Partial Sums of Two Quartic q-Series
The partial sums of two quartic basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several summation and transformation formulae are consequently established.
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2009 |
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| Мова: | Англійська |
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Інститут математики НАН України
2009
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| Цитувати: | Partial Sums of Two Quartic q-Series / W. Chu, C. Wang // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 20 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860079051637522432 |
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| author | Chu, W. Wang, C. |
| author_facet | Chu, W. Wang, C. |
| citation_txt | Partial Sums of Two Quartic q-Series / W. Chu, C. Wang // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 20 назв. — англ. |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The partial sums of two quartic basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several summation and transformation formulae are consequently established.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 050, 19 pages
Partial Sums of Two Quartic q-Series
Wenchang CHU † and Chenying WANG ‡
† Dipartimento di Matematica, Università degli Studi di Salento,
Lecce-Arnesano P. O. Box 193, Lecce 73100, Italy
E-mail: chu.wenchang@unile.it
‡ College of Mathematics and Physics, Nanjing University of Information Science
and Technology, Nanjing 210044, P. R. China
E-mail: wang.chenying@163.com
Received January 20, 2009, in final form April 17, 2009; Published online April 22, 2009
doi:10.3842/SIGMA.2009.050
Abstract. The partial sums of two quartic basic hypergeometric series are investigated
by means of the modified Abel lemma on summation by parts. Several summation and
transformation formulae are consequently established.
Key words: basic hypergeometric series (q-series); well-poised q-series; quadratic q-series;
cubic q-series; quartic q-series; the modified Abel lemma on summation by parts
2000 Mathematics Subject Classification: 33D15; 05A15
1 Introduction and motivation
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)∞ /
(
xqn; q
)
∞.
The product and fraction of shifted factorials are abbreviated respectively 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
.
Following Gasper–Rahman [12], the basic hypergeometric series is defined by
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,
where the base q will be restricted to |q| < 1 for nonterminating q-series. For its connections to
special functions and orthogonal polynomials, the reader can refer, for example, to the mono-
graph written by Andrews–Askey–Roy [2] and the paper by Koornwinder [15].
In the theory of basic hypergeometric series, there are several important classes, for example,
well-poised [3], quadratic [13, 17], cubic [16] and quartic [10, 11] series. To our knowledge, there
are four quartic series which can be displayed as follows:
Fn(a, b, d) :=
n−1∑
k=0
(
1− q5ka
) [
b, d
q3a/b2d2
∣∣∣q]
k
[qa/bd; q]3k
[bd, bd/q, qbd; q2]k
[
b2d2/q2
q4a/b, q4a/d
∣∣∣q4
]
k
qk;
mailto:chu.wenchang@unile.it
mailto:wang.chenying@163.com
http://dx.doi.org/10.3842/SIGMA.2009.050
2 W. Chu and C. Wang
Gn(a, c, e) :=
n−1∑
k=0
(
1− q5ka
)[ c2e2/q2a3
qa/c, qa/e
∣∣∣q]
k
[qa2/ce, q2a2/ce, q3a2/ce; q2]k
(ce/a; q)3k
[
c, e
q6a4/c2e2
∣∣∣q4
]
k
qk;
Un(a, b, d) :=
n−1∑
k=0
(
1−q5ka
)(q2a/bd; q)k
(bd; q2)2k
[
b, d
q5a2/b2d2
∣∣∣q2
]
k
(q3a2/bd; q6)k
(−bd/q3a)k
[
b2d2/q3a
q3a/b, q3a/d
∣∣∣q3
]
k
q(
k
2);
Vn(a, c, e) :=
n−1∑
k=0
(
1− q5ka
)(a2/ce; q2)2k
(qce/a; q)k
[
qc2e2/a2
qa/c, qa/e
∣∣∣q2
]
k
(−a/ce)k
(q5ce; q6)k
[
qc, qe
q2a3/c2e2
∣∣∣q3
]
k
q−(k
2).
By means of the series rearrangement, Gasper and Rahman [10, 11, 12] discovered several
summation and transformation formulae for the nonterminating special cases of Fn(a, b, d) and
Un(a, b, d). The terminating series identities for the last four sums have been established by
Chu [4] and Chu–Wang [8], respectively, through inversion techniques and Abel’s lemma on
summation by parts. For the partial sums Fn(a, b, d) and Gn(a, c, e), the present authors [7]
recently derived several useful reciprocal relations and transformation formulae in terms of well-
poised series.
The purpose of this paper is to investigate the remaining two partial sums Un(a, b, d) and
Vn(a, c, e). By utilizing the modified Abel lemma on summation by parts, we shall show six
unusual transformation formulae with two between them and other four expressing Un(a, b, d)
and Vn(a, c, e) as partial sums of quadratic and cubic series. Several new and known terminating
as well as nonterminating series identities are consequently obtained as particular instances.
In order to make the paper self-contained, we reproduce Abel’s lemma on summation by
parts (cf. [5, 6, 7, 8]) as follows. For an arbitrary complex sequence {τk}, define the backward
and forward difference operators 5 and ∆, respectively, by
5τk = τk − τk−1 and ∆τk = τk+1 − τk.
Then Abel’s lemma on summation by parts can be modified as follows:
n−1∑
k=0
Bk 5Ak =
{
An−1Bn −A−1B0
}
−
n−1∑
k=0
Ak∆Bk.
This can be considered as the discrete counterpart for the integral formula∫ b
a
f(x)g′(x)dx =
{
f(b)g(b)− f(a)g(a)
}
−
∫ b
a
f ′(x)g(x)dx.
In fact, it is almost trivial to check the following expression
n−1∑
k=0
Bk 5Ak =
n−1∑
k=0
Bk
{
Ak −Ak−1
}
=
n−1∑
k=0
AkBk −
n−1∑
k=0
Ak−1Bk.
Replacing k by k + 1 for the last sum, we can reformulate the equation as follows
n−1∑
k=0
Bk 5Ak = An−1Bn −A−1B0 +
n−1∑
k=0
Ak
{
Bk −Bk+1
}
= An−1Bn −A−1B0 −
n−1∑
k=0
Ak∆Bk,
which is exactly the equality stated in the modified Abel lemma.
Partial Sums of Two Quartic q-Series 3
Throughout the paper, if Wn is used to denote the partial sum of some q-series, then the
corresponding letter W without subscript will stand for the limit of Wn (if it exists of course)
when n →∞.
When applying the modified Abel lemma on summation by parts to deal with hypergeometric
series, the crucial step lies in finding shifted factorial fractions {Ak, Bk} so that their differences
are expressible as ratios of linear factors. This has not been an easy task, even though it is
indeed routine to factorize {Ak, Bk} once they are figured out. Specifically for Un(a, b, d) and
Vn(a, b, d), we shall devise three well-poised difference pairs for each partial sum. This is based
on numerous attempts to detect Ak and Bk sequences such that not only their differences turn
out to be factorizable, but also their combinations match exactly the summands displayed in
Un(a, b, d) and Vn(a, b, d).
The contents of the paper will be organized as follows. In the second section, Un(a, b, d) will
be reformulated through the modified Abel lemma on summation by parts, which lead to three
transformations of Un(a, b, d) into partial sums of quadratic, cubic and quartic series. Then the
third section will be devoted to the transformation formulae of Vn(a, b, d) in terms of partial
sums of quadratic, cubic and quartic series. These transformations on Un(a, b, d) and Vn(a, b, d)
will recover several known identities appeared in Chu–Wang [4, 8] and Gasper–Rahman [12],
and yield a few additional new summation formulae.
2 Transformation and summation formulae for Un(a, b, d)
In this section, we shall investigate the partial sum of quartic q-series Un(a, b, d). Three trans-
formation formulae will be established from Un(a, b, d) to quadratic, cubic and another quar-
tic series, unlike those for Fn(a, b, d) and Gn(a, c, e) shown in [7], where reciprocal relations
and transformations into well-poised partial sums have been derived. As particular cases of
Un(a, b, d), four nonterminating series will be evaluated, including two that appeared in Gasper–
Rahman’s book [12].
In order for the reader to gain an immediate insight into the well-poised structure, we refor-
mulate the series Un(a, b, d) in the following manner:
Un(a, b, d) =
n−1∑
k=0
(
1− q5ka
) [b, d; q2]k
[q3a/b, q3a/d; q3]k
(b2d2/q3a; q3)k
(q5a2/b2d2; q2)k
qk
× [q2a/bd, q2a/bd; q]k
[bd, q2bd; q4]k
(q3a2/bd; q6)k
(bd/q2a; q−1)k
.
2.1 Quartic series to quadratic series
Let Ak and Bk be defined by
Ak =
[q3a/bd, q5a/bd; q]k(b3d3/q7a2; q2)k(q9a2/bd; q6)k
[bd, q2bd; q4]k(q12a3/b3d3; q3)k(bd/q4a; q−1)k
,
Bk =
[b, d; q2]k
[q3a/b, q3a/d; q3]k
[b2d2/q6a, q12a3/b3d3; q3]k
[q9a2/b2d2, b3d3/q9a2; q2]k
.
We can easily show the relations
$ := A−1B0 =
a(1− q2/bd)(1− q4/bd)(1− q3a/bd)(1− q9a3/b3d3)
(1− q2a/bd)(1− q4a/bd)(1− q3a2/bd)(1− q9a2/b3d3)
,
R :=
An−1Bn
A−1B0
=
1− q9+3na3/b3d3
1− q9a3/b3d3
[
b, d
q9a2/b2d2
∣∣∣q2
]
n
4 W. Chu and C. Wang
× [q2a/bd, q4a/bd; q]n
(bd/q4; q2)2n
[
b2d2/q6a
q3a/b, q3a/d
∣∣∣q3
]
n
(q3a2/bd; q6)n
(bd/q3a; q−1)n
;
and calculate the finite differences
5Ak =
(1− q5ka)(1− q6−3ka/b2d2)(1− q2k+5a2/b2d2)(1− q2k+7a2/b2d2)
(1− q2a/bd)(1− q4a/bd)(1− q3a2/bd)(1− q9a2/b3d3)
× [q2a/bd, q4a/bd; q]k(b3d3/q9a2; q2)k(q3a2/bd; q6)k
[bd, q2bd; q4]k(q12a3/b3d3; q3)k(bd/q4a; q−1)k
q2k,
∆Bk = −(1− q3+5ka)(1− q3+ka/bd)(1− q9a2/b2d3)(1− q9a2/b3d2)
(1− q3a/b)(1− q3a/d)(1− q9a2/b2d2)(1− q9a2/b3d3)
×
[
b, d
q11a2/b2d2, b3d3/q7a2
∣∣∣q2
]
k
[
b2d2/q6a, q12a3/b3d3
q6a/b, q6a/d
∣∣∣q3
]
k
q2k.
According to the modified Abel lemma on summation by parts, the finite U -sum can be refor-
mulated as follows:
Un(a, b, d)
(1− q5a2/b2d2)(1− q7a2/b2d2)(1− q6a/b2d2)
(1− q2a/bd)(1− q4a/bd)(1− q3a2/bd)(1− q9a2/b3d3)
=
n−1∑
k=0
Bk 5Ak = $
{
R− 1
}
−
n−1∑
k=0
Ak∆Bk.
Observing that there holds the equality for the last partial sum
−
n−1∑
k=0
Ak∆Bk =
Un(q3a, b, d) (1− q3a/bd)(1− q9a2/b2d3)(1− q9a2/b3d2)
(1− q3a/b)(1− q3a/d)(1− q9a2/b2d2)(1− q9a2/b3d3)
,
we derive after some simplification the recurrence relation
Un(a, b, d) = Un(q3a, b, d)
(q2a/bd; q)3(1− q3a2/bd)(1− q9a2/b2d3)(1− q9a2/b3d2)
(q5a2/b2d2; q2)3(1− q3a/b)(1− q3a/d)(1− q6a/b2d2)
− a
{
1− R(a, b, d)
}(1− q2/bd)(1− q4/bd)(1− q3a/bd)(1− q9a3/b3d3)
(1− q5a2/b2d2)(1− q7a2/b2d2)(1− q6a/b2d2)
.
Iterating this equation m-times, we get the following expression
Un(a, b, d) = Un(q3ma, b, d)
(q2a/bd; q)3m[q3a2/bd, q9a2/b2d3, q9a2/b3d2; q6]m
(q5a2/b2d2; q2)3m[q3a/b, q3a/d, q6a/b2d2; q3]m
− a (1− q2/bd)(1− q4/bd)(1− q3a/bd)
(1− q5a2/b2d2)(1− q7a2/b2d2)(1− q6a/b2d2)
×
m−1∑
k=0
(1− q9+9ka3/b3d3)
[
q2a/bd, q4a/bd, q6a/bd
q3a/b, q3a/d, q9a/b2d2
∣∣∣q3
]
k
q3k
×
{
1− R(q3ka, b, d)
} [
q3a2/bd, q9a2/b2d3, q9a2/b3d2
q9a2/b2d2, q11a2/b2d2, q13a2/b2d2
∣∣∣q6
]
k
.
Writing explicitly the R-function by separating k and n factorials
R(q3ka, b, d) =
1− q9+3n+9ka3/b3d3
1− q9+9ka3/b3d3
[
b, d
q9+6ka2/b2d2
∣∣∣q2
]
n
× [q2+3ka/bd, q4+3ka/bd; q]n
(bd/q4; q2)2n
[
q−6−3kb2d2/a
q3+3ka/b, q3+3ka/d
∣∣∣q3
]
n
(q3+6ka2/bd; q6)n
(q−3−3kbd/a; q−1)n
Partial Sums of Two Quartic q-Series 5
=
[q2a/bd, q4a/bd; q]n
(bd/q4; q2)2n
[
b, d
q9a2/b2d2
∣∣∣q2
]
n
[
b2d2/q6a
q3a/b, q3a/d
∣∣∣q3
]
n
(q3a2/bd; q6)n
(bd/q3a; q−1)n
× 1− q9+3n+9ka3/b3d3
1− q9+9ka3/b3d3
[
q3a/b, q3a/d, q9a/b2d2
q2a/bd, q4a/bd, q6a/bd
∣∣∣q3
]
k
(q3+6na2/bd; q6)k
(q3a2/bd; q6)k
× (q9a2/b2d2; q2)3k
(q9+2na2/b2d2; q2)3k
[
q2+na/bd, q4+na/bd, q6+na/bd
q3+3na/b, q3+3na/d, q9−3na/b2d2
∣∣∣q3
]
k
,
and then defining the partial sum of quadratic series in base q3 by
U�
m(a, b, d) =
m−1∑
k=0
(1− q9+9ka3/b3d3)
[
q2a/bd, q4a/bd, q6a/bd
q3a/b, q3a/d, q9a/b2d2
∣∣∣q3
]
k
×
[
q3a2/bd, q9a2/b2d3, q9a2/b3d2
q9a2/b2d2, q11a2/b2d2, q13a2/b2d2
∣∣∣q6
]
k
q3k,
we derive the following transformation formula.
Theorem 1 (Transformation between quartic and quadratic series).
Un(a, b, d)− Un(q3ma, b, d)
(q2a/bd; q)3m[q3a2/bd, q9a2/b2d3, q9a2/b3d2; q6]m
(q5a2/b2d2; q2)3m[q3a/b, q3a/d, q6a/b2d2; q3]m
=
(1− q3a/bd)(1− bd/q2)(1− bd/q4)
(1− q5a2/b2d2)(1− q7a2/b2d2)(1− b2d2/q6a)
×
{
U�
m(a, b, d)− U�
m(q5na, q2nb, q2nd)
(q3a2/bd; q6)n
(bd/q3a; q−1)n
×
[
b, d
q9a2/b2d2
∣∣∣q2
]
n
[q2a/bd, q4a/bd; q]n
(bd/q4; q2)2n
[
b2d2/q6a
q3a/b, q3a/d
∣∣∣q3
]
n
}
.
By means of the Weierstrass M -test on uniformly convergent series (cf. Stromberg [19,
p. 141]), we can compute the following limit
lim
m,n→∞
Un(q3ma, b, d) =
∞∑
k=0
(bd)k [b, d; q2]k
(bd; q2)2k
q4(k
2).
Letting m,n →∞ in Theorem 1, we obtain the transformation formula.
Proposition 2 (Nonterminating series transformation).
U(a, b, d) =
(1− q3a/bd)(1− bd/q2)(1− bd/q4)
(1− q5a2/b2d2)(1− q7a2/b2d2)(1− b2d2/q6a)
U�(a, b, d)
+
(q2a/bd; q)∞[q3a2/bd, q9a2/b2d3, q9a2/b3d2; q6]∞
(q5a2/b2d2; q2)∞[q3a/b, q3a/d, q6a/b2d2; q3]∞
∞∑
k=0
(bd)k [b, d; q2]k
(bd; q2)2k
q4(k
2).
When bd = q2+2δ with δ = 0, 1, this proposition results in
U(a, b, q2+2δ/b) =
[
a, q1−2δa
q3a/b, q1−2δab
∣∣∣q3
]
∞
[
q3−6δa2b, q5−4δa2/b
q3−6δa2, q5−4δa2
∣∣∣q6
]
∞
×
∞∑
k=0
[b, q2+2δ/b; q2]k
(q2+2δ; q2)2k
q4(k
2)+(2+2δ)k.
6 W. Chu and C. Wang
Taking a = q1+4δ in this equation and noting that the initial condition
U(q1+4δ, b, q2+2δ/b) = 1− q1+4δ,
we recover the following formula due to Andrews [1, equation (4.6)] and Ismail–Stanton [14,
Proposition 6]
∞∑
k=0
[b, q2+2δ/b; q2]k
(q2+2δ; q2)2k
q4(k
2)+(2+2δ)k =
[
q2+2δb, q4+4δ/b
q2+2δ, q4+4δ
∣∣∣q6
]
∞
,
which leads, under the replacement a → q2δa, to the nonterminating series formula.
Corollary 3 (Gasper–Rahman [12, Exercise 3.29(ii),(iii)]).
∞∑
k=0
1−q5k+2δa
1− q2δa
[
b, q2+2δ/b
qa2
∣∣∣q2
]
k
(a; q)k(q1+2δ/a; q3)k(q1+2δa2; q6)k
(q2+2δ; q2)2k[qab, q3+2δa/b; q3]k
q(
k+1
2 )(−a)k
=
[
qa, q3+2δa
qab, q3+2δa/b
∣∣∣q3
]
∞
[
q5a2/b, q3−2δa2b, q2+2δb, q4+4δ/b
q5a2, q3−2δa2, q2+2δ, q4+4δ
∣∣∣q6
]
∞
.
2.2 Quartic series to cubic series
Define two sequences by
Ak =
[q3a/bd, bd2/q4a; q]k(q2b; q2)k(q9a2/bd; q6)k
[q2bd, q9a2/bd2; q4]k(q3a/b; q3)k(bd/q4a; q−1)k
,
Bk =
(q4a/bd; q)k(d/q2; q2)k(b2d2/q3a; q3)k(q9a2/bd2; q4)k
(bd; q4)k(q6a/d; q3)k(q7a2/b2d2; q2)k(bd2/q5a; q)k
.
It is not hard to check the relations
$ := A−1B0 =
(1− bd/q2)(1− q5a2/bd2)(1− a/b)(1− bd/q3a)
(1− q2a/bd)(1− bd2/q5a)(1− b)(1− q3a2/bd)
,
R :=
An−1Bn
A−1B0
=
1− q5+4na2/bd2
1− q5a2/bd2
[
b, d/q2
q7a2/b2d2
∣∣∣q2
]
n
× [q2a/bd, q4a/bd; q]n
(bd/q2; q2)2n
[
b2d2/q3a
a/b, q6a/d
∣∣∣q3
]
n
(q3a2/bd; q6)n
(bd/q3a; q−1)n
;
and compute the finite differences
5Ak =
(1− q5ka)(1− q2−2k/d)(1− q5+2ka2/b2d2)(1− q3+3ka/d)
(1− q2a/bd)(1− q5a/bd2)(1− b)(1− q3a2/bd)
× [q2a/bd, bd2/q5a; q]k(b; q2)k(q3a2/bd; q6)k
[q2bd, q9a2/bd2; q4]k(q3a/b; q3)k(bd/q4a; q−1)k
qk,
∆Bk = −(1− q4+5ka)(1− q3+ka/bd)(1− q2+2kb)(1− q9a2/b2d3)
(1− q5a/bd2)(1− q7a2/b2d2)(1− q6a/d)(1− bd)
×
[
q4a/bd
bd2/q4a
∣∣∣q]
k
[
d/q2
q9a2/b2d2
∣∣∣q2
]
k
[
b2d2/q3a
q9a/d
∣∣∣q3
]
k
[
q9a2/bd2
q4bd
∣∣∣q4
]
k
qk.
Then applying the modified Abel lemma on summation by parts, the U -sum can alternatively
be reformulated as follows:
Un(a, b, d)
(1− q2/d)(1− q3a/d)(1− q5a2/b2d2)
(1− q2a/bd)(1− q5a/bd2)(1− b)(1− q3a2/bd)
Partial Sums of Two Quartic q-Series 7
=
n−1∑
k=0
Bk 5Ak = $
{
R− 1
}
−
n−1∑
k=0
Ak∆Bk.
Observing that the last partial sum results in
−
n−1∑
k=0
Ak∆Bk =
Un(q4a, q4b, d/q2) (1− q2b)(1− q3a/bd)(1− q9a2/b2d3)
(1− bd)(1− q6a/d)(1− q5a/bd2)(1− q7a2/b2d2)
,
we derive after some simplification the relation
Un(a, b, d) = Un(q4a, q4b, d/q2)
(b; q2)2(q2a/bd; q)2(1− q3a2/bd)(1− q9a2/b2d3)
(q5a2/b2d2; q2)2(q3a/d; q3)2(1− q2/d)(1− bd)
+
{
1− R(a, b, d)
}(1− bd/q2)(1− q5a2/bd2)(1− a/b)(1− q3a/bd)
(1− d/q2)(1− q3a/d)(1− q5a2/b2d2)
.
Iterating it m-times, we get the following expression
Un(a, b, d) = Un(q4ma, q4mb, q−2md)
(b; q2)2m(q2a/bd; q)2m[q3a2/bd, q9a2/b2d3; q6]m
(q5a2/b2d2; q2)2m(q3a/d; q3)2m[q2/d, bd; q2]m
+
(1− bd/q2)(1− q3a/bd)(1− a/b)
(1− d/q2)(1− q3a/d)(1− q5a2/b2d2)
m−1∑
k=0
(1− q5+8ka2/bd2)(b; q2)2k
(q7a2/b2d2; q2)2k
×
{
1−R(q4ka, q4kb, q−2kd)
}[q2a/bd, q5a/bd
q4/d, bd/q2
∣∣∣q2
]
k
[
q3a2/bd, q9a2/b2d3
q6a/d, q9a/d
∣∣∣q6
]
k
q2k.
Rewriting the R-function explicitly as
R(q4ka, q4kb, q−2kd) =
1− q5+4n+8ka2/bd2
1− q5+8ka2/bd2
[
q4kb, q−2−2kd
q7+4ka2/b2d2
∣∣∣q2
]
n
× [q2+2ka/bd, q4+2ka/bd; q]n
(q2k−2bd; q2)2n
[
b2d2/q3a
a/b, q6+6ka/d
∣∣∣q3
]
n
(q3+6ka2/bd; q6)n
(q−3−2kbd/a; q−1)n
=
[q2a/bd, q4a/bd; q]n
(bd/q2; q2)2n
[
b, d/q2
q7a2/b2d2
∣∣∣q2
]
n
[
b2d2/q3a
a/b, q6a/d
∣∣∣q3
]
n
(q3a2/bd; q6)n
(bd/q3a; q−1)n
× 1− q5+4n+8ka2/bd2
1− q5+8ka2/bd2
[
q2+na/bd, q5+na/bd, q4/d, q−2bd
q2a/bd, q5a/bd, q4−2n/d, q4n−2bd
∣∣∣q2
]
k
× (q6a/d; q3)2k
(q6+3na/d; q3)2k
[
q2nb, q7a2/b2d2
b, q7+2na2/b2d2
∣∣∣q2
]
2k
(q3+6na2/bd; q6)k
(q3a2/bd; q6)k
,
and defining further the finite cubic sum in base q2 by
U4
m(a, b, d) =
m−1∑
k=0
(1− q5+8ka2/bd2)
[
q3a2/bd, q9a2/b2d3
q6a/d, q9a/d
∣∣∣q6
]
k
× (b; q2)2k
(q7a2/b2d2; q2)2k
[
q2a/bd, q5a/bd
q4/d, bd/q2
∣∣∣q2
]
k
q2k,
we find the following transformation formula.
Theorem 4 (Transformation between quartic and cubic series).
Un(a, b, d)− Un(q4ma, q4mb, q−2md)
(b; q2)2m(q2a/bd; q)2m[q3a2/bd, q9a2/b2d3; q6]m
(q5a2/b2d2; q2)2m(q3a/d; q3)2m[q2/d, bd; q2]m
8 W. Chu and C. Wang
=
(1− bd/q2)(1− q3a/bd)(1− a/b)
(1−d/q2)(1−q3a/d)(1−q5a2/b2d2)
{
U4
m(a, b, d)− U4
m(q5na, q2nb, q2nd)
× [q2a/bd, q4a/bd; q]n
(bd/q2; q2)2n
[
b, d/q2
q7a2/b2d2
∣∣∣q2
]
n
[
b2d2/q3a
a/b, q6a/d
∣∣∣q3
]
n
(q3a2/bd; q6)n
(bd/q3a; q−1)n
}
.
By means of the Weierstrass M -test, we can compute the limit
lim
m,n→∞
Un(q4ma, q4mb, q−2md) =
∞∑
k=0
(a
b
)k (b2d2/q3a; q3)k
(q3a/b; q3)k
q3(k+1
2 ).
The limiting case m, n →∞ of Theorem 4 leads to the transformation formula.
Proposition 5 (Nonterminating series transformation).
U(a, b, d) =
(1− bd/q2)(1− q3a/bd)(1− a/b)
(1− d/q2)(1− q3a/d)(1− q5a2/b2d2)
U4(a, b, d)
+
(q2a/bd; q)∞(b; q2)∞[q3a2/bd, q9a2/b2d3; q6]∞
(q3a/d; q3)∞[q2/d, bd, q5a2/b2d2; q2]∞
∞∑
k=0
(q3a
b
)k (b2d2/q3a; q3)k
(q3a/b; q3)k
q3(k
2).
When b = a, this proposition reduces to the following relation
U(a, a, d) =
(q2/d; q)∞(a; q2)∞[q3a/d, q9/d3; q6]∞
(q3a/d; q3)∞[q2/d, ad, q5/d2; q2]∞
∞∑
k=0
(ad2/q3; q3)k
(q3; q3)k
q3(k
2)+3k.
Taking d = 1 in the last equation and noting U(a, b, 1) = 1− a, we get
∞∑
k=0
(a/q3; q3)k
(q3; q3)k
q3(k
2)+3k =
(a; q6)∞
(q3; q6)∞
,
which results also from a limiting case of the q-Bailey–Daum formula (cf. [12, II-9]).
Combining the last two equations leads us to the nonterminating series identity.
Corollary 6 (Gasper–Rahman [12, Exercise 3.29(i)]).
∞∑
k=0
1− q5ka
1− a
[
a, d
q5/d2
∣∣∣q2
]
k
[
ad2/q3
q3, q3a/d
∣∣∣q3
]
k
(q2/d; q)k(q3a/d; q6)k
(ad; q2)2k
q(
k
2)
(
− q3/d
)k
=
[
q2a, q3/d
ad, q5/d2
∣∣∣q2
]
∞
[
ad2, q9/d3
q3, q6a/d
∣∣∣q6
]
∞
.
2.3 Quartic series to quartic series
Finally, for the two sequences given by
Ak =
(q3a/bd; q)k(q2b; q2)k(b2d2/a; q3)k(q5a2/b2d; q4)k
(q2bd; q4)k(q3a/b; q3)k(q5a2/b2d2; q2)k(b2d/a; q)k
,
Bk =
[qa/bd, b2d/a; q]k(d/q2; q2)k(q3a2/bd; q6)k
[bd, qa2/b2d; q4]k(q3a/d; q3)k(bd/q2a; q−1)k
;
we have no difficulty to check the relations
$ := A−1B0 =
(1− b2d/qa)(1− b2d2/q3a2)(1− b/a)(1− bd/q2)
(1− bd/q2a)(1− b)(1− b2d2/q3a)(1− b2d/qa2)
,
Partial Sums of Two Quartic q-Series 9
R :=
An−1Bn
A−1B0
=
1− qn−1b2d/a
1− q−1b2d/a
[
b, d/q2
q3a2/b2d2
∣∣∣q2
]
n
× [qa/bd, q2a/bd; q]n
(bd/q2; q2)2n
[
b2d2/q3a
a/b, q3a/d
∣∣∣q3
]
n
(q3a2/bd; q6)n
(bd/q2a; q−1)n
;
and compute the finite differences
5Ak =
(1− q5ka)(1− q2−2k/d)(1− q1+ka/bd)(1− q3a2/b3d2)
(1− b)(1− q2a/bd)(1− qa2/b2d)(1− q3a/b2d2)
×
[
q2a/bd
b2d/a
∣∣∣q]
k
[
b
q5a2/b2d2
∣∣∣q2
]
k
[
b2d2/q3a
q3a/b
∣∣∣q3
]
k
[
qa2/b2d
q2bd
∣∣∣q4
]
k
q3k,
∆Bk = −(1− q1+5ka)(1− q3ka/b)(1− q2+2kb)(1− q3+2ka2/b2d2)
(1− bd)(1− qa2/b2d)(1− q3a/d)(1− q2a/bd)
× [qa/bd, b2d/a; q]k(d/q2; q2)k(q3a2/bd; q6)k
[q4bd, q5a2/b2d; q4]k(q6a/d; q3)k(bd/q3a; q−1)k
q−k.
Then by means of the modified Abel lemma on summation by parts, the U -sum can be refor-
mulated as follows:
Un(a, b, d)
(1− q2/d)(1− qa/bd)(1− q3a2/b3d2)
(1− b)(1− q2a/bd)(1− qa2/b2d)(1− q3a/b2d2)
=
n−1∑
k=0
Bk 5Ak = $
{
R− 1
}
−
n−1∑
k=0
Ak∆Bk.
Writing the last partial sum in terms of U -sum as
−
n−1∑
k=0
Ak∆Bk =
Un(qa, q4b, d/q2) (1− b/a)(1− q2b)(1− b2d2/q3a2)
(1− bd)(1− b2d/qa2)(1− q3a/d)(1− bd/q2a)
,
we derive after some simplification the following relation
Un(a, b, d) =
{
1− R(a, b, d)
}(1− a/b)(1− bd/q2)(1− b2d/qa)(1− b2d2/q3a2)
(1− d/q2)(1− bd/qa)(1− b3d2/q3a2)
− Un(qa, q4b, d/q2)
(q2a/bd) (b; q2)2(1− b2d2/q3a)(1− b2d2/q3a2)(1− b/a)
(1− bd)(1− q2/d)(1− bd/qa)(1− q3a/d)(1− b3d2/q3a2)
.
Iterating it m-times, we get the following expression
Un(a, b, d) = Un(qma, q4mb, q−2md)
[
b2d2/q3a2
q2/d, bd
∣∣∣q2
]
m
× (b; q2)2m
[bd/qa, bd/q2a; q]m
[
b/a, b2d2/q3a
q3a/d
∣∣∣q3
]
m
(q2a/bd; q−1)m
(b3d2/q3a2; q6)m
+
(1− a/b)(1− bd/q2)(1− b2d2/q3a2)
(1− d/q2)(1− bd/qa)(1− b3d2/q3a2)
m−1∑
k=0
(1− q5k−1b2d/a)
[
b2d2/qa2
q4/d, bd/q2
∣∣∣q2
]
k
×
{
1−R(qka, q4kb, q−2kd)
} (b; q2)2k
(bd/a; q)k
[
q3b/a, b2d2/q3a
q3a/d
∣∣∣q3
]
k
(
qa/bd
)k
q−(k
2)
(q3b3d2/a2; q6)k
.
Separating k and n factorials in the R-function
R(qka, q4kb, q−2kd) =
1− qn+5k−1b2d/a
1− q5k−1b2d/a
[
q4kb, q−2−2kd
q3−2ka2/b2d2
∣∣∣q2
]
n
10 W. Chu and C. Wang
× [q1−ka/bd, q2−ka/bd; q]n
(q2k−2bd; q2)2n
[
q3k−3b2d2/a
q−3ka/b, q3+3ka/d
∣∣∣q3
]
n
(q3a2/bd; q6)n
(qk−2bd/a; q−1)n
=
[qa/bd, q2a/bd; q]n
(bd/q2; q2)2n
[
b, d/q2
q3a2/b2d2
∣∣∣q2
]
n
[
b2d2/q3a
a/b, q3a/d
∣∣∣q3
]
n
(q3a2/bd; q6)n
(bd/q2a; q−1)n
× 1− qn+5k−1b2d/a
1− q5k−1b2d/a
[
q3n−3b2d2/a, q3−3nb/a, q3a/d
b2d2/q3a, q3b/a, q3+3na/d
∣∣∣q3
]
k
qnk
× (bd/a; q)k(q2nb; q2)2k
(q−nbd/a; q)k(b; q2)2k
[
q4/d, bd/q2, q−1−2nb2d2/a2
q4−2n/d, q4n−2bd, b2d2/qa2
∣∣∣q2
]
k
,
and then defining the partial sum of quartic series
U?
m(a, b, d) =
m−1∑
k=0
(1− q5k−1b2d/a)
[
b2d2/qa2
q4/d, bd/q2
∣∣∣q2
]
k
(b; q2)2k
(bd/a; q)k
×
[
q3b/a, b2d2/q3a
q3a/d
∣∣∣q3
]
k
(
− qa/bd
)k
q−(k
2)
(q3b3d2/a2; q6)k
,
we establish the following transformation formula.
Theorem 7 (Transformation between two quartic series).
Un(a, b, d)− Un(qma, q4mb, q−2md)
[
b2d2/q3a2
bd, q2/d
∣∣∣q2
]
m
× (b; q2)2m
[bd/qa, bd/q2a; q]m
[
b2d2/q3a, b/a
q3a/d
∣∣∣q3
]
m
(q2a/bd; q−1)m
(b3d2/q3a2; q6)m
=
(1− a/b)(1−bd/q2)(1−b2d2/q3a2)
(1−d/q2)(1−bd/qa)(1−b3d2/q3a2)
{
U?
m(a, b, d)− U?
m(q5na, q2nb, q2nd)
× [qa/bd, q2a/bd; q]n
(bd/q2; q2)2n
[
b, d/q2
q3a2/b2d2
∣∣∣q2
]
n
[
b2d2/q3a
a/b, q3a/d
∣∣∣q3
]
n
(q3a2/bd; q6)n
(bd/q2a; q−1)n
}
.
In particular for m = n, we have the reduced transformation.
Proposition 8 (Transformation between two quartic series: q3a2 = b2d2).
Un(a, b, d) = U?
n(q5na, q2nb, q2nd)
(a; q3)n(bd; q6)n
(q3a/d; q3)n(q3a/b; q3)n−1
× (qa/bd; q)n(q3a/bd; q)n−1
(qa/bd; q−1)n(bd; q2)2n−1
[
q2b, d
q2
∣∣∣q2
]
n−1
.
In order to examine the limiting case n → ∞ of the last equation, we write U?
n(q5na, q2nb,
q3/2+2na/b) explicitly as follows:
U?
n(q5na, q2nb, q3/2+2na/b) =
n−1∑
k=0
(1− qn+5k+1/2b)
[
q2−2n
q5/2−2nb/a, q4n−1/2a
∣∣∣q2
]
k
×
[
q3−3nb/a, q3na
q3/2+3nb
∣∣∣q3
]
k
(q2nb; q2)2k
(
− qn−1/2
)k
(q3/2−n; q)k(q6b; q6)k
q−(k
2).
Inverting the summation index k → n− 1− k and then applying the relation
(q2−2n; q2)n−1−k(q3−3nb/a; q3)n−1−k
(q5/2−2nb/a; q2)n−1−k(q3/2−n; q)n−1−k
Partial Sums of Two Quartic q-Series 11
=
(q2; q2)n−1(q3a/b; q3)n−1
(q3/2a/b; q2)n−1(q1/2; q)n−1
q1−n2 (q1/2; q)k(q3/2a/b; q2)k
(q2; q2)k(q3a/b; q3)k
qk2+2k,
we can reformulate the finite sum U?
n(q5na, q2nb, q3/2+2na/b) as
(−1)n−1 (q2; q2)n−1(q3a/b; q3)n−1
(q3/2a/b; q2)n−1(q1/2; q)n−1
q
1−n2
2
n−1∑
k=0
(−1)k (q1/2; q)k(q3/2a/b; q2)k
(q2; q2)k(q3a/b; q3)k
q
k(k+2)
2
× (1− q6n−5k−9/2b)(q2nb; q2)2n−2−2k(q3na; q3)n−1−k
(q4n−1/2a; q2)n−1−k(q3/2+3nb; q3)n−1−k(q6b; q6)n−1−k
.
Substituting this expression into Proposition 8 and then letting n →∞, we derive the following
transformation formula
U(a, b, q3/2a/b) =
(b; q2)∞(a; q3)∞(q3/2a; q6)∞
(q3/2a; q2)∞(q3/2b; q3)∞(b; q6)∞
×
∞∑
k=0
(
−q3/2
)k (q1/2; q)k(q3/2a/b; q2)k
(q2; q2)k(q3a/b; q3)k
q(
k
2). (1)
From this transformation, we can derive two new interesting summation formulae. First,
taking b = 1 in this equation and keeping in mind that U(a, 1, d) = 1, we obtain the following
remarkable summation formula.
Corollary 9 (Nonterminating series identity).
∞∑
k=0
(
− q3/2
)k (q1/2; q)k(q3/2a; q2)k
(q2; q2)k(q3a; q3)k
q(
k
2) =
(q3/2a; q2)∞(q3/2; q3)∞(q6; q6)∞
(q2; q2)∞(q3a; q3)∞(q3/2a; q6)∞
.
The special case a = 0 of this corollary recovers an identity of Rogers–Ramanujan type due
to Stanton [18, p. 61]:
∞∑
k=0
(−q; q2)k
(q4; q4)k
qk(k+2) =
(−q; q2)∞
(q2; q2)∞
[q6, q, q5; q6]∞.
Combining (1) with Corollary 9 yields another formula for quartic series.
Corollary 10 (Nonterminating series identity).
∞∑
k=0
(−1)k 1− q5ka
1− a
[
b, q3/2a/b
q2
∣∣∣q2
]
k
[
a
q3a/b, q3/2b
∣∣∣q3
]
k
(q1/2; q)k(q3/2a; q6)k
(q3/2a; q2)2k
q
k2+2k
2
=
[
b, q3/2a/b
q2, q3/2a
∣∣∣q2
]
∞
[
q3a, q3/2
q3a/b, q3/2b
∣∣∣q3
]
∞
[
q6, q3/2a
b, q3/2a/b
∣∣∣q6
]
∞
.
3 Transformation and summation formulae for Vn(a, b, d)
The quartic series Vn(a, c, e) may be considered as dual one to the Un(a, b, d) in the last section
in the sense that the numerator factorials and denominator factorials are inverted. This section
will be devoted analogously to investigation of summation and transformation formulae for
Vn(a, c, e). As the series Un(a, c, e), the following expression for Vn(a, c, e) makes its well-poised
structure more transparent
Vn(a, c, e) =
n−1∑
k=0
(1− q5ka)
(qc2e2/a2; q2)k
(q2a3/c2e2; q3)k
[qc, qe; q3]k
[qa/c, qa/e; q2]k
qk
× [a2/ce, q2a2/ce; q4]k
[qce/a, qce/a; q]k
(q−1a/ce; q−1)k
(q5ce; q6)k
.
12 W. Chu and C. Wang
3.1 Quartic series to quadratic series
Let Ak and Bk be defined by
Ak =
[q3c2e2/a2, a4/qc3e3; q2]k[q4c, q4e; q3]k
[q2a3/c2e2, q6c3e3/a3; q3]k[qa/c, qa/e; q2]k
,
Bk =
(q6c3e3/a3; q3)k(a2/ce; q2)2k(a/q2ce; q−1)k
(a4/q3c3e3; q2)k[qce/a, q3ce/a; q]k(q5ce; q6)k
.
We can easily show the following relations
$ := A−1B0 =
(1− a/qc)(1− a/qe)(1− a3/qc2e2)(1− q3c3e3/a3)
(1− qc)(1− qe)(1− qc2e2/a2)(1− a4/q3c3e3)
,
R :=
An−1Bn
A−1B0
=
1− q3+3nc3e3/a3
1− q3c3e3/a3
[
qc2e2/a2
a/qc, a/qe
∣∣∣q2
]
n
× (a2/ce; q2)2n
[qce/a, q3ce/a; q]n
[
qc, qe
a3/qc2e2
∣∣∣q3
]
n
(a/q2ce; q−1)n
(q5ce; q6)n
;
and calculate the finite differences
5Ak =
(1− q5ka)(1− q2+kce/a)(1− q2c2e3/a3)(1− q2c3e2/a3)
(1− qc)(1− qe)(1− qc2e2/a2)(1− q3c3e3/a4)
×
[
qc2e2/a2, a4/q3c3e3
qa/c, qa/e
∣∣∣q2
]
k
[
qc, qe
q2a3/c2e2, q6c3e3/a3
∣∣∣q3
]
k
q2k,
∆Bk = −(1− q3+5ka)(1− q1−3kc2e2/a3)(1− q3+2kc2e2/a2)(1− q5+2kc2e2/a2)
(1− q3c3e3/a4)(1− qce/a)(1− q3ce/a)(1− q5ce)
× (q6c3e3/a3; q3)k(a2/ce; q2)2k(a/q2ce; q−1)k
(a4/qc3e3; q2)k[q2ce/a, q4ce/a; q]k(q11ce; q6)k
q2k.
Applying the modified Abel lemma on summation by parts, we can manipulate the finite V -sum
as follows:
Vn(a, c, e)
(1− q2ce/a)(1− q2c2e3/a3)(1− q2c3e2/a3)
(1− qc)(1− qe)(1− qc2e2/a2)(1− q3c3e3/a4)
=
n−1∑
k=0
Bk 5Ak = $
{
R− 1
}
−
n−1∑
k=0
Ak∆Bk.
Noting that the last partial sum results in
−
n−1∑
k=0
Ak∆Bk = Vn(q3a, q3c, q3e)
(1− qc2e2/a3)(1− q3c2e2/a2)(1− q5c2e2/a2)
(1− q3c3e3/a4)(1− qce/a)(1− q3ce/a)(1− q5ce)
,
we derive after some simplification the recurrence relation
Vn(a, c, e) = Vn(q3a, q3c, q3e)
(1− qc)(1− qe)(1− qc2e2/a3)(qc2e2/a2; q2)3
(1− q5ce)(1− q2c2e3/a3)(1− q2c3e2/a3)(qce/a; q)3
−
{
1− R(a, c, e)
}a(1− qc/a)(1− qe/a)(1− qc2e2/a3)(1− q3c3e3/a3)
(1− q2ce/a)(1− q2c3e2/a3)(1− q2c2e3/a3)
.
Iterating it m-times, we get the following expression
Vn(a, c, e) = Vn(q3ma, q3mc, q3me)
[qc, qe, qc2e2/a3; q3]m(qc2e2/a2; q2)3m
[q5ce, q2c2e3/a3, q2c3e2/a3; q6]m(qce/a; q)3m
Partial Sums of Two Quartic q-Series 13
− a (1− qc/a)(1− qe/a)(1− qc2e2/a3)
(1− q2ce/a)(1− q2c2e3/a3)(1− q2c3e2/a3)
×
m−1∑
k=0
(1− q3+9kc3e3/a3)
[
qc, qe, q4c2e2/a3
qce/a, q3ce/a, q5ce/a
∣∣∣q3
]
k
q3k
×
{
1− R(q3ka, q3kc, q3ke)
} [
qc2e2/a2, q3c2e2/a2, q5c2e2/a2
q5ce, q8c2e3/a3, q8c3e2/a3
∣∣∣q6
]
k
.
Writing explicitly the R-function by separating k and n factorials
R(q3ka, q3kc, q3ke) =
1− q3+3n+9kc3e3/a3
1− q3+9kc3e3/a3
[
q1+6kc2e2/a2
a/qc, a/qe
∣∣∣q2
]
n
× (a2/ce; q2)2n
[q1+3kce/a, q3+3kce/a; q]n
[
q1+3kc, q1+3ke
q−1−3ka3/c2e2
∣∣∣q3
]
n
(q−2−3ka/ce; q−1)n
(q5+6kce; q6)n
=
(a2/ce; q2)2n
[qce/a, q3ce/a; q]n
[
qc2e2/a2
a/qc, a/qe
∣∣∣q2
]
n
[
qc, qe
a3/qc2e2
∣∣∣q3
]
n
(a/q2ce; q−1)n
(q5ce; q6)n
× 1− q3+3n+9kc3e3/a3
1− q3+9kc3e3/a3
(q1+2nc2e2/a2; q2)3k(q5ce; q6)k
(qc2e2/a2; q2)3k(q5+6nce; q6)k
×
[
q1+3nc, q1+3ne, qce/a, q3ce/a, q5ce/a, q4−3nc2e2/a3
qc, qe, q1+nce/a, q3+nce/a, q5+nce/a, q4c2e2/a3
∣∣∣q3
]
k
,
and then defining the finite quadratic sum in base q3 by
V �
m(a, c, e) =
m−1∑
k=0
(1− q3+9kc3e3/a3)
[
qc, qe, q4c2e2/a3
qce/a, q3ce/a, q5ce/a
∣∣∣q3
]
k
×
[
qc2e2/a2, q3c2e2/a2, q5c2e2/a2
q5ce, q8c2e3/a3, q8c3e2/a3
∣∣∣q6
]
k
q3k,
we obtain the following transformation formula.
Theorem 11 (Transformation between quartic and quadratic series).
Vn(a, c, e)− Vn(q3ma, q3mc, q3me)
[qc, qe, qc2e2/a3; q3]m(qc2e2/a2; q2)3m
[q5ce, q2c2e3/a3, q2c3e2/a3; q6]m(qce/a; q)3m
=
(1− a/qc)(1− a/qe)(1− qc2e2/a3)
(1−a/q2ce)(1−q2c3e2/a3)(1−q2c2e3/a3)
{
V �
m(a, c, e)− V �
m(q5na, q3nc, q3ne)
× (a2/ce; q2)2n
[qce/a, q3ce/a; q]n
[
qc2e2/a2
a/qc, a/qe
∣∣∣q2
]
n
[
qc, qe
a3/qc2e2
∣∣∣q3
]
n
(a/q2ce; q−1)n
(q5ce; q6)n
}
.
Letting n → 1 + m, c → a/q and e → q−1−3m in this theorem, we derive the summation
formula, which does not seem to have explicitly appeared previously.
Corollary 12 (Terminating series identity).
m∑
k=0
1− q5ka
1− a
[
q−3−6m
q2, q2+3ma
∣∣∣q2
]
k
[
a, q−3m
q6+6ma
∣∣∣q3
]
k
(q2+3ma; q2)2k
(q−1−3m; q)k
(−1)kq(2+3m)k−(k
2)
(q3−3ma; q6)k
=
[
q6+3ma, q−3m/a
q2, q4
∣∣∣q3
]
m
[
q5, q7
q9+3ma, q3−3m/a
∣∣∣q6
]
m
.
14 W. Chu and C. Wang
3.2 Quartic series to cubic series
Define two sequences by
Ak =
(a2/qc2e; q)k(q3c2e2/a2; q2)k(q4c; q3)k(q4a2/ce; q4)k
(q6c2e/a; q4)k(q2a3/c2e2; q3)k(qa/c; q2)k(qce/a; q)k
,
Bk =
[q2a2/ce, q6c2e/a; q4]k(qe; q3)k(a/qce; q−1)k
[q2ce/a, a2/q2c2e; q]k(q3a/e; q2)k(q5ce; q6)k
.
It is not hard to check the relations
$ := A−1B0 =
(1− ce/a)(1− a/qc)(1− a3/qc2e2)(1− q2c2e/a)
(1− a2/q2c2e)(1− qc2e2/a2)(1− qc)(1− a2/ce)
,
R :=
An−1Bn
A−1B0
=
1− q2+4nc2e/a
1− q2c2e/a
[
qc2e2/a2
a/qc, q3a/e
∣∣∣q2
]
n
× (a2/ce; q2)2n
[ce/a, q2ce/a; q]n
[
qc, qe
a3/qc2e2
∣∣∣q3
]
n
(a/qce; q−1)n
(q5ce; q6)n
;
and compute the finite differences
5Ak =
(1− q5ka)(1− q1+kce/a)(1− q1+2ka/e)(1− q2c3e2/a3)
(1− q2c2e/a2)(1− qc2e2/a2)(1− qc)(1− a2/ce)
×
[
a2/q2c2e
qce/a
∣∣∣q]
k
[
qc2e2/a2
qa/c
∣∣∣q2
]
k
[
qc
q2a3/c2e2
∣∣∣q3
]
k
[
a2/ce
q6c2e/a
∣∣∣q4
]
k
qk,
∆Bk = −(1− q4+5ka)(1− q1−2kc/a)(1− q3+2kc2e2/a2)(1− q4+3kc)
(1− q2ce/a)(1− q2c2e/a2)(1− q3a/e)(1− q5ce)
× [q2a2/ce, q6c2e/a; q4]k(qe; q3)k(a/qce; q−1)k
[q3ce/a, a2/qc2e; q]k(q5a/e; q2)k(q11ce; q6)k
qk.
Then by means of the modified Abel lemma on summation by parts, the V -sum can be refor-
mulated as follows:
Vn(a, c, e)
(1− qce/a)(1− qa/e)(1− q2c3e2/a3)
(1− q2c2e/a2)(1− qc2e2/a2)(1− qc)(1− a2/ce)
=
n−1∑
k=0
Bk 5Ak = $
{
R− 1
}
−
n−1∑
k=0
Ak∆Bk.
By expressing the last partial sum in terms of V -sum as
−
n−1∑
k=0
Ak∆Bk =
Vn(q4a, q6c, e) (1− qc/a)(1− q3c2e2/a2)(1− q4c)
(1− q2ce/a)(1− q2c2e/a2)(1− q3a/e)(1− q5ce)
,
we derive after some simplification the following relation
Vn(a, c, e) = Vn(q4a, q6c, e)
(qc; q3)2(qc2e2/a2; q2)2(1− qc/a)(1− a2/ce)
(qce/a; q)2(qa/e; q2)2(1− q5ce)(1− q2c3e2/a3)
+
{
1− R(a, c, e)
}(1− ce/a)(1− qc/a)(1− a3/qc2e2)(1− q2c2e/a)
(1− a/qce)(1− qa/e)(1− q2c3e2/a3)
.
Iterating it m-times, we get the following expression
Vn(a, c, e) = Vn(q4ma, q6mc, e)
(qc; q3)2m(qc2e2/a2; q2)2m[qc/a, a2/ce; q2]m
(qce/a; q)2m(qa/e; q2)2m[q5ce, q2c3e2/a3; q6]m
Partial Sums of Two Quartic q-Series 15
+
(1− ce/a)(1− qc/a)(1− a3/qc2e2)
(1− a/qce)(1− qa/e)(1− q2c3e2/a3)
m−1∑
k=0
(1−q2+8kc2e/a)(qc; q3)2k
[q5ce, q8c3e2/a3; q6]k
×
{
1− R(q4ka, q6kc, e)
} [
q3c/a, a2/ce
ce/a, q3ce/a
∣∣∣q2
]
k
(qc2e2/a2; q2)2k
(q3a/e; q2)2k
q2k.
Rewriting the R-function explicitly as
R(q4ka, q6kc, e) =
1− q2+4n+8kc2e/a
1− q2+8kc2e/a
[
q1+4kc2e2/a2
q−1−2ka/c, q3+4ka/e
∣∣∣q2
]
n
× (q2ka2/ce; q2)2n
[q2kce/a, q2+2kce/a; q]n
[
q1+6kc, qe
a3/qc2e2
∣∣∣q3
]
n
(q−1−2ka/ce; q−1)n
(q5+6kce; q6)n
=
(a2/ce; q2)2n
[ce/a, q2ce/a; q]n
[
qc2e2/a2
a/qc, q3a/e
∣∣∣q2
]
n
[
qc, qe
a3/qc2e2
∣∣∣q3
]
n
(a/qce; q−1)n
(q5ce; q6)n
× 1− q2+4n+8kc2e/a
1− q2+8kc2e/a
[
ce/a, q3ce/a, q4na2/ce, q3−2nc/a
qnce/a, qn+3ce/a, a2/ce, q3c/a
∣∣∣q2
]
k
× (q1+3nc; q3)2k(q5ce; q6)k
(qc; q3)2k(q5+6nce; q6)k
[
q1+2nc2e2/a2, q3a/e
qc2e2/a2, q3+2na/e
∣∣∣q2
]
2k
,
and defining further the finite cubic sum in base q2 by
V 4
m (a, c, e) =
m−1∑
k=0
(1− q2+8kc2e/a)
[
q3c/a, a2/ce
ce/a, q3ce/a
∣∣∣q2
]
k
(qc; q3)2k(qc2e2/a2; q2)2k
(q3a/e; q2)2k[q5ce, q8c3e2/a3; q6]k
q2k,
we establish the following transformation formula.
Theorem 13 (Transformation between quartic and cubic series).
Vn(a, c, e)− Vn(q4ma, q6mc, e)
(qc; q3)2m(qc2e2/a2; q2)2m[qc/a, a2/ce; q2]m
(qce/a; q)2m(qa/e; q2)2m[q5ce, q2c3e2/a3; q6]m
=
(1− ce/a)(1− qc/a)(1− a3/qc2e2)
(1− a/qce)(1− qa/e)(1− q2c3e2/a3)
{
V 4
m (a, c, e)− V 4
m (q5na, q3nc, q3ne)
× (a2/ce; q2)2n
[ce/a, q2ce/a; q]n
[
qc2e2/a2
a/qc, q3a/e
∣∣∣q2
]
n
[
qc, qe
a3/qc2e2
∣∣∣q3
]
n
(a/qce; q−1)n
(q5ce; q6)n
}
.
When n → 1 + δ + 2m, e → q−1/2a3/2/c and c = q−1−3δ−6m with δ = 0, 1, the last theorem
recovers the following summation formula.
Corollary 14 (Chu [4, Equation (4.8d)]).
n∑
k=0
1− q5ka
1− a
[
a
q2+3na, q
1
2
−3n/a
1
2
∣∣∣q2
]
k
(q
1
2 a
1
2 ; q2)2k(−q
1
2 a−
1
2 )k
(q
1
2 a
1
2 ; q)k(q
9
2 a
3
2 ; q6)k
[
q
3
2
+3na
3
2 , q−3n
q3
∣∣∣q3
]
k
q−(k
2)
=
0, n− odd;[
q2a
q3/2a3/2
∣∣∣q2
]
3m
[
q3
q9/2a3/2
∣∣∣q6
]
m
, n = 2m.
Instead, letting n → 1 + m, e → q−1/2a3/2/c and a = q−1−4m in Theorem 13, we recover
another terminating series identity.
16 W. Chu and C. Wang
Corollary 15 (Chu and Wang [8, Corollary 40]).
m∑
k=0
1− q5k−1−4m
1− q−1−4m
[
q−1−4m
q−4m/c, q2+2mc
∣∣∣q2
]
k
(q−2m; q2)2k(−q1+2m)k
(q−2m; q)k(q3−6m; q6)k
[
qc, q−1−6m
q3
∣∣∣q3
]
k
q−(k
2)
=
[
q4c
q3
∣∣∣q6
]
m
[
q
q2c
∣∣∣q2
]
2m
[
q2c
q
∣∣∣q2
]
m
.
3.3 Quartic series to quartic series
Finally, for the two sequences given by
Ak =
[q4a2/ce, q2ce2/a; q4]k(q4c; q3)k(a/ce; q−1)k
[qce/a, q3a2/ce2; q]k(qa/c; q2)k(q5ce; q6)k
,
Bk =
(q3a2/ce2; q)k(c2e2/qa2; q2)k(e/q2; q3)k(q2a2/ce; q4)k
(ce2/q2a; q4)k(q2a3/c2e2; q3)k(q3a/e; q2)k(ce/qa; q)k
;
we have no difficulty to check the relations
$ := A−1B0 =
(1− ce/a)(1− q2a2/ce2)(1− a/qc)(1− ce/q)
(1− a2/ce)(1− ce2/q2a)(1− qc)(1− qa/ce)
,
R :=
An−1Bn
A−1B0
=
1− q2+na2/ce2
1− q2a2/ce2
[
c2e2/qa2
a/qc, q3a/e
∣∣∣q2
]
n
× (a2/ce; q2)2n
[ce/qa, ce/a; q]n
[
qc, e/q2
q2a3/c2e2
∣∣∣q3
]
n
(qa/ce; q−1)n
(ce/q; q6)n
;
and compute the finite differences
5Ak =
(1− q5ka)(1− q3k−2e)(1− q1+2ka/e)(1− q2k−1c2e2/a2)
(1− a2/ce)(1− ce2/q2a)(1− qc)(1− ce/qa)
× [a2/ce, ce2/q2a; q4]k(qc; q3)k(qa/ce; q−1)k
[qce/a, q3a2/ce2; q]k(qa/c; q2)k(q5ce; q6)k
q−k,
∆Bk = −(1− q1+5ka)(1− qkce/a)(1− q2k−1a/c)(1− q4a3/c2e3)
(1− qa/ce)(1− q3a/e)(1− q2a3/c2e2)(1− ce2/q2a)
×
[
q3a2/ce2
ce/a
∣∣∣q]
k
[
c2e2/qa2
q5a/e
∣∣∣q2
]
k
[
e/q2
q5a3/c2e2
∣∣∣q3
]
k
[
q2a2/ce
q2ce2/a
∣∣∣q4
]
k
qk.
Then according to the modified Abel lemma on summation by parts, the V -sum can be refor-
mulated as follows:
Vn(a, c, e)
(1− e/q2)(1− qa/e)(1− c2e2/qa2)
(1− a2/ce)(1− ce2/q2a)(1− qc)(1− ce/qa)
=
n−1∑
k=0
Bk 5Ak = $
{
R− 1
}
−
n−1∑
k=0
Ak∆Bk.
Observing that the last partial sum results in
−
n−1∑
k=0
Ak∆Bk =
Vn(qa, q3c, e/q3) (1− ce/a)(1− a/qc)(1− q4a3/c2e3)
(1− qa/ce)(1− q3a/e)(1− q2a3/c2e2)(1− ce2/q2a)
,
we derive after some simplification the following relation
Vn(a, c, e) = a
(1− a/ce)(1− q2a2/ce2)(1− qc/a)(1− q/ce)
(1− qa2/c2e2)(1− q2/e)(1− qa/e)
{
R(a, c, e)− 1
}
Partial Sums of Two Quartic q-Series 17
− Vn(qa, q3c, e/q3)
(1− qc)(1− qc/a)(1− a/ce)(1− a2/ce)(1− q4a3/c2e3)
(ce/qa)(1− q2a3/c2e2)(1− qa2/c2e2)(1− q2/e)(qa/e; q2)2
.
Iterating it m-times, we get the following expression
Vn(a, c, e) = Vn(qma, q3mc, q−3me)
[
qc/a, a2/ce
qa2/c2e2
∣∣∣q2
]
m
× [a/ce, qa/ce; q]m
(qa/e; q2)2m
[
qc
q2a3/c2e2, q2/e
∣∣∣q3
]
n
(q4a3/c2e3; q6)m
(ce/qa; q−1)m
− a
(1− a/ce)(1− qc/a)(1− q/ce)
(1− qa2/c2e2)(1− q2/e)(1− qa/e)
×
m−1∑
k=0
(1− q2+5ka2/ce2)
[
q3c/a, a2/ce
q3a2/c2e2
∣∣∣q2
]
k
(qa/ce; q)k
(q3a/e; q2)2k
q(
k
2)
×
{
1− R(qka, q3kc, q−3ke)
}(qc; q3)k(q4a3/c2e3; q6)k
[q2a3/c2e2, q5/e; q3]k
(
− q2a
ce
)k
.
Writing explicitly the R-function as
R(qka, q3kc, q−3ke) =
1− q2+n+5ka2/ce2
1− q2+5ka2/ce2
[
q−1−2kc2e2/a2
q−1−2ka/c, q3+4ka/e
∣∣∣q2
]
n
× (q2ka2/ce; q2)2n
[q−1−kce/a, q−kce/a; q]n
[
q1+3kc, q−2−3ke
q2+3ka3/c2e2
∣∣∣q3
]
n
(q1+ka/ce; q−1)n
(ce/q; q6)n
=
(a2/ce; q2)2n
[ce/qa, ce/a; q]n
[
c2e2/qa2
a/qc, q3a/e
∣∣∣q2
]
n
[
qc, e/q2
q2a3/c2e2
∣∣∣q3
]
n
(qa/ce; q−1)n
(ce/q; q6)n
× 1− q2+n+5ka2/ce2
1− q5k+2a2/ce2
[
q3a2/c2e2, q4na2/ce, q3−2nc/a
q3−2na2/c2e2, a2/ce, q3c/a
∣∣∣q2
]
k
q−nk
× (q1−na/ce; q)k(q3a/e; q2)2k
(qa/ce; q)k(q3+2na/e; q2)2k
[
q1+3nc, q5/e, q2a3/c2e2
qc, q5−3n/e, q2+3na3/c2e2
∣∣∣q3
]
k
,
and then defining the finite sum of quartic series
V ?
m(a, c, e) =
m−1∑
k=0
(1− q2+5ka2/ce2)
[
q3c/a, a2/ce
q3a2/c2e2
∣∣∣q2
]
k
(qa/ce; q)k
(q3a/e; q2)2k
q(
k
2)
×
(
− q2a
ce
)k
[
qc
q2a3/c2e2, q5/e
∣∣∣q3
]
k
(q4a3/c2e3; q6)k,
we find the following transformation formula.
Theorem 16 (Transformation between two quartic series).
Vn(a, c, e)− Vn(qma, q3mc, q−3me)
[
qc/a, a2/ce
qa2/c2e2
∣∣∣q2
]
m
× [a/ce, qa/ce; q]m
(qa/e; q2)2m
[
qc
q2a3/c2e2, q2/e
∣∣∣q3
]
n
(q4a3/c2e3; q6)m
(ce/qa; q−1)m
=
(1− a/qc)(1− a/ce)(1− ce/q)
(1− qa2/c2e2)(1− e/q2)(1− qa/e)
{
V ?
m(a, c, e)− V ?
m(q5na, q3nc, q3ne)
× (a2/ce; q2)2n
[ce/qa, ce/a; q]n
[
c2e2/qa2
a/qc, q3a/e
∣∣∣q2
]
n
[
qc, e/q2
q2a3/c2e2
∣∣∣q3
]
n
(qa/ce; q−1)n
(ce/q; q6)n
}
.
18 W. Chu and C. Wang
Letting n → 1+m, c = q−1−3m and e → q2+3m, we obtain from the last theorem the following
terminating series identity.
Corollary 17 (Terminating series identity).
m∑
k=0
1− q5ka
1− a
[
q3/a2
q2+3ma, q−1−3ma
∣∣∣q2
]
k
[
q3+3m, q−3m
a3
∣∣∣q3
]
k
(a2/q; q2)2k(−a)k
(q2/a; q)k(q6; q6)k
q−(1+k
2 )
=
[a/q, qa; q]m(q−3m/a; q2)m(q−3ma3; q6)m
(q−1−3ma; q2)2m(a3; q3)m(1/a; q−1)m
.
4 Concluding remarks
Recently, hypergeometric series has been found to have elliptic analogue after the pioneering work
of Frenkel–Turaev [9]. Warnaar [20] derived several terminating elliptic series identities. Further
summation formulae have been established by Chu–Jia [6] through Abel’s lemma on summation
by parts. It is plausible that the same approach works also for the elliptic analogue of the quartic
series. All what we have gotten are two terminating elliptic analogues for Corollary 6 plus one
for Corollary 17. Following the notations of [6], they are produced below for reader’s reference.
Theorem 18 (Terminating elliptic series identity).
m∑
k=0
θ(q5ka; p)
θ(a; p)
[
a, q2+m
q1−2m
∣∣∣q2, p
]
k
[q−m; q, p]k[q1−ma; q6, p]k
[q2+ma; q2, p]2k
q(
1+k
2 )−mk
× (−1)k
[
q1+2ma
q3, q1−ma
∣∣∣q3, p
]
k
=
χ(m = 2n)[q2a; q2, p]n[q3; q6, p]n
[q1+2n; q2, p]n[q4−2na; q6, p]n
.
Theorem 19 (Terminating elliptic series identity).
m∑
k=0
θ(q5ka; p)
θ(a; p)
[
a, q−2m
q5+4m
∣∣∣q2, p
]
k
[q2+2m; q, p]k[q3+2ma; q6, p]k
[q−2ma; q2, p]2k
q(
k
2)+(3+2m)k
× (−1)k
[
q−3−4ma
q3, q3+2ma
∣∣∣q3, p
]
k
=
[q2a; q2, p]m[q5; q2, p]2m[q−4ma; q6, p]m
[q3; q2, p]m[q−2ma; q2, p]2m[q9; q6, p]m
.
Theorem 20 (Terminating elliptic series identity).
m∑
k=0
θ(q5ka; p)
θ(a; p)
[
q3/a2
q2+3ma, q−1−3ma
∣∣∣q2, p
]
k
[a2/q; q2, p]2k(−a)k
[q2/a; q, p]k[q6; q6, p]k
q−(1+k
2 )
×
[
q3+3m, q−3m
a3
∣∣∣q3, p
]
k
=
[a/q, qa; q, p]m[q−3m/a; q2, p]m[q−3ma3; q6, p]m
[q−1−3ma; q2, p]2m[a3; q3, p]m[1/a; q−1, p]m
.
Acknowledgments
The work of the second author was partially supported by National Science Foundation of China
(Youth grant 10801026).
Partial Sums of Two Quartic q-Series 19
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18 (2002), 479–502, math.QA/0001006.
http://arxiv.org/abs/math.QA/0001006
1 Introduction and motivation
2 Transformation and summation formulae for U_n(a,b,d)
2.1 Quartic series to quadratic series
2.2 Quartic series to cubic series
2.3 Quartic series to quartic series
3 Transformation and summation formulae for V_n(a,b,d)
3.1 Quartic series to quadratic series
3.2 Quartic series to cubic series
3.3 Quartic series to quartic series
4 Concluding remarks
References
|
| id | nasplib_isofts_kiev_ua-123456789-149151 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:15:35Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chu, W. Wang, C. 2019-02-19T17:42:18Z 2019-02-19T17:42:18Z 2009 Partial Sums of Two Quartic q-Series / W. Chu, C. Wang // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 20 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 33D15; 05A15 https://nasplib.isofts.kiev.ua/handle/123456789/149151 The partial sums of two quartic basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several summation and transformation formulae are consequently established. The work of the second author was partially supported by National Science Foundation of China (Youth grant 10801026). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Partial Sums of Two Quartic q-Series Article published earlier |
| spellingShingle | Partial Sums of Two Quartic q-Series Chu, W. Wang, C. |
| title | Partial Sums of Two Quartic q-Series |
| title_full | Partial Sums of Two Quartic q-Series |
| title_fullStr | Partial Sums of Two Quartic q-Series |
| title_full_unstemmed | Partial Sums of Two Quartic q-Series |
| title_short | Partial Sums of Two Quartic q-Series |
| title_sort | partial sums of two quartic q-series |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/149151 |
| work_keys_str_mv | AT chuw partialsumsoftwoquarticqseries AT wangc partialsumsoftwoquarticqseries |