Volume 2009, Article ID 370390,9pages doi:10.1155/2009/370390
Research Article
A Note on Four-Variable Reciprocity Theorem
Chandrashekar Adiga and P. S. Guruprasad
Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore 570 006, India
Correspondence should be addressed to Chandrashekar Adiga,adiga [email protected]
Received 12 September 2009; Accepted 7 December 2009
Recommended by Teodor Bulboac˘a
We give new proof of a four-variable reciprocity theorem using Heine’s transformation, Watson’s transformation, and Ramanujan’s 1ψ1-summation formula. We also obtain a generalization of Jacobi’s triple product identity.
Copyrightq2009 C. Adiga and P. S. Guruprasad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Throughout the paper, we let|q|<1 and we employ the standard notation:
a0 :
a;q01,
a∞:a;q∞
∞
n0
1−aqn,
an :
a;qn
a;q∞
aqn;q
∞
, −∞< n <∞.
1.1
Ramanujan1stated severalq-series identities in his “lost” notebook. One of the beautiful identities is the two-variable reciprocity theorem.
Theorem 1.1see2. Forab /0,
ρa, b−ρb, a
1
b−
1
a
aq/b
∞
bq/a∞q∞
where
ρa, b:
1 1
b ∞
n0
−1nqnn1/2anb−n
−aqn . 1.3
In the recent past many new proofs of 1.2 have been found. The first proof of
1.2was given by Andrews3using four-free-variable identity and Jacobi’s triple product identity. Further, Andrews 4 applied 1.2 in proving Euler partition identity analogues stated in1. Somashekara and Fathima5established an equivalent version of1.2using Ramanujan’s1ψ1summation formula6and Heine’s transformation7,8. Berndt et al.9 also derived1.2using the same above mentioned two transformations. In fact, Berndt et al.
9in the same paper have given two more proofs of 1.2one employing the Rogers-Fine identity10and the other is purely combinatorial. Using theq-binomial theorem:
∞
n0
an
qnt
n at∞
t∞ , |t|<1, q<1, 1.4
Kim et al. 11 gave a much different proof of 1.2. Guruprasad and Pradeep 12 also devised a proof of 1.2using theq-binomial theorem. Adiga and Anitha13 established
1.2along the lines of Ismail’s proof14of Ramanujan’s1ψ1summation formula. Further, they showed that the reciprocity theorem1.2leads to aq-integral extension of the classical gamma function. Kang2constructed a proof of1.2along the lines of Venkatachaliengar’s proof of the Ramanujan1ψ1summation formula6,15.
In2Kang proved the following three- and four-variable generalizations of1.2. For|c|<|a|<1 and|c|<|b|<1,
ρ3a, b, c−ρ3b, a, c
1
b−
1
a
c∞aq/b∞bq/a∞q∞
−c/a∞−c/b∞−aq∞−bq∞, 1.5
where
ρ3a, b, c:
11
b ∞
n0
cn−1nqnn1/2anb−n −aqn−c/bn1
, a,c b/ −q
−n, 1.6
and for|c|,|d|<|a|,|b|<1,
ρ4a, b, c, d−ρ4b, a, c, d
1
b−
1
a
d∞c∞cd/ab∞aq/b∞bq/a∞q∞ −d/a∞−d/b∞−c/a∞−c/b∞−aq∞−bq∞,
where
ρ4a, b, c, d
:
11
b ∞
n0
dncncd/abn1cdq2n/b−1nqnn1/2anb−n
−aqn−c/bn1−d/bn1 , a, c b,
d b/ −q
−n.
1.8
Kang2established1.5on employing Ramanujan’s1ψ1summation formula and Jackson’s transformation of2φ1and2φ2-series. Recently1.5was derived by Adiga and Guruprasad
16usingq-binomial theorem and Gauss summation formula. Somashekara and Mamta17,
18obtained1.5using the two-variable reciprocity theorem1.2, Jackson’s transformation, and again two-variable reciprocity theorem by parameter augmentation. Zhang 19 also established1.5.
Kang2established1.7on employing Andrews’s generalization of1ψ1summation formula, Sears’s transformation of3φ2-series, and a limiting case of Watson’s transformation for a terminating very well-poised8φ7-series8:
∞
n0
αn
βnγnδn n
1−αq2nqnn3/2
αq/βnαq/γnαq/δnαq/ nqn1−α −α2
βγδ n
αq∞αq/δ ∞
αq/δ∞αq/ ∞
∞
n0
δn n
αq/βγn
αq/βnαq/γnqn
αq δ
n .
1.9
Recently Ma20,21proved a six-variable generalization and a five-variable generalization of1.2. The main purpose of this paper is to provide a new proof of1.7using1.9, Heine’s transformation:
∞
n0
αn
βn
qnγnz n
γ/β∞βz∞
γ∞z∞
∞
n0
αβz/γnβn
βznqn
γ β
n
, q<1, |z|<1, γ<β<1
1.10
and Ramanujan’s1ψ1summation formula:
1ψ1a;b;z:
∞
n−∞
an bn
zn b/a∞az∞
q/az∞q∞
q/a∞b/az∞b∞z∞ , q<1,
ba<|z|<1. 1.11
Jacobi’s triple product identity states that
∞
n−∞q
nn1/2znq
∞
−zq∞
−1
z
∞, z /0,
Andrews22gave a proof of1.12using Euler identities. Combinatorial proofs of Jacobi’s triple product identity were given by Wright23, Cheema24, and Sudler25. We can also find a proof of1.12in26. Using1.12, Hirschhorn27,28established Jacobi’s two-square and four-square theorems.
Somashekara and Fathima5and Kim et al.11established
∞
n0
−1nanb−nqnn1/2
−an1 −
∞
n0
−1na−n1bn1qnn1/2
−bn1
aq/b∞b/a∞q∞
−a∞−b∞ . 1.13
Note that 1.13 which is equivalent to 1.2 may be considered as a two-variable generalization of1.12. Corteel and Lovejoy29, equation1.5have given a bijective proof of1.13using representations of over partitions. All the reciprocity theorems1.2,1.5, and
1.7are generalizations of Jacobi’s triple product identity1.12.
We also obtain a generalization of Jacobi’s triple product identity1.12which is due to Kang2.
2. Proof of
1.7
—The Four-Variable Reciprocity Theorem
On employingq-binomial theorem, we have
∞
n0
−cqn−dqn
−aqn−bqnqn
−dq∞
−bq∞
∞
n0
−cqn−bqn1
∞
−aqn−dqn1
∞
qn
−dq∞
−bq∞
∞
m0
b/d m qm
−dqm∞ n0
−cqn
−aqn
qm1n.
2.1
On using Heine’s transformation1.10withα−cq,βq,γ−aq,zqm1, we have
∞
n0
−cqn
−aqn
qm1n
qm2
∞−a∞
qm1
∞
−aq∞
∞
n0
cqm2/a
n
qm2
n
−an
qm1a−a−m−1
cq/am1
∞
n0
cq/anm1
qnm1 −a nm1
qm1a−a−m−1
cq/am1
∞
n0
cq/an
qn −a n−m
n0
cq/an
qn −a n
qm−a−m−1−cq∞
cq/am1−aq∞ −
qm1a−a−m−1
cq/am1 m
n0
cq/an
qn −a n
Substituting this in2.1, we obtain
∞
n0
−cqn−dqn
−aqn−bqnqn
−dq∞−cq∞ −a−bq∞−aq∞
∞
m0
b/dm
cq/am1 dq a m
1a−1−dq
∞
−bq∞
∞
m0
m
n0
b/dm
dq/amcq/an−an
cq/am1qn .
2.3
Now,
1a−1−dq
∞
−bq∞
∞
m0
m
n0
b/dm
dq/amcq/an−an
cq/am1qn
1a−1−dq
∞
−bq∞
∞
n0
b/dn
−dqn
qn1−cqn1/a
∞
m0
bqn/d
m
dq/am
cqn2/a
m
1a−1−dq∞
−bq∞
∞
n0
b/dn
−dqn
qn1−cqn1/a·
1−cqn1/a
1−dq/a
∞
m0
b/cm
cqn1/am
dq2/a
m ,
on using1.10withα bq n
d , βq, γ cqn2
a , z dq
a
1a−1−dq
∞
−bq∞
∞
m0
b/cm
cq/am
dq/am1
∞
n0
b/dn
−dqm1n
qn
1a−1−dq
∞
−bq∞
∞
m0
b/cm
cq/am
dq/am1 ·
−bqm1
∞
−dqm1
∞
1a−1
∞
m0
b/cm
−dqm
−bqmdq/am1 cq
a m
.
2.4
Substituting2.4in2.3, we obtain
∞
n0
−cqn−dqn
−aqn−bqnqn
−dq∞−cq∞ −a−bq∞−aq∞
∞
m0
b/dm
cq/am1
dq a
m
1a−1
∞
m0
b/cm
−dqm
−bqmdq/am1 cq
a m
−dq∞−cq∞ −a−bq∞−aq∞
∞
m0
b/cm
dq/am1 cq
a m
1a−1
∞
m0
b/cm
−dqm
−bqmdq/am1 cq
a m
.
2.5
Changingcto−c/q,dto−d/qin2.5, we get
∞
n0
cndn
−aqn−bqnq
n d∞c∞
−a−bq∞−aq∞
∞
m0
−bq/cm −d/am1
−c
a m
1a−1
∞
m0
−bq/cmdm
−bqm−d/am1
−c a m . 2.6
Interchangingaandbin2.6, we have
∞
n0
cndn
−aqn−bqnqn
d∞c∞ −b−bq∞−aq∞
∞
m0
−aq/cm −d/bm1
−c
b m
1b−1∞
m0
−aq/cmdm
−aqm−d/bm1
−c b m . 2.7
Subtracting2.6from2.7, we deduce that
d∞c∞
−bq∞−aq∞
1
b
∞
m0
−aq/cm −d/bm1 −c b m − 1 a ∞
m0
−bq/cm −d/am1
−c
a m
1b−1
∞
m0
−aq/cmdm
−aqm−d/bm1
−c
b m
−1a−1
∞
m0
−bq/cmdm
−bqm−d/am1
−c a m . 2.8
Now changeato−b/d,bto−c/a, andzto−d/ain1.11to obtain
∞
n1
−b/dn −c/an
−d
a n
∞ n0
−aq/cn
−dq/bn
−c b
n
cd/ab∞b/a∞
aq/b∞q∞
−c/a∞−c/b∞−d/a∞−dq/b∞. 2.9
Changingnton1 in the first summation of the above identity and then multiplying both sides by1d/b−1, we find that
1
1d/b
∞
n0
−b/dn1 −c/an1
−d
a n1
∞ n0
−aq/cn −d/bn1 −c b n
1−b
a
cd/ab
∞bq/a∞
aq/b∞q∞ −c/a∞−c/b∞−d/a∞−d/b∞.
Using1.10withα −bq/c,β q,γ −dq/a, andz −c/ain the first summation of the above identity and then multiplying both sides by 1/b, we get
1
b
∞
n0
−aq/cn −d/bn1
−c
b n
− 1 a
∞
n0
−bq/cn −d/an1
−c
a n
1
b−
1
a
cd/ab
∞
bq/a∞aq/b∞q∞ −c/a∞−c/b∞−d/a∞−d/b∞.
2.11
Substituting2.11in2.8, we see that
1
b−
1
a
cd/ab
∞c∞d∞bq/a∞
aq/b∞q∞
−aq∞−bq∞−c/a∞−c/b∞−d/a∞−d/b∞
1b−1
∞
m0
−aq/cmdm
−aqm−d/bm1
−c b
m
−1a−1
∞
m0
−bq/cmdm
−bqm−d/am1
−c a
m .
2.12
Now settingα−cd/b,βcd/ab,γc,δq, and din1.9and then multiplying both sides by 1/1d/b1c/b, we obtain
∞
n0
cd/abncndn
1cdq2n/bqnn1/2−1nanb−n
−aqn−c/bn1−d/bn1
∞ n0
−aq/cndn
−aqn−d/bn1
−c
b n
.
2.13
Interchangingaandbin2.13, we have
∞
n0
cd/abncndn
1cdq2n/aqnn1/2−1nbna−n
−bqn−c/an1−d/an1
∞ n0
−bq/cndn
−bqn−d/an1
−c
a n
. 2.14
Substituting2.13and2.14in2.12, we deduce1.7.
Theorem 2.1A four-variable generalization of Jacobi’s triple product identity. For|c|,|d|< |a|,|b|<1,
cd/ab∞c∞d∞b/a∞aq/b∞q∞ −a∞−b∞−c/a∞−c/b∞−d/a∞−d/b∞
∞ m0
dm
−cq−m/a m−1
m
amb−mqmm1/2 −am1−d/bm1
−∞ m0
dm
−cq−m/b m−1
m
a−m1bm1qmm1/2
−bm1−d/am1
.
Proof. Employing
−aq
c
m
a c
m
qmm1/2
−cq−m
a
m ,
−bq
c
m
b c
m
qmm1/2
−cq−m b
m
2.16
in the right side of 2.12 and then multiplying both sides by b/1a1b, we obtain
2.15.
Acknowledgment
The authors thank the anonymous referee for several helpful comments.
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