(aqzi/bd, aqzi/cd)Ni
(aqzi/d, aqzi/bcd)Ni
. (2.5)
3 The elliptic Gustafson–Rakha summation
Our main result is the following identity. It is easy to see that the case p = 0 is equivalent
to [7, Theorem 1.2] and the general case to the conjecture of [16, p. 953]. Recall the notation
Z = z1 · · · zn.
Theorem 3.1. For parameters subject to qN−1b1 · · · b4z21 · · · z2n = 1,
∑
x1,...,xn≥0,
x1+···+xn=N
∆(zqx)
∆(z)
∏
1≤i<j≤n
qxixj (zizj)xi+xj
n∏
i=1
4∏
j=1
(zibj)xi
zxii
n∏
j=1
(qzi/zj)xi
4 H. Rosengren
=
(Zb1, Zb2, Zb3, Zb4)N
ZN (q)N
, n odd,
(Z,Zb1b2, Zb1b3, Zb1b4)N
(Zb1)N (q)N
, n even.
(3.1)
We will prove Theorem 3.1 by induction on N . In the case N = 1, we have xi = δik for
some k. Using k as summation index, Theorem 3.1 reduces to the following theta function
identity.
Lemma 3.2. For parameters subject to b1b2b3b4z
2
1 · · · z2n = 1,
n∑
k=1
4∏
j=1
θ(zkbj)
zk
n∏
j=1, j 6=k
θ(zkzj)
θ(zk/zj)
=
{
θ(Zb1, Zb2, Zb3, Zb4)/Z, n odd,
θ(Z,Zb1b2, Zb1b3, Zb1b4)/Zb1, n even.
(3.2)
Proof. We apply induction on n, starting from the trivial case n = 1. Let b1 = vw and
b2 = v/w, with v and w free parameters. As a function of w, each term in the sum, as well as
the right-hand side, has the form f(w) = Cθ(aw, a/w) with C and a independent of w. It is
a classical fact that any such function is determined by its values at two generic points. Indeed,
Weierstrass’ identity (which is equivalent to the case n = 2 of (2.2)) states that
f(w) = f(b)
θ(cw, c/w)
θ(cb, c/b)
+ f(c)
θ(bw, b/w)
θ(bc, b/c)
,
provided that bc, b/c /∈ pZ. Thus, it suffices to verify (3.2) for two independent values of b1.
Assuming n ≥ 2, we choose b1 = 1/zn−1 and b1 = 1/zn. By symmetry, it is enough to consider
the second case. Then, the term corresponding to k = n cancels and we are reduced to an
identity equivalent to (3.2), with n replaced by n− 1 and b1 by zn. �
We mention that it is not hard to deduce (3.2) from classical theta function identities. Indeed,
let t = bn+1 in (2.2) (so that the right-hand side vanishes) and then make the substitutions
n 7→ n+ 4, (z1, . . . , zn) 7→ (z1, . . . , zn, 1,−1,
√
p,−1/
√
p), (b1, . . . , bn) 7→ (z−11 , . . . , z−1n , b−11 , b−12 ,
b−13 , b−14 ). Using the elementary identities θ(z2) = θ(z,−z,√pz,−√pz) and 2 = θ(−1,
√
p,−√p),
one may deduce that the left-hand side of (3.2) is equal to
1
2
(−1)n+1Z
4∏
j=1
θ(bj) + Z
4∏
j=1
θ(−bj) +
1
√
p
4∏
j=1
θ
(√
pbj
)
− 1
√
p
4∏
j=1
θ
(
−√pbj
) .
The fact that this equals the right-hand side of (3.2) follows from Jacobi’s fundamental formulae
[21, Section 21.22].
The inductive step in the proof of Theorem 3.1 is almost identical to that of [12, Theorem 5.1].
Denoting the right-hand side of (3.1) by RN (Z; b1, b2, b3, b4) (where we for a moment consider Z
as a free variable) we observe that, regardless of the parity of n,
RN+1(Z; b1, b2, b3, b4) =
qNθ(q)
θ(qN+1)
R1
(
qNZ; q−Nb1, b2, b3, b4
)
RN (Z; qb1, b2, b3, b4).
Assuming (3.1) for fixed N , it follows that
RN+1(Z; b1, b2, b3, b4) =
qNθ(q)
θ(qN+1)
R1
(
qNZ; q−Nb1, b2, b3, b4
)
Gustafson–Rakha-Type Elliptic Hypergeometric Series 5
×
∑
x1,...,xn≥0,
x1+···+xn=N
∆(zqx)
∆(z)
∏
1≤i<j≤n
qxixj (zizj)xi+xj
n∏
i=1
(qzib1)xi
4∏
j=2
(zibj)xi
zxii
n∏
j=1
(qzi/zj)xi
,
where Z = z1 · · · zn and qNBZ2 = 1. We pull the factor R1 inside the sum and expand it
using (3.1), with zi replaced by qxizi. This gives
RN+1(Z; b1, b2, b3, b4)
=
qNθ(q)
θ(qN+1)
∑
x1,...,xn≥0,
x1+···+xn=N
∑
y1,...,yn≥0,
y1+···+yn=1
∆(zqx+y)
∆(z)
∏
1≤i<j≤n
qxixj (zizj)xi+xj+yi+yj
×
n∏
i=1
(qzib1)xi
(
qxi−Nzib1
)
yi
4∏
j=2
(zibj)xi+yi
zxii (ziqxi)yi
n∏
j=1
(qzi/zj)xi
(
q1+xi−xjzi/zj
)
yi
=
qNθ(q)
θ(qN+1)
∑
x1,...,xn≥0,
x1+···+xn=N+1
∑
y1,...,yn≥0,
y1+···+yn=1
∆(zqx)
∆(z)
∏
1≤i<j≤n
qxixj−xiyj−xjyi(zizj)xi+xj
×
n∏
i=1
(qzib1)xi−yi
(
qxi−yi−Nzib1
)
yi
4∏
j=2
(zibj)xi
zxii q
(xi−yi)yi
n∏
j=1
(qzi/zj)xi−yi
(
q1+xi−xj−yi+yjzi/zj
)
yi
,
where we replaced each xi by xi − yi and used that yiyj = 0 for i 6= j.
By elementary manipulations, using∏
i<j
qxiyj+xjyi
∏
i
q(xi−yi)yi = q(x1+···+xn)(y1+···+yn)−(y21+···+y2n) = q(N+1)·1−1 = qN ,
the expression above can be rewritten
RN+1(Z; b1, b2, b3, b4)
=
1
θ(qN+1)
∑
x1,...,xn≥0,
x1+···+xn=N+1
∆(zqx)
∆(z)
∏
1≤i<j≤n
qxixj (zizj)xi+xj
n∏
i=1
(qzib1)xi
4∏
j=2
(zibj)xi
zxii
n∏
j=1
(qzi/zj)xi
×
∑
y1,...,yn≥0,
y1+···+yn=1
n∏
i=1
(
qxi−yi−Nzib1
)
yi
n∏
j=1
(
q1+xi−yizi/zj
)
yi(
q1+xi−yizib1
)
yi
n∏
j=1, j 6=i
(
qxi−xjzi/zj
)
yi
.
Writing yi = δik, the inner sum takes the form
n∑
k=1
θ
(
qxk−N−1zkb1
) n∏
j=1
θ(qxkzk/zj)
θ(qxkzkb1)
n∏
j=1, j 6=k
θ(qxk−xjzk/zj)
.
6 H. Rosengren
By (2.2), this can be evaluated as
θ
(
qN+1
) n∏
i=1
θ(zib1)
θ(qxizib1)
= θ
(
qN+1
) n∏
i=1
(zib1)xi
(qzib1)xi
and we arrive at (3.1) with N replaced by N + 1. This completes the proof of Theorem 3.1.
We will now rewrite (3.1) in a way that hides some of its symmetry but makes it clear that
it generalizes the Frenkel–Turaev summation (2.3). To this end, we replace n by n + 1, zn+1
by q−Na−1 and eliminate xn+1 from the summation. After routine simplification, we arrive at
the following identity.
Corollary 3.3. Assuming a2qN+1 = b1b2b3b4z
2
1 · · · z2n,
∑
x1,...,xn≥0,
x1+···+xn≤N
∆(zqx)
∆(z)
n∏
i=1
θ(aziq
|x|+xi)
θ(azi)
∏
1≤i<j≤n
(zizj)xi+xj
n∏
i=1
(aq/zi)|x|−xi
(
q−N
)
|x|
n∏
i=1
(azi)|x|
4∏
j=1
(aq/bj)|x|
q|x|
×
n∏
i=1
4∏
j=1
(zibj)xi(
aqN+1zi
)
xi
n∏
j=1
(qzi/zj)xi
=
n∏
j=1
(aqzj)N(
aq/b1, aq/b2, aq/b3, aq/b1b2b3Z2
)
N
n∏
j=1
(aq/zj)N
×
(aq/Z, aq/b1b2Z, aq/b1b3Z, aq/b2b3Z)N , n odd,
(aq/b1Z, aq/b2Z, aq/b3Z, aq/b1b2b3Z)N , n even.
(3.3)
4 Applications
The elliptic Bailey transformation (2.4) can be derived from the elliptic Jackson summation (2.3).
Similar arguments can be used in multivariable situations, see, e.g., [1, 2, 8] for the trigonometric
and [12] for the elliptic case. We will use this method to derive a new multivariable elliptic Bailey
transformation by combining the two multivariable elliptic Jackson summations (2.5) and (3.3).
Theorem 4.1. Suppose that a3qN+2 = bcdefgz21 · · · z2n and let λ = a2q/bcd. Then,
∑
x1,...,xn≥0,
x1+···+xn≤N
∆(zqx)
∆(z)
n∏
i=1
θ(aziq
|x|+xi)
θ(azi)
∏
1≤i<j≤n
(zizj)xi+xj
n∏
i=1
(aq/zi)|x|−xi
×
(
q−N , b
)
|x|
n∏
i=1
(azi)|x|
(aq/c, aq/d, aq/e, aq/f, aq/g)|x|
q|x|
n∏
i=1
(czi, dzi, ezi, fzi, gzi)xi(
aqN+1zi, aqzi/b
)
xi
n∏
j=1
(qzi/zj)xi
=
zN
n∏
i=1
(aqzi)N
(λq, aq/e, aq/f, aq/g)N
n∏
i=1
(aq/zi)N
Gustafson–Rakha-Type Elliptic Hypergeometric Series 7
×
∑
x1,...,xn≥0,
x1+···+xn≤N
∆(zqx)
∆(z)
θ
(
λq2|x|
)
θ(λ)
∏
1≤i<j≤n
(zizj)xi+xj
n∏
i=1
(λb/azi)|x|−xi
×
(
λ, q−N , λc/a, λd/a
)
|x|
n∏
i=1
(λb/azi)|x|(
λqN+1, aq/c, aq/d
)
|x|
q|x|
n∏
i=1
(
ezi, fzi, gzi, q
−Nzi/a
)
xi
(aqzi/b)xi
n∏
j=1
(qzi/zj)xi
×
(a
λ
)N (aq/Z, λq/eZ, λq/fZ, λq/gZ)N
(q−NZ/a, λq/eZ, λq/fZ, λq/gZ)|x|
, n odd,
(λq/Z, aq/eZ, aq/fZ, aq/gZ)N
(λq/Z, λq/efZ, λq/egZ, λq/fgZ)|x|
, n even.
(4.1)
Proof. If we substitute
(N1, . . . , Nn, a, b, c, d, e) 7→
(
x1, . . . , xn, λ, λc/a, λd/a, λb/a, aq
|x|)
in (2.5), the right-hand side takes the form
(λq, b)|x|
(aq/c, aq/d)|x|
n∏
i=1
(czi, dzi)xi
(aqzi/b, azi/λ)xi
.
Thus, the left-hand side of (4.1) can be expressed as
∑
x1,...,xn≥0,
x1+···+xn≤N
∆(zqx)
∆(z)
n∏
i=1
θ(aziq
|x|+xi)
θ(azi)
∏
1≤i<j≤n
(zizj)xi+xj
n∏
i=1
(aq/zi)|x|−xi
×
(
q−N
)
|x|
n∏
i=1
(azi)|x|
(λq, aq/e, aq/f, aq/g)|x|
q|x|
n∏
i=1
(azi/λ, ezi, fzi, gzi)xi(
aqN+1zi
)
xi
n∏
j=1
(qzi/zj)xi
×
x1,...,xn∑
y1,...,yn=0
∆(zqy)
∆(z)
θ
(
λq2|y|
)
θ(λ)
(λ, λc/a, λd/a)|y|
n∏
i=1
(λb/azi)|y|(
aq/c, aq/d, λq|x|+1
)
|y|
n∏
i=1
(
λq1−xi/azi
)
|y|
q|y|
×
n∏
i=1
(λq/azi)|y|−yi
(
aq|x|zi
)
yi
n∏
j=1
(q−xjzi/zj)yi
(λb/azi)|y|−yi(aqzi/b)yi
n∏
j=1
(qzi/zj)yi
.
We change the order of summation and replace the vector x by x+ y. Some elementary manip-
ulation, using in particular (2.1), gives
∑
y1,...,yn≥0,
y1+···+yn≤N
∆(zqy)
∆(z)
θ
(
λq2|y|
)
θ(λ)
∏
1≤i<j≤n
q−yiyj (zizj)yi+yj
n∏
i=1
(aq/zi, λb/azi)|y|−yi
n∏
i=1
(aqzi)|y|+yi
(λq)2|y|
×
(
q−N , λ, λc/a, λd/a
)
|y|
n∏
i=1
(λb/azi)|y|
(aq/c, aq/d, aq/e, aq/f, aq/g)|y|
(aq
λ
)|y| n∏
i=1
(ezi, fzi, gzi)yiz
yi
i(
aqN+1zi, aqzi/b
)
yi
n∏
j=1
(qzi/zj)yi
8 H. Rosengren
×
∑
x1,...,xn≥0,
x1+···+xn≤N−|y|
∆(zqy+x)
∆(zqy)
n∏
i=1
θ
(
aziq
|y|+yi+|x|+xi
)
θ
(
aziq|y|+yi
)
∏
1≤i<j≤n
(
zizjq
yi+yj
)
xi+xj
n∏
i=1
(
aq|y|+1−yi/zi
)
|x|−xi
×
(
q|y|−N
)
|x|
n∏
i=1
(
aziq
|y|+yi
)
|x|(
aq|y|+1/e, aq|y|+1/f, aq|y|+1/g, λq2|y|+1
)
|x|
q|x|
×
n∏
i=1
(
eziq
yi , fziq
yi , gziq
yi , qyi−|y|azi/λ
)
xi(
aqN+1+yizi
)
xi
n∏
j=1
(
q1+yi−yjzi/zj
)
xi
.
We observe that the inner sum is as in Corollary 3.3, with the substitutions
(z1, . . . , zn, N, a, b1, b2, b3, b4) 7→
(
z1q
y1 , . . . , znq
yn , N − |y|, aq|y|, e, f, g, q−|y|a/λ
)
.
When n is odd, the value of this sum can be rewritten
(aq/Z, aq/efZ, aq/egZ, aq/fgZ)N−|y|
n∏
i=1
(
aq|y|+1+yizi
)
N−|y|(
aq|y|+1/e, aq|y|+1/f, aq|y|+1/g, q−N−|y|/λ
)
N−|y|
n∏
i=1
(
aq|y|+1−yi/zi
)
N−|y|
= zN
(a
λ
)N−|y| (aq/Z, λq/eZ, λq/fZ, λq/gZ)N
n∏
i=1
(aqzi)N
(λq, aq/e, aq/f, aq/g)N
n∏
i=1
(aq/zi)N
×
q
∑
i<j
yiyj
(λq)2|y|(aq/e, aq/f, aq/g)|y|
n∏
i=1
(aq/zi)|y|−yi
(
aqN+1zi, q
−Nzi/a
)
yi(
λqN+1, q−NZ/a, λq/eZ, λq/fZ, λq/gZ
)
|y|
n∏
i=1
zyii (aqzi)|y|+yi
,
which leads to the right-hand side of (4.1). The case of even n is treated similarly. �
One may obtain further transformation formulas by iterating Theorem 4.1. We will only give
one example, exploiting the fact that the left-hand side of (4.1) is invariant under interchan-
ging c and e. In the identity expressing the corresponding symmetry of the right-hand side, we
make the substitutions (λ, a, b, c, d) 7→ (a, a2q/bcd, aq/cd, aq/bd, aq/bc), keeping e, f, g, z1, . . . , zn
fixed. This leads to another multivariable elliptic Bailey transformation.
Corollary 4.2. Suppose that a3qN+2 = bcdefgz21 · · · z2n and let λ = a2q/bde. Then,
∑
x1,...,xn≥0,
x1+···+xn≤N
∆(zqx)
∆(z)
θ
(
aq2|x|
)
θ(a)
∏
1≤i<j≤n
(zizj)xi+xj
n∏
i=1
(b/zi)|x|−xi
×
(
a, q−N , c, d
)
|x|
n∏
i=1
(b/zi)|x|(
aqN+1, aq/c, aq/d
)
|x|
q|x|
n∏
i=1
(
ezi, fzi, gzi, aqzi/efgZ
2
)
xi
(aqzi/b)xi
n∏
j=1
(qzi/zj)xi
×
1
(aq/eZ, aq/fZ, aq/gZ, aq/efgZ)|x|
, n odd,
1
(aq/Z, aq/efZ, aq/egZ, aq/fgZ)|x|
, n even
Gustafson–Rakha-Type Elliptic Hypergeometric Series 9
=
(aq, λq/c)N
(λq, aq/c)N
∑
x1,...,xn≥0,
x1+···+xn≤N
∆(zqx)
∆(z)
θ
(
λq2|x|
)
θ(λ)
∏
1≤i<j≤n
(zizj)xi+xj
n∏
i=1
(λb/azi)|x|−xi
×
(
λ, q−N , c, λd/a
)
|x|
n∏
i=1
(λb/azi)|x|(
λqN+1, λq/c, aq/d
)
|x|
q|x|
n∏
i=1
(
λezi/a, fzi, gzi, aqzi/efgZ
2
)
xi
(aqzi/b)xi
n∏
j=1
(qzi/zj)xi
×
(aq/cfZ, λq/fZ)N
(aq/fZ, λq/cfZ)N (aq/eZ, λq/fZ, λq/gZ, aq/efgZ)|x|
, n odd,
(aq/cZ, λq/Z)N
(aq/Z, λq/cZ)N (λq/Z, aq/efZ, aq/egZ, λq/fgZ)|x|
, n even.
Theorem 4.1 reduces to Corollary 3.3 when aq = bc. More interestingly, when b = 1 the
left-hand side of (4.1) reduces to 1. After a change of parameters, this leads to the following
new multivariable elliptic Jackson summation.
Corollary 4.3. If a2qN+1 = bcdez21 · · · z2n, then
∑
x1,...,xn≥0
x1+···+xn≤N
∆(zqx)
∆(z)
θ
(
aq2|x|
)
θ(a)
∏
1≤i<j≤n
(zizj)xi+xj
n∏
i=1
(e/zi)|x|−xi
(
a, q−N
)
|x|
n∏
i=1
(e/zi)|x|(
aqN+1
)
|x|
q|x|
×
n∏
i=1
(
bzi, czi, dzi, q
−Nezi/a
)
xi
(aqzi/e)xi
n∏
j=1
(qzi/zj)xi
·
1(
q−NeZ/a, aq/bZ, aq/cZ, aq/dZ
)
|x|
, n odd,
1
(aq/Z, aq/bcZ, aq/bdZ, aq/cdZ)|x|
, n even
=
(aq, aq/be, aq/ce, aq/de)N
n∏
i=1
(aq/ezi)N
ZN
n∏
i=1
(aqzi/e)N
×
eN
(aq/bZ, aq/cZ, aq/dZ, aq/eZ)N
, n odd,
1
(aq/Z, aq/beZ, aq/ceZ, aq/deZ)N
, n even.
Examining the proof of Theorem 4.1, we see that Corollary 4.3 is obtained by combining (3.3)
with the special case aq = bc of (2.5), when the right-hand side is equal to
n∏
i=1
δNi,0. The latter
identity can be viewed as a matrix inversion [14], so Corollary 4.3 is an inverted version of Corol-
lary 3.3, just as (2.5) is an inverted version of the standard A-type summation [12, Corollary 5.2].
If we let λd/a = 1 in Corollary 4.2, we obtain yet another multivariable elliptic Jackson
summation. After a change of parameters, it takes the form
∑
x1,...,xn≥0
x1+···+xn≤N
∆(zqx)
∆(z)
θ
(
aq2|x|
)
θ(a)
∏
1≤i<j≤n
(zizj)xi+xj
n∏
i=1
(t/zi)|x|−xi
(
a, q−N , b, c
)
|x|
n∏
i=1
(t/zi)|x|
(aqN+1, aq/b, aq/c)|x|
q|x|
10 H. Rosengren
×
n∏
i=1
(
dzi, ezi, tzi/deZ
2
)
xi
n∏
j=1
(qzi/zj)xi
·
1
(aq/dZ, aq/eZ, t/Z, t/deZ)|x|
, n odd,
1
(aq/Z, aq/deZ, t/dZ, t/eZ)|x|
, n even
=
(aq, aq/bc, aq/bdZ, aq/cdZ)N
(aq/b, aq/c, aq/dZ, aq/bcdZ)N
, n odd,
(aq, aq/bc, aq/bZ, aq/cZ)N
(aq/b, aq/c, aq/Z, aq/bcZ)N
, n even,
(4.2)
valid for a2qN+1 = bcdez21 · · · z2n and t arbitrary. This identity is less novel than Corollary 4.3,
as it can be deduced from Theorem 3.1 in a more direct manner. Indeed, writing the sum as
N∑
k=0
∑
x1,...,xn≥0
x1+···+xn=k
(· · · ),
the inner sum is computed by Theorem 3.1 and the outer sum by (2.3). In fact, the same
proof gives the following more general result, which reduces to (4.2) when (d, e) = (fZ, gZ) or
(Z, fgZ) if n is odd or even, respectively.
Corollary 4.4. For parameters subject to a2qN+1 = bcde, fghz21 · · · z2n = t,
∑
x1,...,xn≥0
x1+···+xn≤N
∆(zqx)
∆(z)
θ
(
aq2|x|
)
θ(a)
∏
1≤i<j≤n
(zizj)xi+xj
(
a, q−N , b, c, d, e
)
|x|(
aqN+1, aq/b, aq/c, aq/d, aq/e
)
|x|
q|x|
×
n∏
i=1
(t/zi)|x|(fzi, gzi, hzi)xi
(t/zi)|x|−xi
n∏
j=1
(qzi/zj)xi
·
1
(fZ, gZ, hZ, t/Z)|x|
, n odd,
1
(Z, fgZ, fhZ, ghZ)|x|
, n even
=
(aq, aq/bc, aq/bd, aq/cd)N
(aq/b, aq/c, aq/d, aq/bcd)N
.
Acknowledgements
This research is supported by the Swedish Science Research Council (Vetenskapsr̊adet). I would
like to thank the anonymous referee for a very careful reading of the manuscript, leading to
many improvements.
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1 Introduction
2 Preliminaries
3 The elliptic Gustafson–Rakha summation
4 Applications
References
|