Iteration process for multiple Rogers-Ramanujan identities
Replacing the monomials by an arbitrary sequence in the recursive lemma found by Bressoud (1983), we establish several general transformation formulas from unilateral multiple basic hypergeometric series to bilateral univariate ones, which are then used for the derivation of numerous multiple serie...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508473638256640 |
|---|---|
| author | Chu, W. Wang, C. Чу, У. Ван, С. |
| author_facet | Chu, W. Wang, C. Чу, У. Ван, С. |
| author_sort | Chu, W. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:29:28Z |
| description | Replacing the monomials by an arbitrary sequence in the recursive lemma found by Bressoud (1983), we establish several general transformation formulas
from unilateral multiple basic hypergeometric series to bilateral univariate ones, which are then used for the derivation of numerous multiple series identities of Rogers-Ramanujan type. |
| first_indexed | 2026-03-24T02:25:46Z |
| format | Article |
| fulltext |
UDC 517.5
W. Chu (Hangzhou Normal Univ., Inst. Combinat. Math., China),
C. Wang (Nanjing Univ. Inform. Sci. and Technol., China)
ITERATION PROCESS FOR MULTIPLE ROGERS – RAMANUJAN IDENTITIES
IТЕРАЦIЙНИЙ ПРОЦЕС ДЛЯ КРАТНИХ ТОТОЖНОСТЕЙ
РОДЖЕРСА – РАМАНУДЖАНА
Replacing the monomials by an arbitrary sequence in the recursive lemma found by Bressoud (1983), we establish several
general transformation formulas from unilateral multiple basic hypergeometric series to bilateral univariate ones, which are
then used for the derivation of numerous multiple series identities of Rogers – Ramanujan type.
За допомогою замiни мономiв довiльною послiдовнiстю в рекурентнiй лемi Брессо (1983) встановлено декiлька
загальних формул перетворення однобiчних кратних основних гiпергеометричних рядiв у двобiчнi одновимiрнi
ряди, якi потiм використовуються для виведення численних тотожностей типу Роджерса – Рамануджана для кратних
рядiв.
1. Introduction and motivation. By means of the following well-known q-analogue of the binomial
theorem
1
(qa; q)n
=
n∑
m=0
[
n
m
]
qm
2
am
(qa; q)m
, where
[
n
m
]
=
(q; q)n
(q; q)m(q; q)n−m
. (1)
Bressoud [19] (Lemma 2) devised ingeniously the recursive lemma on finite sums
n∑
k=−n
qλk
2
xk
(q; q)n+k(q; q)n−k
=
n∑
m=0
qm
2
(q; q)n−m
m∑
k=−m
q(λ−1)k
2
xk
(q; q)m+k(q; q)m−k
. (2)
Iterating the last equation `-times and then putting λ = `+1/2, he discovered the following multiple
series transformation theorem [19] (Theorem):∑
n≥m1≥m2≥...≥m`≥0
qm
2
1+m
2
2+...+m
2
` (x; q)m`(q/x; q)m`
(q; q)n−m1(q; q)m1−m2 . . . (q; q)m`−1−m`(q; q)2m`
= (3a)
=
n∑
k=−n
(−1)k
[
2n
n+ k
]
qk
2`+(k2)
(q; q)2n
xk. (3b)
The limiting case n→∞ of the last formula yields easy proofs and generalizations of the celebrated
Rogers – Ramanujan identities (cf. Watson [59], Slater [52] (Eqs 14), and [18]):
∞∑
n=0
qn
2
(q; q)n
=
[
q5, q2, q3; q5
]
∞
(q; q)∞
, (4a)
∞∑
n=0
qn
2+n
(q; q)n
=
[
q5, q, q4; q5
]
∞
(q; q)∞
. (4b)
The similar identities in q-series are generally called identities of Rogers – Ramanujan type (RR-type),
which express infinite series in terms of infinite products. The well-known Bailey lemma [13, 14] has
c© W. CHU, C. WANG, 2012
100 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
ITERATION PROCESS FOR MULTIPLE ROGERS – RAMANUJAN IDENTITIES 101
been shown powerful in proving these identities. Slater [51, 52] exploited this technique extensively
and collected 130 identities of RR-type. By iterative use of Bailey lemma, Andrews [5] showed how
to embed each of classical identities of RR-type in an infinite family of multiple series identities. The
typical approaches to them together with the main contributors may be sketched as follows:
Bailey lemma: Andrews and Bressoud et al. [1, 5, 6, 21, 43, 41, 54, 10, 48, 58].
Lattice path enumeration: Agarwal and Bressoud [2, 20].
Hall – Littlewood functions: Stembridge [55].
Partition bijections: Bressoud, Zeilberger [22], Garvan [35] and Lovejoy [40].
Multiple series transformations: Bressoud [17], Singh [50] and Chu [26, 27].
Observing carefully the proof of Bressoud [19], we notice that the monomials {xk}k≥0 appeared
in (1) can be replaced by an arbitrary sequence {Wk}k∈Z. This suggests us to consider a more general
iteration process. Following the same approach of Bressoud [19], we shall establish a remarkably
useful transformation theorem involving an arbitrary sequence {Wk}k≥0. By specifying the W -
sequence, several transformation formulae from unilateral multiple series to bilateral univariate one
will be derived. Their limiting cases lead us to several known and numerous new multiple series
identities of RR-type.
Even though there is no technical difficulties to realize what is just described by means of Bailey
lemma, we have opted to proceed along Bressoud’s way. This is justified mainly by two reasons.
Firstly, there is more freedom and transparency to manipulate the universal sequence {Wk}k≥0
appearing in Lemma 1 and Theorem 1 than the two sequences {αk}k≥0 and {βk}k≥0 that are tied
by a linear relation in Bailey lemma. Secondly, in the final phase of constructing multiple series
identities of RR-type, the factorization process through the identities of Jacobi’s triple product and
quintuple product will be facilitated by the bilateral sum displayed in (5b). This is traditionally done
by incorporating two unilateral series into a bilateral one, which results often in small errors.
Following Bailey [11], Gasper, Rahman [36] and Slater [53], we shall utilize the following
notations for basic hypergeometric series throughout the paper. Denote by N, N0 and Z the sets
of natural numbers, nonnegative integers and integers, respectively. For two indeterminate x and q,
the shifted factorial of x with base q is defined by
(x; q)0 = 1 and (x; q)n = (1− x)(1− xq) . . . (1− xqn−1) for n ∈ N.
When |q| < 1, we have two well-defined infinite products
(x; q)∞ =
∞∏
k=0
(1− qkx) and (x; q)n = (x; q)∞ / (xq
n; q)∞ .
With the multiparameter forms of shifted factorials being abbreviated to
[ α, β, . . . , γ; q ]n = (α; q)n (β; q)n . . . (γ; q)n ,[
α, β, . . . , γ
A, B, . . . , C
∣∣∣q]
n
=
(α; q)n (β; q)n . . . (γ; q)n
(A; q)n (B; q)n . . . (C; q)n
;
we define the unilateral and bilateral basic hypergeometric series, respectively, by
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
102 W. CHU, C. WANG
1+rφs
[
a0, a1, . . . , ar
b1, . . . , bs
∣∣∣q; z] = ∞∑
n=0
{
(−1)nq(
n
2)
}s−r [ a0, a1, . . . , ar
q, b1, . . . , bs
∣∣∣q]
n
zn,
rψs
[
a1, a2, . . . , ar
b1, b2, . . . , bs
∣∣∣q; z] = +∞∑
n=−∞
{
(−1)nq(
n
2)
}s−r [ a1, a2, . . . , ar
b1, b2, . . . , bs
∣∣∣q]
n
zn,
where the base q will be restricted to |q| < 1 for nonterminating q-series.
The paper will be organized as follows. The next section will be devoted to the main theorem
of this paper, which transforms a unilateral multiple series to a bilateral univariate series. Then we
shall present, in the third section, its applications to multiple series transformation formulae and
multiple series identities of RR-type. From the identities examined in this paper, one can see that the
iteration process is powerful and simple for dealing with multiple series identities of RR-type, just
like Bressoud’s approach to the classical Rogers – Ramanujan identities. Finally, the paper ends with
a table, putting, in evidence, the connections between the multiple series identities of RR-type and
their ` = 1 counterparts of single sum cases.
2. The main theorem and proof. Performing the replacements m → m − k, n → n − k and
a→ q2k+δ with δ = 0, 1, we can reformulate the equation displayed in (1) as
1
(q; q)n+k+δ
=
n∑
m=k
[
n− k
m− k
]
q(m−k)(m+k+δ)
(q; q)m+k+δ
.
Let {Wk}k∈Z be an arbitrary sequence. Multiplying by Wk/(q; q)n−k across the last equation, we
may manipulate the following bilateral finite sum with respect to k over −n− δ ≤ k ≤ n
n∑
k=−n−δ
Wk
(q; q)n−k(q; q)n+k+δ
=
n∑
k=−n−δ
Wk
(q; q)n−k
n∑
m=k
[
n− k
m− k
]
q(m−k)(m+k+δ)
(q; q)m+k+δ
=
=
n∑
m=−n−δ
qm
2+mδ
(q; q)n−m
m∑
k=−n−δ
q−k(k+δ)Wk
(q; q)m−k(q; q)m+k+δ
=
=
n∑
m=0
qm
2+mδ
(q; q)n−m
m∑
k=−m−δ
q−k(k+δ)Wk
(q; q)m−k(q; q)m+k+δ
,
where the last line has been justified by the fact that the innermost summand vanishes for m < 0 and
k < −m − δ. Replacing k by −k in the two extreme sums with respect to k, we find the following
generalized recursive lemma.
Lemma 1 (Recursive sums).
n+δ∑
k=−n
Wk
(q; q)n+k(q; q)n−k+δ
=
n∑
m=0
qm
2+mδ
(q; q)n−m
m+δ∑
k=−m
qk(δ−k)Wk
(q; q)m+k(q; q)m−k+δ
.
When δ = 0 and Wk → qλk
2
xk, this lemma reduces clearly to Bressoud’s Lemma (2). However,
with the presence of an arbitrary sequence Wk, our lemma has more flexibility in application and
will be more useful.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
ITERATION PROCESS FOR MULTIPLE ROGERS – RAMANUJAN IDENTITIES 103
Iterating `-times the recursion in Lemma 1 leads to the following equation:
n+δ∑
k=−n
Wk
(q; q)n+k(q; q)n−k+δ
=
∑
n≥r1≥r2≥...≥r`≥0
qr1(r1+δ)+r2(r2+δ)+...+r`(r`+δ)
(q; q)n−r1(q; q)r1−r2 . . . (q; q)r`−1−r`
×
×
r`+δ∑
k=−r`
qk`(δ−k)Wk
(q; q)r`+k(q; q)r`−k+δ
.
In order to shorten the long expressions, we make the replacements on summation indices and fix the
compact notations as follows:
n− r1 → m0,
r1 − r2 → m1,
. . . . . .
r`−1 − r` → m`−1,
r` → m`,
and
m̃ = (m1,m2, . . . ,m`),
Mk =
∑̀
ι=k
mι, 0 ≤ k ≤ `.
Further replacing Wk by (−1)kq(
k
2)+k`(k−δ)Wk in the last finite series transformation, we may refor-
mulate the result as the following main theorem of this paper.
Theorem 1 (Multiple series transformation). For an arbitrary bilateral sequence {Wk}k∈Z, there
holds the multiple series transformation
∑
M0=n
(q; q)2n+δ
(q; q)m0(q; q)m`+δ
∏̀
ι=1
qMι(Mι+δ)
(q; q)mι
m`+δ∑
k=−m`
qk(m`+δ)
(q−m`−δ; q)k
(qm`+1; q)k
Wk = (5a)
=
n+δ∑
k=−n
(−1)k
[
2n+ δ
n+ k
]
qk`(k−δ)+(
k
2)Wk, (5b)
where the multiple sum on the left runs over (m0,m1, . . . ,m`) ∈ N1+`
0 subject to the condition
M0 = m0 +m1 + . . .+m` = n.
This theorem is remarkably useful for deriving concrete multiple transformation formulae and
multiple series identities of RR-type. We first examine Bressoud’s work now. More examples will be
presented in the next section.
Letting Wk = xk in Theorem 1 and then evaluating the sum with respect to k displayed in (5a)
by means of the bilateral q-binomial theorem (cf. Chu [29] (Eq. 5)
m+δ∑
k=−m
qk(m+δ) (q
−m−δ; q)k
(qm+1; q)k
xk =
(q; q)m(q; q)m+δ(x; q)m+δ(q/x; q)m
(q; q)2m+δ
we derive the following variant of Bressoud’s theorem stated in (3).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
104 W. CHU, C. WANG
Theorem 2 (Terminating series transformation).
∑
M0=n
(q; q)2n+δ(q; q)m`(x; q)m`+δ(q/x; q)m`
(q; q)m0(q; q)2m`+δ
∏̀
ι=1
qMι(Mι+δ)
(q; q)mι
=
=
n+δ∑
k=−n
(−1)k
[
2n+ δ
n+ k
]
qk`(k−δ)+(
k
2)xk.
For δ = 0, it is not hard to see that the last theorem coincides with multiple series transformation
formula (3). Letting n→∞ and evaluating the last sum through Jacobi’s triple product identity [39]
(see [8, p. 497] for historical notes)
[q, x, q/x; q]∞ =
+∞∑
k=−∞
(−1)kq(
k
2)xk (6)
we obtain the following multiple nonterminating series identity.
Proposition 1 (Multiple series identity).
∑
m̃∈N`0
(q; q)m`(x; q)m`+δ(q/x; q)m`
(q; q)2m`+δ
∏̀
ι=1
qMι(Mι+δ)
(q; q)mι
=
[
q1+2`, q`(1−δ)x, q1+`(1+δ)/x; q1+2`
]
∞
(q; q)∞
.
This identity may be considered as a common generalization of the multiple series identities of
RR-type displayed in the following corollaries.
Corollary 1 (δ = 0 and x = 1: Andrews [4] (Eq. 2.14) and Bressoud [18] (Eq. 6.1)).
∑
m̃∈N`0
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q3+2`, q1+`, q2+`; q3+2`
]
∞
(q; q)∞
.
Corollary 2 (δ = 1 and x = q: Andrews [6] (Eq. 3.46) and Stembridge [55] (Eq. c)).
∑
m̃∈N`0
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q3+2`, q, q2+2`; q3+2`
]
∞
(q; q)∞
.
These two corollaries are multiple series generalizations of the classical Rogers – Ramanujan
identities (4a) and (4b). They have been covered extensively in literature. Additional information may
further be found in [5, 10, 18, 21, 22, 34, 35, 54 – 57] for Corollary 1 and [10, 34] for Corollary 2.
More comments will be made after Corollary 12.
3. Transformations and multiple series identities. Following the example illustrated in the
last section, we shall derive sixteen multiple finite series transformations corresponding to different
settings of W -sequence. Then their limiting cases will yield several known and numerous new
multiple series identities of RR-type, that will be displayed as corollaries.
For properly chosen Wk, the corresponding finite sums displayed in (5a) will essentially be
evaluated by Bailey’s summation formula of very well-poised bilateral 6ψ6-series [12] (see also [28]
and [36] (II-33)) with |qa2/bcde| < 1
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
ITERATION PROCESS FOR MULTIPLE ROGERS – RAMANUJAN IDENTITIES 105
6ψ6
[
q
√
a, −q
√
a, b, c, d, e√
a, −
√
a, qa/b, qa/c, qa/d, qa/e
∣∣∣q; qa2
bcde
]
= (7a)
=
[
q, qa, q/a, qa/bc, qa/bd, qa/be, qa/cd, qa/ce, qa/de
qa/b, qa/c, qa/d, qa/e, q/b, q/c, q/d, q/e, qa2/bcde
∣∣∣q]
∞
(7b)
together with two particular cases due to Bailey [15] (cf. Chu [25] (Eqs 3.16a-b-c))
4ψ4
[
qw, b, c, d
w, q/b, q/c, q/d
∣∣∣q; q
bcd
]
=
[
q, q/bc, q/bd, q/cd
q/b, q/c, q/d, q/bcd
∣∣∣q]
∞
, (8)
5ψ5
[
qu, qv, b, c, d
u, v, 1/b, 1/c, 1/d
∣∣∣q; q−1
bcd
]
=
uv − 1/q
(1− u)(1− v)
[
q, 1/bc, 1/bd, 1/cd
q/b, q/c, q/d, q−1/bcd
∣∣∣q]
∞
. (9)
The limiting cases of the multiple series transformations will be simplified through Jacobi’s triple
product identity (6) and its variant, in view of the parity of summation index, which is originally due
to Bailey [16] (Eq. 4.1)
[
q2, qy, q/y; q2
]
∞ =
+∞∑
n=−∞
{1− yq1+4n} q4n2
y2n (10)
as well as the quintuple product identity [23, 30, 31, 33, 60]
[q, z, q/z; q]∞×
[
qz2, q/z2; q2
]
∞ =
+∞∑
n=−∞
{1− zqn} q3(
n
2)
(
qz3
)n
. (11)
Considering that the computations from multiple series transformations to multiple series identities of
RR-type are entirely routine, we shall not reproduce them in details. Instead, the specific parameter
settings will briefly be indicated in the headers of corollaries. Throughout this section, the Gaussian
binomial coefficient
[
n
k
]
will be denoted by
[
n
k
]
qm
under the base change q → qm for m ∈ N. In
addition, we shall also fix ε = ±1 and δm,n, the usual Kronecker symbol.
3.1. Letting δ = 0 and
Wk =
1− qkw
1− w
[
b, d
q/b, q/d
∣∣∣q]
k
( q
bd
)k
we may evaluate the sum with respect to k displayed in (5a) by means of the bilateral 4ψ4-series
identity (8) as
m∑
k=−m
qkm
(q−m; q)k
(qm+1; q)k
1− qkw
1− w
[
b, d
q/b, q/d
∣∣∣q]
k
( q
bd
)k
=
[
q, q/bd
q/b, q/d
∣∣∣q]
m
.
In view of Theorem 1, this leads to the following transformation formula.
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106 W. CHU, C. WANG
Theorem 3 (Terminating series transformation).∑
M0=n
(q; q)2n
(q; q)m0
[
q/bd
q/b, q/d
∣∣∣q]
m`
∏̀
ι=1
qM
2
ι
(q; q)mι
:=
:=
n∑
k=−n
(−1
bd
)k 1− qkw
1− w
[
2n
n+ k
] [
b, d
q/b, q/d
∣∣∣q]
k
q`k
2+(k+1
2 ).
Letting n→∞, we establish the nonterminating multiple transformation.
Proposition 2 (Nonterminating series transformation).∑
m̃∈N`0
[
q/bd
q/b, q/d
∣∣∣q]
m`
∏̀
ι=1
qM
2
ι
(q; q)mι
=
=
1
(q; q)∞
+∞∑
k=−∞
(−1
bd
)k 1− qkw
1− w
[
b, d
q/b, q/d
∣∣∣q]
k
q`k
2+(k+1
2 ).
Five multiple series identities of RR-type are derived from this proposition.
Corollary 3 (w = b = −1, d→ 0).∑
m̃∈N`0
(−1)m`
(−q; q)m`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q2`, q`, q`; q2`
]
∞
(q; q)∞
.
Corollary 4 (b, d→ 0).∑
m̃∈N`0
(−1)m`q−(
1+m`
2 )
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q2`−1, q`−1, q`; q2`−1
]
∞
(q; q)∞
.
Corollary 5 (w = b = −1, d→∞).∑
m̃∈N`0
1
(−q; q)m`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q2+2`, q1+`, q1+`; q2+2`
]
∞
(q; q)∞
.
Corollary 6 (b→ 0, d = q1/2 | q → q2).∑
m̃∈N`0
q−m`
(q; q2)m`
∏̀
k=1
q2M
2
k
(q2; q2)mk
=
[
q4`,−q2`−1,−q1+2`; q4`
]
∞
(q2; q2)∞
.
Corollary 7 (b→∞, d = q1/2 | q → q2).∑
m̃∈N`0
1
(q; q2)m`
∏̀
k=1
q2M
2
k
(q2; q2)mk
=
[
q4+4`,−q1+2`,−q3+2`; q4+4`
]
∞
(q2; q2)∞
.
The identities displayed in Corollaries 5 and 7 were found, through iterative use of Bailey lemma,
by Paule [43] (Eqs 44 and 54), where the first one contains printing errors. For different proofs, refer
to Warnaar [57, 56] for the first and Andrews [9] (Eq. 7.26), Bressoud [21] and Chu [27] (Example 16)
for the second.
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ITERATION PROCESS FOR MULTIPLE ROGERS – RAMANUJAN IDENTITIES 107
3.2. For δ = 0, first take
Wk =
1− q−kw
1− w
[
b, d
1/b, 1/d
∣∣∣q]
k
( q
bd
)k
in Theorem 1. Then evaluate the sum with respect to k displayed in (5a) by means of the bilateral
5ψ5-series identity (9) as
m∑
k=−m
qkm
(q−m; q)k
(qm+1; q)k
1−q−kw
1− w
[
b, d
1/b, 1/d
∣∣∣q]
k
( q
bd
)k
=
1−qmw
1− w
[
q, 1/bd
q/b, q/d
∣∣∣q]
m
.
Therefore we derive the following terminating series transformation formula.
Theorem 4 (Terminating series transformation).
∑
M0=n
1− qm`w
1− w
(q; q)2n
(q; q)m0
[
1/bd
q/b, q/d
∣∣∣q]
m`
∏̀
ι=1
qM
2
ι
(q; q)mι
=
=
n∑
k=−n
(−1
bd
)k 1− q−kw
1− w
[
2n
n+ k
] [
b, d
1/b, 1/d
∣∣∣q]
k
q`k
2+(k+1
2 ).
The limiting case n→∞ leads to the nonterminating series transformation.
Proposition 3 (Nonterminating series transformation).
∑
m̃∈N`0
1− qm`w
1− w
[
1/bd
q/b, q/d
∣∣∣q]
m`
∏̀
ι=1
qM
2
ι
(q; q)mι
=
=
1
(q; q)∞
+∞∑
k=−∞
(−1
bd
)k 1− q−kw
1− w
[
b, d
1/b, 1/d
∣∣∣q]
k
q`k
2+(k+1
2 ).
Five multiple series identities of RR-type are derived from this proposition.
Corollary 8 (w = 0, b = −1, d→ 0).
∑
m̃∈N`0
(−q)−m`
(−q; q)m`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q2`, q`−1, q1+`; q2`
]
∞
(q; q)∞
.
Corollary 9 (w = 0, b, d→ 0).
∑
m̃∈N`0
(−1)m`q−(
m`
2 )−2m`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q2`−1, q`−2, q1+`; q2`−1
]
∞
(q; q)∞
.
Corollary 10 (w, d→∞, b = −1: Stanton [54, p. 63]).
∑
m̃∈N`0
qm`
(−q; q)m`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q2+2`, q`, q2+`; q2+2`
]
∞
(q; q)∞
.
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108 W. CHU, C. WANG
Corollary 11 (b = −1, w = −d = q1/2 | q → q2).
∑
m̃∈N`0
1− q2m`+1
1− q
(q−1; q2)m`
(−q; q)2m`
∏̀
k=1
q2M
2
k
(q2; q2)mk
=
[
q2+4`, q2`−1, q3+2`; q2+4`
]
∞
(q2; q2)∞
.
Corollary 12 (b, d, w →∞).
∑
m̃∈N`0
qm`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q3+2`, q`, q3+`; q3+2`
]
∞
(q; q)∞
.
There is a more general multiple series identity due to Andrews [3] and Gordon [37]
∑
m̃∈N`0
q
∑`
ι=nmι
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q3+2`, qn, q3−n+2`; q3+2`
]
∞
(q; q)∞
which results in a common extension of Corollaries 1, 2 and 12. This important identity has been
studied extensively in literature. Different proofs may be found in [27, 1, 2, 7, 10, 17, 18, 20, 21, 44,
34, 35, 43, 24, 41], just for examples.
3.3. For δ = 1 and
Wk =
(1− qku)(1− qkv)
uv − q−1
[
b, d
1/b, 1/d
∣∣∣q]
k
(q−1
bd
)k
evaluate the sum displayed in (5a) through the bilateral 5ψ5-series identity (9) as
m+1∑
k=−m
qkm
(q−m−1; q)k
(qm+1; q)k
(1−qku)(1−qkv)
(uv−q−1)(bd)k
[
b, d
1/b, 1/d
∣∣∣q]
k
=
(q; q)m+1(1/bd; q)m
(q/b; q)m(q/d; q)m
.
According to Theorem 1, we have the following transformation formula.
Theorem 5 (Terminating series transformation).
∑
M0=n
(q; q)2n+1
(q; q)m0
[
1/bd
q/b, q/d
∣∣∣q]
m`
∏̀
ι=1
qM
2
ι +Mι
(q; q)mι
=
=
n+1∑
k=−n
(−1)k
[
2n+ 1
n+ k
]
(1− qku)(1− qkv)
(uv − q−1)(qbd)k
[
b, d
1/b, 1/d
∣∣∣q]
k
q(2`+1)(k2).
Letting n→∞ in this theorem leads to the nonterminating series transformation.
Proposition 4 (Nonterminating series transformation).
∑
m̃∈N`0
[
1/bd
q/b, q/d
∣∣∣q]
m`
∏̀
ι=1
qM
2
ι +Mι
(q; q)mι
=
=
1
(q; q)∞
+∞∑
k=−∞
(1− qku)(1− qkv)
(uv − q−1)(−qbd)k
[
b, d
1/b, 1/d
∣∣∣q]
k
q(2`+1)(k2).
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ITERATION PROCESS FOR MULTIPLE ROGERS – RAMANUJAN IDENTITIES 109
Five multiple series identities of RR-type are derived from this proposition.
Corollary 13 (b = −1, d→ 0).
∑
m̃∈N`0
(−q)−m`
(−q; q)m`
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q2`, q, q2`−1; q2`
]
∞
(q; q)∞
.
Corollary 14 (b, d→ 0).
∑
m̃∈N`0
(−1)m`q−(
m`
2 )−2m`
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q2`−1, q, q2`−2; q2`−1
]
∞
(q; q)∞
.
Corollary 15 (u = −v = b = −d = q−1/2: Chu [27] (Example 2)).
∑
m̃∈N`0
(−q; q)m`
(q; q2)m`+1
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q1+2`,−q1+2`,−q1+2`; q1+2`
]
∞
(q; q)∞
.
Corollary 16 (d→ 0, b = u = q−1/2 | q → q2).
∑
m̃∈N`0
q−m`
(q; q2)m`+1
∏̀
k=1
q2M
2
k+2Mk
(q2; q2)mk
=
[
q4`,−q,−q4`−1; q4`
]
∞
(q2; q2)∞
.
Corollary 17 (d→∞, b = u = q−1/2 | q → q2: Paule [43] (Eq. 53)).
∑
m̃∈N`0
1
(q; q2)m`+1
∏̀
k=1
q2M
2
k+2Mk
(q2; q2)mk
=
[
q4+4`,−q,−q3+4`; q4+4`
]
∞
(q2; q2)∞
.
3.4. Letting δ = 0 and
W2k+1 = 0 and W2k =
1− q2kw
1− w
(c; q2)k
(q2/c; q2)k
(q
c
)k
in Theorem 1, then evaluating the sum with respect to k displayed in (5a) by means of the bilateral
4ψ4-series identity (8) as∑
k
q2km
(q−m; q)2k
(qm+1; q)2k
1− q2kw
1− w
(c; q2)k
(q2/c; q2)k
(q
c
)k
=
(q; q)m(q/c; q
2)m
(q/c; q)m(q; q2)m
we establish the following terminating series transformation formula.
Theorem 6 (Terminating series transformation).
∑
M0=n
(q; q)2n(q/c; q
2)m`
(q; q)m0(q; q
2)m`(q/c; q)m`
∏̀
ι=1
qM
2
ι
(q; q)mι
=
=
∑
k
1− q2kw
1− w
[
2n
n+ 2k
]
(c; q2)k
(q2/c; q2)k
q(4`+2)k2
ck
.
Its limiting case n→∞ yields the nonterminating multiple series transformation.
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110 W. CHU, C. WANG
Proposition 5 (Nonterminating series transformation).∑
m̃∈N`0
(q/c; q2)m`
(q; q2)m`(q/c; q)m`
∏̀
ι=1
qM
2
ι
(q; q)mι
=
1
(q; q)∞
+∞∑
k=−∞
1−q2kw
1− w
(c; q2)k
(q2/c; q2)k
q(4`+2)k2
ck
.
Four multiple series identities of RR-type are derived from this proposition.
Corollary 18 (c→ 0: Agarwal, Bressoud [2] (Eq. 1.8) and [20] (Eq. 1.8)).∑
m̃∈N`0
q(
m`
2 )
(q; q2)m`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q2+8`, q4`, q2+4`; q2+8`
]
∞
(q; q)∞
.
Corollary 19 (w = c = −1).∑
m̃∈N`0
(−q; q2)m`
(q; q2)m`(−q; q)m`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q4+8`, q2+4`, q2+4`; q4+8`
]
∞
(q; q)∞
.
Corollary 20 (c = εq).∑
m̃∈N`0
(ε; q2)m`
(q; q2)m`(ε; q)m`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q4+8`,−εq1+4`,−εq3+4`; q4+8`
]
∞
(q; q)∞
.
Corollary 21 (c→∞: Agarwal, Bressoud [2, 20] and Warnaar [58]).∑
m̃∈N`0
1
(q; q2)m`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q6+8`, q2+4`, q4+4`; q6+8`
]
∞
(q; q)∞
.
We remark that when ` = 1, the last corollary reduces to a classical identity of RR-type due to
Rogers [45] (see also Slater [52] (Eq. 61)).
3.5. Let δ = 1 and
W2k+1 = 0 and W2k =
1− q−2kw
1− w
(c; q2)k
(1/c; q2)k
(q
c
)k
in Theorem 1. Then evaluate the sum with respect to k displayed in (5a) by means of the bilateral
5ψ5-series identity (9) as∑
k
q2km
(q−m−1; q)2k
(qm+1; q)2k
1−q−2kw
1− w
(c; q2)k
(1/c; q2)k
(q3
c
)k
=
1−qmw
1− w
(q; q)m+1(q/c; q
2)m
(q; q2)m+1(q/c; q)m
.
We therefore find the following terminating series transformation formula.
Theorem 7 (Terminating series transformation).∑
M0=n
1− qm`w
1− w
(q; q)2n+1(q/c; q
2)m`
(q; q)m0(q; q
2)m`+1(q/c; q)m`
∏̀
ι=1
qM
2
ι +Mι
(q; q)mι
=
=
∑
k
1− q−2kw
1− w
[
2n+ 1
n+ 2k
]
(c; q2)k
(1/c; q2)k
q(2`+1)(2k2 )+k
ck
.
Letting n→∞ in this theorem gives the nonterminating series transformation.
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ITERATION PROCESS FOR MULTIPLE ROGERS – RAMANUJAN IDENTITIES 111
Proposition 6 (Nonterminating series transformation).
∑
m̃∈N`0
1− qm`w
1− w
(q/c; q2)m`
(q; q2)m`+1(q/c; q)m`
∏̀
ι=1
qM
2
ι +Mι
(q; q)mι
=
=
1
(q; q)∞
+∞∑
k=−∞
1− q−2kw
1− w
(c; q2)k
(1/c; q2)k
q(2`+1)(2k2 )+k
ck
.
Five multiple series identities of RR-type are derived from this proposition.
Corollary 22 (w = 0, c→ 0).
∑
m̃∈N`0
q(
m`
2 )
(q; q2)m`+1
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q2+8`, q2+2`, q6`; q2+8`
]
∞
(q; q)∞
.
Corollary 23 (c = −1, w →∞).
∑
m̃∈N`0
qm`
(−q; q2)m`
(q; q2)m`+1(−q; q)m`
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q4+8`, q2`, q4+6`; q4+8`
]
∞
(q; q)∞
.
Corollary 24 (c = εq−1, w = εq ).
∑
m̃∈N`0
(εq2; q2)m`
(q; q2)m`+1(εq; q)m`
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q4+8`,−εq1+2`,−εq3+6`; q4+8`
]
∞
(q; q)∞
.
Corollary 25 (w = 0, c = −1).
∑
m̃∈N`0
(−q; q2)m`
(q; q2)m`+1(−q; q)m`
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q4+8`, q2+2`, q2+6`; q4+8`
]
∞
(q; q)∞
.
Corollary 26 (c, w →∞).
∑
m̃∈N`0
qm`
(q; q2)m`+1
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q6+8`, q2`, q6+6`; q6+8`
]
∞
(q; q)∞
.
3.6. Letting δ = 1 and
W2k+1 = 0 and W2k =
q4k − q
(qbd)k
[
b, d
q/b, q/d
∣∣∣q2]
k
we may evaluate the sum with respect to k displayed in (5a) by means of the bilateral 6ψ6-series
identity (7a), (7b) as∑
k
q2k(m+1) (q
−m−1; q)2k
(qm+1; q)2k
[
b, d
q/b, q/d
∣∣∣q2]
k
q4k−q
(qbd)k
=
(q; q)m+1(q/bd; q
2)m
(−q; q)m[q/b, q/d; q]m
.
In view of Theorem 1, this leads to the following transformation formula.
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112 W. CHU, C. WANG
Theorem 8 (Terminating series transformation).
∑
M0=n
(q; q)2n+1(q/bd; q
2)m`
(q; q)m0(−q; q)m` [q/b, q/d; q]m`
∏̀
ι=1
qM
2
ι +Mι
(q; q)mι
=
=
∑
k
q4k − q
(qbd)k
[
2n+ 1
n+ 2k
] [
b, d
q/b, q/d
∣∣∣q2]
k
q(2`+1)(2k2 ).
Letting n→∞, we obtain the nonterminating multiple series transformation.
Proposition 7 (Nonterminating series transformation).
∑
m̃∈N`0
(q/bd; q2)m`
(−q; q)m` [q/b, q/d; q]m`
∏̀
ι=1
qM
2
ι +Mι
(q; q)mι
=
=
1
(q; q)∞
+∞∑
k=−∞
q4k − q
(qbd)k
[
b, d
q/b, q/d
∣∣∣q2]
k
q(2`+1)(2k2 ).
Five multiple series identities of RR-type are derived from this proposition.
Corollary 27 (b = −d = q1/2).
∑
m̃∈N`0
(−1; q2)m`
(−q; q)m`(q; q2)m`
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
−q1+2`, q,−q2`;−q1+2`
]
∞
(q; q)∞
.
Corollary 28 (b = −d = q−1/2).
∑
m̃∈N`0
(−q2; q2)m`
(−q; q)m`(q; q2)m`+1
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q4+8`, q1+2`, q3+6`; q4+8`
]
∞
(q; q)∞
.
Corollary 29 (b, d→∞).
∑
m̃∈N`0
1
(−q; q)m`
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q2+2`, q, q1+2`; q2+2`
]
∞
(q; q)∞
.
Corollary 30 (d→ 0, b = −q1/2 | q → q2).
∑
m̃∈N`0
(−1)m` q
m2
`−2m`
(−q; q)2m`
∏̀
k=1
q2M
2
k+2Mk
(q2; q2)mk
=
[
q1+4`, q2, q4`−1; q1+4`
]
∞
(q2; q2)∞
.
Corollary 31 (d→∞, b = −q1/2 | q → q2).
∑
m̃∈N`0
1
(−q; q)2m`
∏̀
k=1
q2M
2
k+2Mk
(q2; q2)mk
=
[
q3+4`, q2, q1+4`; q3+4`
]
∞
(q2; q2)∞
.
The univariate cases of the last two identities can be found in Rogers [45], where the latter is
usually called the second Rogers, Selberg identity [47].
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ITERATION PROCESS FOR MULTIPLE ROGERS – RAMANUJAN IDENTITIES 113
3.7. For δ = 0 and
W3k+1 =W3k+2 = 0 and W3k =
1− q3kw
1− w
in Theorem 1, evaluate the sum displayed in (5a) by means of (8) as
∑
k
q3km
(q−m; q)3k
(qm+1; q)3k
1− q3kw
1− w
=
(q; q)m(q
3; q3)m−1+δ0,m
(q; q)2m−1+δ0,m
.
We derive consequently the following multiple series transformation formula.
Theorem 9 (Terminating series transformation).
∑
M0=n
(q; q)2n(q
3; q3)m`−1+δ0,m`
(q; q)m0(q; q)2m`−1+δ0,m`
∏̀
k=1
qM
2
k
(q; q)mk
=
∑
k
(−1)k 1−q
3kw
1− w
[
2n
n+ 3k
]
q9`k
2+(3k2 ).
Letting n→∞ in this theorem leads us to the multiple series identity of RR-type.
Corollary 32.
∑
m̃∈N`0
(q3; q3)m`−1+δ0,m`
(q; q)2m`−1+δ0,m`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q9+18`, q3+9`, q6+9`; q9+18`
]
∞
(q; q)∞
.
The last corollary may be considered as a multiple series generalization of a classical identity of
RR-type due to Bailey, Dyson [13] (Eq. B4).
3.8. Letting δ = 1 and
W3k+1 =W3k+2 = 0 and W3k = 1
we may evaluate the sum displayed in (5a) through (9) as∑
k
q3k(m+1) (q
−m−1; q)3k
(qm+1; q)3k
=
(q; q)m+1(q
3; q3)m
(q; q)2m+1
.
In view of Theorem 1, we derive the following transformation formula.
Theorem 10 (Terminating series transformation).
∑
M0=n
(q; q)2n+1(q
3; q3)m`
(q; q)m0(q; q)2m`+1
∏̀
ι=1
qM
2
ι +Mι
(q; q)mι
=
∑
k
(−1)k
[
2n+ 1
n+ 3k
]
q(2`+1)(3k2 ).
For n→∞, this theorem becomes the multiple series identity of RR-type.
Corollary 33.
∑
m̃∈N`0
(q3; q3)m`
(q; q)2m`+1
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q9+18`, q3+6`, q6+12`; q9+18`
]
∞
(q; q)∞
.
This corollary extends another identity due to Bailey, Dyson [13] (Eq. B3).
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114 W. CHU, C. WANG
3.9. Taking δ = 1 and
W3k+1 =W3k+2 = 0 and W3k =
(c; q3)k
(q2/c; q3)k
q6k − q
(q2c)k
in Theorem 1 and then evaluating the sum with respect to k displayed in (5a) by means of the bilateral
6ψ6-series identity (7a), (7b) as
∑
k
q3k(m+1) (q
−m−1; q)3k
(qm+1; q)3k
(c; q3)k
(q2/c; q3)k
q6k−q
(q2c)k
=
(q; q)m(q; q)m+1(q/c; q
3)m
(q; q)2m(q/c; q)m
we establish the following multiple series transformation formula.
Theorem 11 (Terminating series transformation).
∑
M0=n
(q; q)2n+1(q; q)m`(q/c; q
3)m`
(q; q)m0(q; q)2m`(q/c; q)m`
∏̀
ι=1
qM
2
ι +Mι
(q; q)mι
=
=
∑
k
q6k − q
(−q2c)k
[
2n+ 1
n+ 3k
]
(c; q3)k
(q2/c; q3)k
q(2`+1)(3k2 ).
The limiting case n→∞ leads us to the nonterminating series transformation.
Proposition 8 (Nonterminating series transformation).
∑
m̃∈N`0
(q; q)m`(q/c; q
3)m`
(q; q)2m`(q/c; q)m`
∏̀
ι=1
qM
2
ι +Mι
(q; q)mι
=
1
(q; q)∞
+∞∑
k=−∞
q6k−q
(−q2c)k
(c; q3)k
(q2/c; q3)k
q(2`+1)(3k2 ).
Four multiple series identities of RR-type are derived from this proposition.
Corollary 34 (c→ 0).
∑
m̃∈N`0
q2(
m`
2 )
(q; q)m`
(q; q)2m`
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q2+6`, q, q1+6`; q2+6`
]
∞
(q; q)∞
[
q6`, q4+6`; q4+12`
]
∞
.
Corollary 35 (c = −q).
∑
m̃∈N`0
(q; q)m`(−1; q3)m`
(q; q)2m`(−1; q)m`
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q3+6`, q, q2+6`; q3+6`
]
∞
(q; q)∞
[
q1+6`, q5+6`; q6+12`
]
∞
.
Corollary 36 (c→∞).
∑
m̃∈N`0
(q; q)m`
(q; q)2m`
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q4+6`, q, q3+6`; q4+6`
]
∞
(q; q)∞
[
q2+6`, q6+6`; q8+12`
]
∞
.
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Corollary 37 (c = −q−1/2 | q → q2).
∑
m̃∈N`0
(q2; q2)m`(−q3; q6)m`
(q2; q2)2m`(−q; q2)m`+1
∏̀
k=1
q2M
2
k+2Mk
(q2; q2)mk
=
=
[
q6+12`, q, q5+12`; q6+12`
]
∞
(q2; q2)∞
[
q4+12`, q8+12`; q12+24`
]
∞
.
The ` = 1 cases of Corollaries 34 and 36 have been discovered by Jackson [38] (1928) and
Rogers [45] (1894) respectively. Instead, the ` = 1 case of Corollary 37 has recently been found
respectively by Chu, Zhang [32] (No. 197) and McLaughlin, Sills [42] (Eq. 4.17).
3.10. For δ = 0, first replace q by q2 in Theorem 1. Then put
Wk =
1− qkw
1− w
(c; q)k
(q/c; q)k
(
− q
c
)k
.
The corresponding finite sum displayed in (5a) may be evaluated by means of the bilateral 4ψ4-series
identity (8) as
m∑
k=−m
q2km
(q−2m; q2)k
(q2m+2; q2)k
1− qkw
1− w
(c; q)k
(q/c; q)k
(
− q
c
)k
=
(−q/c; q)2m(q2; q2)m
(−q; q)2m(q2/c2; q2)m
.
We therefore derive the following transformation formula.
Theorem 12 (Terminating series transformation).
∑
M0=n
(q2; q2)2n(−q/c; q)2m`
(q2; q2)m0(−q; q)2m`(q2/c2; q2)m`
∏̀
ι=1
q2M
2
ι
(q2; q2)mι
=
=
n∑
k=−n
1− qkw
1− w
[
2n
n+ k
]
q2
(c; q)k
(q/c; q)k
q(2`+1)k2
ck
.
Letting n→∞, we have the nonterminating multiple series transformation.
Proposition 9 (Nonterminating series transformation).
∑
m̃∈N`0
(−q/c; q)2m`
(−q; q)2m`(q2/c2; q2)m`
∏̀
ι=1
q2M
2
ι
(q2; q2)mι
=
=
1
(q2; q2)∞
+∞∑
k=−∞
1− qkw
1− w
(c; q)k
(q/c; q)k
q(2`+1)k2
ck
.
Two multiple series identities of RR-type are derived from this proposition.
Corollary 38 (c→ 0).
∑
m̃∈N`0
(−1)m`qm2
`
(−q; q)2m`
∏̀
k=1
q2M
2
k
(q2; q2)mk
=
[
q1+4`, q2`, q1+2`; q1+4`
]
∞
(q2; q2)∞
.
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116 W. CHU, C. WANG
Corollary 39 (c→∞: Warnaar [58] (Eq. 5.14)).
∑
m̃∈N`0
1
(−q; q)2m`
∏̀
k=1
q2M
2
k
(q2; q2)mk
=
[
q3+4`, q1+2`, q2+2`; q3+4`
]
∞
(q2; q2)∞
.
The last corollary generalizes the first Rogers, Selberg identity [45, 47].
3.11. Let δ = 0 and q → q2 in Theorem 1. Then for
Wk = (1 + q2k)
[
b, d
−q/b, −q/d
∣∣∣q]
k
(
− q
bd
)k
the corresponding sum displayed in (5a) can be evaluated through (7a), (7b) as
m∑
k=−m
qk(2m+1) (q
−2m; q2)k
(q2m+2; q2)k
[
b, d
−q/b,−q/d
∣∣∣q]
k
1+q2k
(−bd)k
=
2(q2; q2)m(−q/bd; q)2m
(q; q2)m[q2/b2, q2/d2; q2]m
.
We establish consequently the following transformation formula.
Theorem 13 (Terminating series transformation).
∑
M0=n
(q2; q2)2n(−q/bd; q)2m`
(q2; q2)m0(q; q
2)m` [q
2/b2, q2/d2; q2]m`
∏̀
ι=1
q2M
2
ι
(q2; q2)mι
=
=
n∑
k=−n
1 + q2k
2(bd)k
[
2n
n+ k
]
q2
[
b, d
−q/b, −q/d
∣∣∣q]
k
q(2`+1)k2 .
Letting n→∞ leads us to the nonterminating multiple series transformation.
Proposition 10 (Nonterminating series transformation).
∑
m̃∈N`0
(−q/bd; q)2m`
(q; q2)m` [q
2/b2, q2/d2; q2]m`
∏̀
ι=1
q2M
2
ι
(q2; q2)mι
=
=
1
(q2; q2)∞
+∞∑
k=−∞
1 + q2k
2(bd)k
[
b, d
−q/b, −q/d
∣∣∣q]
k
q(2`+1)k2 .
Three multiple series identities of RR-type are derived from this proposition.
Corollary 40 (b = −d =
√
−q).
∑
m̃∈N`0
(−1; q)m`
(q; q2)m`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q1+2`,−q`,−q1+`; q1+2`
]
∞
(q; q)∞
.
Corollary 41 (b = −d =
√
−1: Chu [27] (Example 15)).
∑
m̃∈N`0
(q; q2)m`
(−q; q)2m`
∏̀
k=1
q2M
2
k
(q2; q2)mk
=
[
q2+4`, q1+2`, q1+2`; q2+4`
]
∞
(q2; q2)∞
.
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Corollary 42 (b =
√
−1, d =
√
−q | q → q2).
∑
m̃∈N`0
(q; q2)2m`
(q2; q4)m`(−q2; q2)2m`
∏̀
k=1
q4M
2
k
(q4; q4)mk
=
[
q4+8`, q1+4`, q3+4`; q4+8`
]
∞
(q4; q4)∞
.
For the last identity, its univariate version has been found by Slater [52] (Eq. 53).
3.12. In Theorem 1, let δ = 1 and q → q2. Then for
Wk =
(1− qku)(1− qkv)
uv − q−1
(c; q)k
(1/c; q)k
(
− q−1
c
)k
we may compute the corresponding sum displayed in (5a) via (9) as
m+1∑
k=−m
q2km+k (q
−2m−2; q2)k
(q2m+2; q2)k
(1−qku)(1−qkv)
(uv−q−1)(−c)k
(c; q)k
(1/c; q)k
=
(q2; q2)m+1(−q/c; q)2m
(q2/c2; q2)m(−q; q)2m+1
.
This establishes the following terminating series transformation formula.
Theorem 14 (Terminating series transformation).
∑
M0=n
(q2; q2)2n+1(−q/c; q)2m`
(q2; q2)m0(−q; q)2m`+1(q2/c2; q2)m`
∏̀
ι=1
q2M
2
ι +2Mι
(q2; q2)mι
=
=
n+1∑
k=−n
(1− qku)(1− qkv)
(uv − q−1)(qc)k
[
2n+ 1
n+ k
]
q2
(c; q)k
(1/c; q)k
q(4`+2)(k2).
Letting n→∞, we get the nonterminating multiple series transformation.
Proposition 11 (Nonterminating series transformation).
∑
m̃∈N`0
(−q/c; q)2m`
(−q; q)2m`+1(q2/c2; q2)m`
∏̀
ι=1
q2M
2
ι +2Mι
(q2; q2)mι
=
=
1
(q2; q2)∞
+∞∑
k=−∞
(1− qku)(1− qkv)
(uv − q−1)(qc)k
(c; q)k
(1/c; q)k
q(4`+2)(k2).
Three multiple series identities of RR-type are derived from this proposition.
Corollary 43 (c→ 0: Paule [43] (Eq. 58)).
∑
m̃∈N`0
(−1)m`qm2
`
(−q; q)2m`+1
∏̀
k=1
q2M
2
k+2Mk
(q2; q2)mk
=
[
q1+4`, q, q4`; q1+4`
]
∞
(q2; q2)∞
.
Corollary 44 (c→∞: Paule [43] (Eq. 46)).
∑
m̃∈N`0
1
(−q; q)2m`+1
∏̀
k=1
q2M
2
k+2Mk
(q2; q2)mk
=
[
q3+4`, q, q2+4`; q3+4`
]
∞
(q2; q2)∞
.
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Corollary 45 (c = u = q−1/2 | q → q2).
∑
m̃∈N`0
(q; q2)2m`+1
(−q2; q2)2m`+1(q2; q4)m`+1
∏̀
k=1
q4M
2
k+4Mk
(q4; q4)mk
=
[
q4+8`, q, q3+8`; q4+8`
]
∞
(q4; q4)∞
.
When ` = 1, the last two corollaries reduce respectively to the third Rogers, Selberg identity [46,
47] and an identity due to Slater [52] (Eq. 55).
3.13. First let δ = 1 and q → q2 in Theorem 1. Then for
Wk = (q + q2k)
[
b, d
−1/b, −1/d
∣∣∣q]
k
(
− q−1
bd
)k
the corresponding finite sum displayed in (5a) can be evaluated by means of the bilateral 6ψ6-series
identity (7a), (7b) as
m+1∑
k=−m
qk(2m+1) (q
−2m−2; q2)k
(q2m+2; q2)k
[
b, d
−1/b, −1/d
∣∣∣q]
k
q + q2k
(−bd)k
=
=
2(1 + bd)(q2; q2)m+1(−q/bd; q)2m
(1 + b)(1 + d)(q; q2)m+1[q2/b2, q2/d2; q2]m
.
We therefore derive the following transformation formula.
Theorem 15 (Terminating series transformation).
∑
M0=n
(q2; q2)2n+1(−q/bd; q)2m`
(q2; q2)m0(q; q
2)m`+1[q2/b2, q2/d2; q2]m`
∏̀
ι=1
q2M
2
ι +2Mι
(q2; q2)mι
=
=
(1 + b)(1 + d)
2(1 + bd)
n+1∑
k=−n
q + q2k
(qbd)k
[
2n+ 1
n+ k
]
q2
[
b, d
−1/b, −1/d
∣∣∣q]
k
q(4`+2)(k2).
Letting n→∞ yields the nonterminating multiple series transformation.
Proposition 12 (Nonterminating series transformation).
∑
m̃∈N`0
(−q/bd; q)2m`
(q; q2)m`+1[q2/b2, q2/d2; q2]m`
∏̀
ι=1
q2M
2
ι +2Mι
(q2; q2)mι
=
=
(1 + b)(1 + d)
2(1 + bd)(q2; q2)∞
+∞∑
k=−∞
q + q2k
(qbd)k
[
b, d
−1/b, −1/d
∣∣∣q]
k
q(4`+2)(k2).
Four multiple series identities of RR-type are derived from this proposition.
Corollary 46 (b = −d =
√
−q−1 | q → q1/2).
∑
m̃∈N`0
(−q; q)m`
(q; q2)m`+1
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q1+2`,−q1+2`,−q1+2`; q1+2`
]
∞
(q; q)∞
.
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Corollary 47 (b = −d =
√
−1).
∑
m̃∈N`0
(q; q2)m`
(−q; q)2m`+1
∏̀
k=1
q2M
2
k+2Mk
(q2; q2)mk
=
[
q2+4`, q, q1+4`; q2+4`
]
∞
(q2; q2)∞
.
Corollary 48 (b =
√
−q−1, d→ 0 | q → q1/2).
∑
m̃∈N`0
q(
1+m`
2 )
(q; q2)m`+1
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q2+8`, q2`, q2+6`; q2+8`
]
∞
(q; q)∞
.
Corollary 49 (b =
√
−q−1, d→∞ | q → q1/2).
∑
m̃∈N`0
1
(q; q2)m`+1
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
[
q6+8`, q2+2`, q4+6`; q6+8`
]
∞
(q; q)∞
.
The univariate case of Corollary 49 is due to Rogers [46] (1917) (cf. Slater [52] (Eq. 60).
3.14. For δ = 0, replace q by q3 in Theorem 1. Let further
Wk =
1− q2kω
1− ω
(ω2c; q)k
(qω2/c; q)k
(q
c
)k
where ω 6= 1 is a cubic root of unity. Then we can evaluate the corresponding sum with respect to k
displayed in (5a) by means of (7a), (7b) as
m∑
k=−m
q3km
(q−3m; q3)k
(q3m+3; q3)k
1− q2kω
1− ω
(ω2c; q)k
(qω2/c; q)k
(q
c
)k
=
(q3; q3)2m(q/c; q)3m
(q3; q3)2m(q3/c3; q3)m
.
This leads us to the following transformation formula.
Theorem 16 (Terminating series transformation).
∑
M0=n
(q3; q3)2n(q
3; q3)m`(q/c; q)3m`
(q3; q3)m0(q
3; q3)2m`(q
3/c3; q3)m`
∏̀
ι=1
q3M
2
ι
(q3; q3)mι
=
=
n∑
k=−n
(−1
c
)k 1− q2kω
1− ω
[
2n
n+ k
]
q3
(ω2c; q)k
(qω2/c; q)k
q3`k
2+3(k2)+k.
Letting n→∞, we have the nonterminating multiple series transformation.
Proposition 13 (Nonterminating series transformation).
∑
m̃∈N`0
(q3; q3)m`(q/c; q)3m`
(q3; q3)2m`(q
3/c3; q3)m`
∏̀
ι=1
q3M
2
ι
(q3; q3)mι
=
=
1
(q3; q3)∞
+∞∑
k=−∞
(−1
c
)k 1− q2kω
1− ω
(ω2c; q)k
(qω2/c; q)k
q3`k
2+3(k2)+k.
Four multiple series identities of RR-type are derived from this proposition.
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Corollary 50 (c = ε).
∑
m̃∈N`0
(q3; q3)m`(εq; q)3m`
(q3; q3)2m`(εq
3; q3)m`
∏̀
k=1
q3M
2
k
(q3; q3)mk
=
[
q3+6`, εq1+3`, εq2+3`; q3+6`
]
∞
(q3; q3)∞
.
Corollary 51 (c = q1/2 | q → q2).
∑
m̃∈N`0
(q6; q6)m`(q; q
2)3m`
(q6; q6)2m`(q
3; q6)m`
∏̀
k=1
q6M
2
k
(q6; q6)mk
=
[
q6+12`, q1+6`, q5+6`; q6+12`
]
∞
(q6; q6)∞
.
Corollary 52 (c→ 0 | q → q1/3: McLaughlin, Sills [41] (Eq. 5.8)).
∑
m̃∈N`0
qm
2
`
(q; q)m`
(q; q)2m`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q2+6`, q`, q2+5`; q2+6`
]
∞
(q; q)∞
[
q2+4`, q2+8`; q4+12`
]
∞
.
Corollary 53 (c→∞ | q → q1/3: Andrews [5] (Eq. 3.4) and [41] (Eq. 5.10)).
∑
m̃∈N`0
(q; q)m`
(q; q)2m`
∏̀
k=1
qM
2
k
(q; q)mk
=
[
q4+6`, q1+`, q3+5`; q4+6`
]
∞
(q; q)∞
[
q2+4`, q6+8`; q8+12`
]
∞
.
When ε = 1, the univariate case of Corollary 50 has been discovered by Bailey [13] (Eq. 1.6)
(cf. Slater [52] (Eq. 42)). However, when ε = −1, it seems new even for ` = 1.
3.15. Similarly, for δ = 1, replace q by q3 in Theorem 1. Then for
Wk =
1− q2k−1ω
1− q−1ω
(ω2c; q)k
(ω2/c; q)k
(q−1
c
)k
we can evaluate the corresponding sum displayed in (5a) through (7a), (7b) as
m+1∑
k=−m
q3k(m+1) (q
−3m−3; q3)k
(q3m+3; q3)k
1− q2k−1ω
1− q−1ω
(ω2c; q)k
(ω2/c; q)k
(q−1
c
)k
=
=
(1− ω)(q3; q3)m(q3; q3)m+1(q/c; q)3m+1
(1− ω2/c)(1− qω2)(q3; q3)2m+1(q3/c3; q3)m
which yields consequently the following transformation formula.
Theorem 17 (Terminating series transformation).
∑
M0=n
(q3; q3)2n+1(q
3; q3)m`(q/c; q)3m`+1
(q3; q3)m0(q
3; q3)2m`+1(q3/c3; q3)m`
∏̀
ι=1
q3M
2
ι +3Mι
(q3; q3)mι
=
= (qω/c−qω2)
n+1∑
k=−n
(−1
qc
)k 1− q2k−1ω
1− ω
[
2n+ 1
n+ k
]
q3
(ω2c; q)k
(ω2/c; q)k
q(6`+3)(k2).
Letting n→∞, we obtain the nonterminating multiple series transformation.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
ITERATION PROCESS FOR MULTIPLE ROGERS – RAMANUJAN IDENTITIES 121
Proposition 14 (Nonterminating series transformation).
∑
m̃∈N`0
(q3; q3)m`(q/c; q)3m`+1
(q3; q3)2m`+1(q3/c3; q3)m`
∏̀
ι=1
q3M
2
ι +3Mι
(q3; q3)mι
=
=
qω/c−qω2
(q3; q3)∞
+∞∑
k=−∞
(−1
qc
)k 1− q2k−1ω
1− ω
(ω2c; q)k
(ω2/c; q)k
q(6`+3)(k2).
Four multiple series identities of RR-type are derived from this proposition.
Corollary 54 (c = ε).
∑
m̃∈N`0
(q3; q3)m`(εq; q)3m`+1
(q3; q3)2m`+1(εq3; q3)m`
∏̀
k=1
q3M
2
k+3Mk
(q3; q3)mk
=
=
[
q3+6`, εq, εq2+6`; q3+6`
]
∞
(q3; q3)∞
.
Corollary 55 (c = q−1/2 | q → q2).
∑
m̃∈N`0
(q6; q6)m`(q; q
2)3m`+2
(q6; q6)2m`+1(q3; q6)m`+1
∏̀
k=1
q6M
2
k+6Mk
(q6; q6)mk
=
=
[
q6+12`, q, q5+12`; q6+12`
]
∞
(q6; q6)∞
.
Corollary 56 (c→ 0 | q → q1/3).
∑
m̃∈N`0
qm
2
`+m`
(q; q)m`
(q; q)2m`+1
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
=
[q2+6`, q1+2`, q1+4`;q2+6`]∞
(q; q)∞
[
q2`, q4+10`; q4+12`
]
∞
.
Corollary 57 (c→∞ | q → q1/3: Andrews [5] (Eq. 3.13)).
∑
m̃∈N`0
(q; q)m`
(q; q)2m`+1
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
=
[q4+6`, q1+2`, q3+4`;q4+6`]∞
(q; q)∞
[
q2+2`, q6+10`; q8+12`
]
∞
.
For ε = 1, the univariate case of Corollary 54 has been discovered by Bailey [13] (Eq. 1.7)
(cf. Slater [52] (Eq. 40)). However, Corollary 55 seems new, whose ` = 1 case has recently been
discovered by Chu Zhang [32] (No. 134).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
122 W. CHU, C. WANG
3.16. In Theorem 1, let δ = 1 and
W4k+1 =W4k+2 =W4k+3 = 0 and W4k = q4k − q1−4k.
Evaluating the corresponding sum displayed in (5a) by means of Bailey’s bilateral 6ψ6-series identity
(7a), (7b) as
∑
k
q4k(m+1) (q
−m−1; q)4k
(qm+1; q)4k
(q4k − q1−4k) =
(q; q)m+1(−q2; q2)m−1+δ0,m
(−q; q)m(q; q2)m
we therefore establish the following transformation formula.
Theorem 18 (Terminating series transformation).
∑
M0=n
(q; q)2n+1(−q2; q2)m`−1+δ0,m`
(q; q)m0(−q; q)m`(q; q2)m`
∏̀
ι=1
qM
2
ι +Mι
(q; q)mι
=
=
∑
k
(q4k − q1−4k)
[
2n+ 1
n+ 4k
]
q(2`+1)(4k2 ).
Its limiting case n→∞ yields the nonterminating multiple series identity.
Corollary 58 (Difference of infinite products).
∑
m̃∈N`0
(−q2; q2)m`−1+δ0,m`
(−q; q)m`(q; q2)m`
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
=
1
(q; q)∞
{ [
q16+32`,−q10+12`,−q6+20`; q16+32`
]
∞
−q
[
q16+32`,−q2+12`,−q14+20`; q16+32`
]
∞
}
.
There exist other multiple series identities with the right members being differences of infinite
products. For example, letting c = q in Proposition 13 leads us to the following identity.
Corollary 59 (Difference of infinite products).
∑
m̃∈N`0
(q; q)m`(q
3; q3)m`−1+δ0,m`
(q; q)2m`(q; q)m`−1+δ0,m`
∏̀
k=1
qM
2
k+Mk
(q; q)mk
=
=
1
(q; q)∞
{ [
q9+18`, q6+6`, q3+12`; q9+18`
]
∞
−q
[
q9+18` , q6` , q9+12`; q9+18`
]
∞
}
.
However, we shall not pursue further along this direction due to the space limitation.
Concluding comments. Most of the multiple series identities displayed in this paper generalize
classical single sum identities. The comparisons are summarized in the following table, where most
of the references to the reduced ` = 1 cases of the corollaries are positioned in Slater’s list [52].
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
ITERATION PROCESS FOR MULTIPLE ROGERS – RAMANUJAN IDENTITIES 123
Corollary Case ` = 1 Corollary Case ` = 1
1 Slater [52], (18) 35 McLaughlin, Sills [41], (1.3)
2 Slater [52], (14) 36 Slater [52], (99)
6 Slater [52], (85) 37 McLaughlin, Sills [42], (4.17)
7 Slater [52], (39) 38 Slater [52], (19)
11 Stanton [54], (D3)(S1) 39 Slater [52], (136)
15 Bailey[11], §8.6(10) 40 Slater [52], (6)
16 Slater [52], (9, 84) 41 Sills [49], (4.6)
17 Slater [52], (38) 42 Slater [52], (53)
18 Slater [52], (46) 43 Slater [52], (15)
19 Slater [52], (29) 44 Slater [52], (31)
20 Slater [52], (54) 45 Slater [52], (55, 57)
21 Slater [52], (61) 46 Bailey[11], §8.6(10)
22 Agarwal, Bressoud [2], (2.3) 47 Slater [52], (27)
23 Slater [52], (50) 48 Slater [52], (44)
24 Slater [52], (28) 49 Slater [52], (60)
25 Slater [52], (11, 51) 50 Slater [52], (42)
26 Slater [52], (59) 51 Chu, Zhang [32], No.135
27 Slater [52], (48) 52 Slater [52], (83)
28 Slater [52], (28) 53 Slater [52], (98)
29 Slater [52], (7) 54 Slater [52], (40)
30 Slater [52], (19) 55 Chu, Zhang [32], No.134
31 Slater [52], (32) 56 Slater [52], (86)
32 Slater [52], (93) 57 Slater [52], (94)
33 Slater [52], (92) 58 Slater [52], (120)
34 Slater [52], (83) 59 Slater [52], (89)
Given a known identity of Rogers – Ramanujan type, there may exist several multiple counter-
parts. For example, we have ten multiple series identities displayed in Corollaries 3 – 5, 8 – 10, 12 – 14
and 29 that reduce, when ` = 1, to special cases of Euler’s second q-exponential function.
1. Agarwal A. K., Andrews G. E., Bressoud D. M. The Bailey lattice // J. Indian Math. Soc. (N.S.). – 1987. – 51. –
P. 57 – 73.
2. Agarwal A. K., Bressoud D. M. Lattice paths and multiple basic hypergeometric series // Pacif. J. Math. – 1989. –
136. – P. 209 – 228..
3. Andrews G. E. An analytic generalization of the Rogers – Ramanujan identities for odd moduli // Proc. Nat. Acad.
Sci. U.S.A. – 1974. – 71. – P. 4082 – 4085.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
124 W. CHU, C. WANG
4. Andrews G. E. Problems and prospects for basic hypergeometric functions // Theory and Application of Special
Functions / Ed. R. Askey. – New York: Acad. Press, 1975. – P. 191 – 224.
5. Andrews G. E. Multiple series Rogers – Ramanujan type identities // Pacif. J. Math. – 1984. – 114. – P. 267 – 283.
6. Andrews G. E. q-Series: their development and application in analysis, number theory, combinatorics, physics, and
computer algebra // CBMS Region. Conf. Ser. Math. – 1986. – № 66.
7. Andrews G. E., Askey R. Enumeration of partitions: The role of Eulerian series and q-orthogonal polynomials //
Higher Combinatorics / Ed. M. Aigner. – Dordrecht: D. Reidel Publ. Comp., 1977. – P. 3 – 26.
8. Andrews G. E., Askey R., Roy R. Special functions. – Cambridge: Cambridge Univ. Press, 1999.
9. Andrews G. E., Berkovich A. The WP-Bailey tree and its implications // J. London Math. Soc. – 2002. – 66, №. 3. –
P. 529 – 549.
10. Andrews G. E., Schilling A., Warnaar S. O. An A2 Bailey lemma and Rogers – Ramanujan-type identities // J. Amer.
Math. Soc. – 1999. – 12, № 3. – P. 677 – 702.
11. Bailey W. N. Generalized hypergeometric series. – Cambridge: Cambridge Univ. Press, 1935.
12. Bailey W. N. Series of hypergeometric type which are infinite in both directions // Quart. J. Math. – 1936. – 7. –
P. 105 – 115.
13. Bailey W. N. Some identities in combinatory analysis // Proc. London Math. Soc. – 1947. – 49. – P. 421 – 435.
14. Bailey W. N. Identities of the Rogers – Ramanujan type // Proc. London Math. Soc. – 1948. – 50. – P. 1 – 10.
15. Bailey W. N. On the analogue of Dixon’s theorem for bilateral basic hypergeometric series // Quart. J. Math., Oxford
Ser. – 1950. – 1. – P. 318 – 320.
16. Bailey W. N. On the simplification of some identities of the Rogers – Ramanujan type // Proc. London Math. Soc. –
1951. – 1. – P. 217 – 221.
17. Bressoud D. M. Analytic and combinatorial generalizations of the Rogers – Ramanujan identities // Mem. Amer. Math.
Soc. – 1980. – 24, № 227. – P. 54.
18. Bressoud D. M. On partitions, orthogonal polynomials and the expansion of certain infinite products // Proc. London
Math. Soc. – 1981. – 42. – P. 478 – 500.
19. Bressoud D. M. An easy proof of the Rogers – Ramanujan identities // J. Number Theory. – 1983. – 16. – P. 235 – 241.
20. Bressoud D. M. Lattice paths and the Rogers – Ramanujan identities // Number Тheory, Madras. – 1987. – P. 140 – 172;
Lect. Notes Math. – 1989. – 1395.
21. Bressoud D. M., Ismail M., Stanton D. Change of base in Bailey pairs // Ramanujan J. – 2000. – 4, № 4. – P. 435 – 453.
22. Bressoud D. M., Zeilberger D. Generalized Rogers – Ramanujan bijections // Adv. Math. – 1989. – 78, № 1. –
P. 42 – 75.
23. Carlitz L., Subbarao M. V. A simple proof of the quintuple product identity // Proc. Amer. Math. Soc. – 1972. – 32,
№ 1. – P. 42 – 44.
24. Chapman R. A probabilistic proof of the Andrews – Gordon identities // Discrete Math. – 2005. – 290. – P. 79 – 84.
25. Chu W. Almost-poised hypergeometric series // Mem. Amer. Math. Soc. – 1998. – 135, № 642. – P. 99+iv.
26. Chu W. The Saalschüz chain reactions and bilateral basic hypergeometric series // Constr. Approxim. – 2002. – 18,
№ 4. – P. 579 – 597.
27. Chu W. The Saalschütz chain reactions and multiple q-series transformations // Theory and Applications of Special
Functions dedicated to Mizan Rahman: Developments in Mathematics / Eds Ismail and Koelink. – 2005. – Vol. 13. –
P. 99 – 122.
28. Chu W. Bailey’s very well-poised 6ψ6-series identity // J. Combin. Theory (Ser. A). – 2006. – 113, № 6. – P. 966 – 979.
29. Chu W. Abel’s Lemma on summation by parts and Basic Hypergeometric Series // Adv. Appl. Math. – 2007. – 39,
№ 4. – P. 490 – 514.
30. Chu W. Jacobi’s triple product identity and the quintuple product identity // Boll. Unione mat. ital. – 2007. – B10,
№. 8. – P. 867 – 874
31. Chu W., Yan Q. L. Unification of the Quintuple and Septuple Product Identities // Electron. J. Combinatorics. – 2007.
– 14, № 7.
32. Chu W., Zhang W. Bilateral Bailey lemma and Rogers – Ramanujan identities // Adv. Appl. Math. – 2009. – 42. –
P. 358 – 391.
33. Cooper S. The quintuple product identity // Int. J. Number Theory. – 2006. – 2, № 1. – P. 115 – 161.
34. Fulman J. A probabilistic proof of the Rogers – Ramanujan identities // Bull. London Math. Soc. – 2001. – 33, № 4.
– P. 397 – 407.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 1
ITERATION PROCESS FOR MULTIPLE ROGERS – RAMANUJAN IDENTITIES 125
35. Garvan F. G. Generalizations of Dyson’s rank and non-Rogers – Ramanujan partitions // Manuscr. Math. – 1994. –
84, № 3-4. – P. 343 – 359.
36. Gasper G., Rahman M. Basic hypergeometric series. – 2 nd ed. – Cambridge: Cambridge Univ. Press, 2004.
37. Gordon B. A combinatorial generalization of the Rogers – Ramanujan identities // Amer. J. Math. – 1961. – 83. –
P. 393 – 399.
38. Jackson F. H. Examples of a generalization of Euler’s transformation for power series // Messenger Math. – 1928. –
57. – P. 169 – 187.
39. Jacobi C. G. J. Fundamenta Nova Theoriae Functionum Ellipticarum // Fratrum Bornträger Regiomonti. – 1829;
Gesammelte werke. – Berlin: G. Reimer, 1881. – Bd 1.
40. Lovejoy J. Overpartition theorems of the Rogers – Ramanujan type // J. London Math. Soc. (2). – 2004. – 69, №. 3.
– P. 562 – 574.
41. McLaughlin J., Sills A. V. Ramanujan – Slater type identities related to the moduli 18 and 24 // J. Math. Anal. and
Appl. – 2008. – 344. – P. 765 – 777.
42. McLaughlin J., Sills A. V. Combinatorics of Ramanujan – Slater type identities // Integers 9 Supplement. – 2009. –
Art#10.
43. Paule P. On identities of the Rogers – Ramanujan type // J. Math. Anal. and Appl. – 1985. – 107, №. 1. – P. 255 – 284.
44. Sills A. V. On identities of the Rogers – Ramanujan type // Ramanujan J. – 2006. – 11, № 3. – P. 403 – 429.
45. Rogers L. J. Second memoir on the expansion of certain infinite products // Proc. London Math. Soc. – 1894. – 25.
– P. 318 – 343.
46. Rogers L. J. On two theorems of combinatory analysis and some allied identities // Proc. London Math. Soc. – 1917.
– 16. – P. 315 – 336.
47. Selberg A. Über einige arithmetische identitäten // Avh. Norske. Vidensk. Akad. Oslo l. Mat. Naturvidensk., Kl. –
1936. – 8. – P. 2 – 23.
48. Schilling A., Warnaar S. O. A higher level Bailey lemma: proof and application // Ramanujan J. – 1998. – 2. –
P. 327 – 349.
49. Sills A. V. A partition bijection related to the Rogers – Selberg identities and Gordon’s theorem // J. Combin. Theory
(Ser. A). – 2008. – 115. – P. 67 – 83.
50. Singh U. B. Certain bibasic hypergeometric transformation formulae and their application to Rogers – Ramanujan
identities // J. Math. Anal. and Appl. – 1996. – 198, №. 3. – P. 671 – 684.
51. Slater L. J. A new proof of Rogers’s transformations of infinite series // Proc. London Math. Soc. (2). – 1951. – 53.
– P. 460 – 475.
52. Slater L. J. Further identities of the Rogers – Ramanujan type // Proc. London Math. Soc. (2). – 1952. – 54. –
P. 147 – 167.
53. Slater L. J. Generalized hypergeometric functions. – Cambridge: Cambridge Univ. Press, 1966.
54. Stanton D. The Bailey – Rogers – Ramanujan group // q-Series with Applications to Combinatorics, Number Theory,
and Physics (Urbana, IL, 2000). – P. 55 – 70; Contemp. Math. – Providence, RI: Amer. Math. Soc., 2001. – 291.
55. Stembridge J. R. Hall – Littlewood functions, plane partitions, and the Rogers – Ramanujan identities // Trans. Amer.
Math. Soc. – 1990. – 319. – P. 469 – 498.
56. Warnaar S. O. Supernomial coefficients, Bailey’s lemma and Rogers – Ramanujantype identities. A survey of results
and open problems // The Andrews Festschrift (Maratea, 1998); Séminaire Lotharingien de Combinatoire. – 1999. –
42. – Art. B42n. – P. 22.
57. Warnaar S. O. The generalized Borwein conjecture: I. The Burge transform // q-Series with Applications to
Combinatorics, Number Theory, and Physics (Urbana, IL, 2000). – P. 243 – 267; Contemp. Math. – Providence,
RI: Amer. Math. Soc., 2001. – 291.
58. Warnaar S. O. The generalized Borwein conjecture: II. Refined q-trinomial coefficients // Discrete Math. – 2003. –
272, №. 2-3. – P. 215 – 258.
59. Watson G. N. A new proof of the Rogers – Ramanujan identities // J. London Math. Soc. – 1929. – 4. – P. 4 – 9.
60. Watson G. N. Theorems stated by Ramanujan: VII. Theorems on continued fractions // J. London Math. Soc. – 1929.
– 4. – P. 39 – 48.
Received 14.02.11
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| id | umjimathkievua-article-2559 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:46Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3c/64a03f5fb7c16a3187a2b4f76d5e233c.pdf |
| spelling | umjimathkievua-article-25592020-03-18T19:29:28Z Iteration process for multiple Rogers-Ramanujan identities Ітерацiйний процес для кратних тотожностей Роджерса – Рамануджана Chu, W. Wang, C. Чу, У. Ван, С. Replacing the monomials by an arbitrary sequence in the recursive lemma found by Bressoud (1983), we establish several general transformation formulas from unilateral multiple basic hypergeometric series to bilateral univariate ones, which are then used for the derivation of numerous multiple series identities of Rogers-Ramanujan type. За допомогою замiни мономiв довiльною послiдовнiстю в рекурентнiй лемi Брессо (1983) встановлено декiлька загальних формул перетворення однобiчних кратних основних гiпергеометричних рядiв у двобiчнi одновимiрнi ряди, якi потiм використовуються для виведення численних тотожностей типу Роджерса – Рамануджана для кратних рядiв. Institute of Mathematics, NAS of Ukraine 2012-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2559 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 1 (2012); 100-125 Український математичний журнал; Том 64 № 1 (2012); 100-125 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2559/1877 https://umj.imath.kiev.ua/index.php/umj/article/view/2559/1878 Copyright (c) 2012 Chu W.; Wang C. |
| spellingShingle | Chu, W. Wang, C. Чу, У. Ван, С. Iteration process for multiple Rogers-Ramanujan identities |
| title | Iteration process for multiple Rogers-Ramanujan identities |
| title_alt | Ітерацiйний процес для кратних тотожностей Роджерса – Рамануджана |
| title_full | Iteration process for multiple Rogers-Ramanujan identities |
| title_fullStr | Iteration process for multiple Rogers-Ramanujan identities |
| title_full_unstemmed | Iteration process for multiple Rogers-Ramanujan identities |
| title_short | Iteration process for multiple Rogers-Ramanujan identities |
| title_sort | iteration process for multiple rogers-ramanujan identities |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2559 |
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