AN IDENTITY RELATING A THETA FUNCTION TO A SUM OF LAMBERT SERIES GEORGE E. ANDREWS, RICHARD LEWIS and ZHI-GUO LIU Abstract We derive an identity connecting a theta function and a sum of Lambert series. As a consequence of this identity, we deduce a number of results of Jacobi, Dirichlet, Lorenz, Ramanujan and Rademacher.
1. Introduction Suppose, throughout, that q is a complex number of modulus < 1. We shall use the familiar notation ∞ Y (1 − zq n ), (z; q)∞ := n=0
and (for z 6= 0) we also set [z; q]∞ := (z; q)∞ (z −1 q; q)∞ and [a, b, . . . , y, z; q]∞ := [a; q]∞ [b; q]∞ · · · [y; q]∞ [z; q]∞ . It is easy to see that [z −1 ; q]∞ = [zq; q]∞ = −z −1 [z; q]∞ . Our main purpose is to prove the following. Theorem 1. Suppose that a, b, c 6= q n ( for any n ∈ Z) are non-zero complex numbers with abc 6= q n . Then ∞
∞
X aq n X q n /a [bc, ca, ab; q]∞ (q; q)2∞ =1+ − [a, b, c, abc; q]∞ 1 − aq n 1 − q n /a n=0
+
∞ X n=0
+
∞ X n=0
−
∞ X n=0
n=1
n
bq − 1 − bq n
∞ X
n
∞ X
cq − 1 − cq n
n=1
n=1
q n /b 1 − q n /b q n /c 1 − q n /c
∞
X q n /abc abcq n . + 1 − abcq n 1 − q n /abc n=1
Received 1 October 1999; revised 4 February 2000. 2000 Mathematics Subject Classification 11F11. Bull. London Math. Soc. 33 (2001) 25–31
(1)
26
george e. andrews, richard lewis and zhi-guo liu
We prove Theorem 1 in Section 2. In Section 3, we use this theorem to give proofs of the following two theorems. Theorem 2. ∞ X
qm
2
+n2
∞ X
=1+4
m,n=−∞ ∞ X
q
∞
X q 4n+3 q 4n+1 − 4 , 1 − q 4n+1 1 − q 4n+3
n=0 m2 +2n2
m,n=−∞
! ∞ X q 8n+1 q 8n+3 =1+2 + 1 − q 8n+1 1 − q 8n+3 n=0 n=0 ! ∞ ∞ X X q 8n+5 q 8n+7 + , −2 1 − q 8n+5 1 − q 8n+7 ∞ X
n=0
∞ X
2
q m +3n = 1 + 2
n=0 ∞ X
+4
n=0
qm
2
+7n2
=1+2
∞ X
m,n=−∞
n=0
− −4
q − 1 − q 12n+4
∞ X n=0
q 12n+8 1 − q 12n+8
! ,
q − 1 − q 7n+3
28n+2
n=0 ∞ X
28n+6
n=0
∞ X
∞ X
n=0
n=0
∞
X q 7n+6 q − 1 − q 7n+5 1 − q 7n+6 7n+5
!
n=0
28n+18
q + 1 − q 28n+18
n=0 ∞ X
q − 1 − q 28n+6
(4)
∞
n=0
7n+3
q + 1 − q 28n+2
n=0
!
n=0
12n+4
∞
∞ X
−
∞
X q 3n+2 q 3n+1 − 1 − q 3n+1 1 − q 3n+2
X q 7n+2 X q 7n+4 q 7n+1 + + 1 − q 7n+1 1 − q 7n+2 1 − q 7n+4
n=0 ∞ X
(3)
n=0
∞ X
2
m,n=−∞
∞ X
(2)
n=0
28n+10
∞ X
q 28n+22 1 − q 28n+22
n=0 ∞ X
q − 1 − q 28n+10
n=0
! q 28n+26 . (5) 1 − q 28n+26
In Theorem 2, (2) is due to Jacobi (see [8]), and (3) and (4) are due to Dirichlet and Lorenz, respectively (see [7]). (5) was found by Ramanujan; it is Entry 17(ii) in Chapter 19 of his second Notebook [4, p. 304]. The proof given in [4] uses a modular equation of the seventh order given in Entry 19(i); we believe that our proof of (5) is somewhat more transparent. Theorem 3. ∞ X
!4 q
n2
n=−∞ ∞ X
!6 q
n2
n=−∞ ∞ X n=−∞
=1+8
∞ X n=1
= 1 + 16
q
= 1 + 16
∞ ∞ X X n2 q n (2n − 1)2 q 2n−1 + 4 (−1)n , 2n 1+q 1 − q 2n−1 ∞ X n=1
(6)
n=1
n=1
!8 n2
∞
X nq n nq 2n −8 (−1)n , 2n 1−q 1 − q 2n
(7)
n=1
(−1)n
∞
X n3 q n n3 q 2n + 16 . 2n 1−q 1 − q 2n n=1
(8)
relating a theta function to a sum of lambert series
27
In Theorem 3, (6), (7) and (8) are all due to Jacobi (see [8, Sections 90.2–90.4]). None of (2)–(8) is new. What we believe are new are the uniform proofs of these identities provided by Theorem 1. 2. Proof of Theorem 1 Our original proof of Theorem 1 [6] made use of complex analysis, but then one of us came up with the more elementary proof, using Bailey’s 6 Ψ6 summation [9, Section 7.1.1], that we now give. We shall use the elementary identity (see [2]) b c abc (1 − bc)(1 − ac)(1 − ab) a + + − = . (9) 1+ 1 − a 1 − b 1 − c 1 − abc (1 − a)(1 − b)(1 − c)(1 − abc) We have ∞ ∞ ∞ ∞ X X X X aq n q n /a bq n q n /b + − − 1+ 1 − aq n 1 − q n /a 1 − bq n 1 − q n /b n=0
+
n=1
∞ X n=0
n
cq − 1 − cq n
n=0
∞ X n=1
n
q /c − 1 − q n /c
n=1
∞ X n=0
∞
X q n /abc abcq + n 1 − abcq 1 − q n /abc n
n=1
b c abc a + + − =1+ 1 − a 1 − b 1 − c 1 − abc ∞ X bq n c−1 q n abcq n aq n + − − + 1 − aq n 1 − bq n 1 − c−1 q n 1 − abcq n n=1 ∞ X q n /b cq n q n /a q n /abc − + − − + 1 − q n /a 1 − q n /b 1 − cq n 1 − q n /abc n=1
∞ X
(1 − bc)(1 − ac)(1 − abq 2n )q n by (9) (1 − aq n )(1 − bq n )(1 − c−1 q n )(1 − abcq n ) n=−∞ # " √ √ q ab, −q ab, a, b, 1/c, abc (1 − bc)(1 − ac)(1 − ab) √ × 6 Ψ6 √ ; q, q = (1 − a)(1 − b)(1 − c)(1 − abc) ab, − ab, bq, aq, abcq, q/c
= −c−1
=
[bc, ca, ab; q]∞ (q; q)2∞ [a, b, c, abc; q]∞
by Bailey’s summation.
3. Proofs of Theorems 2 and 3 We now apply Theorem 1 to give uniform proofs of Theorems 2 and 3. We first note that it follows from Jacobi’s Triple Product Identity [1, Theorem 2.8] which, in our notation, states ∞ X z n q n(n−1)/2 , (10) [z; q]∞ (q; q)∞ = that
n=−∞ ∞ X
2
q n = [−q; q 2 ]∞ (q 2 ; q 2 )∞
(11)
n=−∞
(under q 7→ q 2 , z 7→ −q). Then ∞ X follows from (11).
n=−∞
2
(−1)n q n =
(q; q)∞ (−q; q)∞
(12)
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george e. andrews, richard lewis and zhi-guo liu
Proof of Theorem 2. We now establish (2), (3), (4) and (5). For (2), take q 4 for q in (1), and then take a, b, c = q, q, q. The right-hand side of (1) is the right-hand side of (2), and the left-hand side of (1) becomes [q 2 ; q 4 ]2∞ (q 2 ; q 2 )2∞ [q 2 , q 2 , q 2 ; q 4 ]∞ (q 4 ; q 4 )2∞ = [q, q, q, q 3 ; q 4 ]∞ [q; q 4 ]4∞ 2 [q ; q 4 ]2∞ (q 2 ; q 2 )2∞ = [q; q 2 ]2∞
= [−q; q 2 ]2∞ (q 2 ; q 2 )2∞ ,
which, by (11), is equal to
∞ X
2
2
q m +n .
m,n=−∞
To prove (3), we replace q by q 8 in (1), and take for parameters a, b, c = q, q, q 3 . The right-hand side of (1) is the right-hand side of (3), and the left-hand side is [q 2 , q 4 ; q 8 ]∞ (q 4 ; q 4 )2∞ [q 4 , q 4 , q 2 ; q 8 ]∞ (q 8 ; q 8 )2∞ = 3 5 8 [q, q, q , q ; q ]∞ [q; q 2 ]∞ [q 2 , q 4 ; q 8 ]∞ (q 2 ; q 2 )∞ (q 4 ; q 4 )∞ = [q; q 2 ]∞ (q 2 ; q 4 )∞ [q 2 ; q 4 ]∞ [q 4 ; q 8 ]∞ (q 2 ; q 2 )∞ (q 4 ; q 4 )∞ = [q; q 2 ]∞ [q 2 ; q 4 ]∞ = [−q; q 2 ]∞ (q 2 ; q 2 )∞ [−q 2 ; q 4 ]∞ (q 4 ; q 4 )∞ ∞ X 2 2 = q m +2n . m,n=−∞
Now we prove (4). First take q 6 for q in (1), and set a, b, c = −q, −q 2 , −q 3 . We obtain ∞ X
2
(−1)m+n q m +3n
2
m,n=−∞
(q; q)∞ (q 3 ; q 3 )∞ by (12) (−q; q)∞ (−q 3 ; q 3 )∞ 2 [q, q 2 , q 3 ; q 6 ]∞ (q 6 ; q 6 )∞ = [−q, −q 2 , −q 3 , −q 6 ; q 6 ]∞ ∞ ∞ ∞ ∞ X X X X q 6n+1 q 6n+5 q 6n+2 q 6n+4 =2 1− + − + 6n+1 6n+5 6n+2 1+q 1+q 1+q 1 + q 6n+4 n=0 n=0 n=0 n=0 ! ∞ ∞ ∞ ∞ X X X X q 6n+3 q 6n+3 q 6n+6 q 6n − + + − 1 + q 6n+3 1 + q 6n+3 1 + q 6n+6 1 + q 6n =
=1−2
n=0 ∞ X n=0
6n+1
n=0 ∞ X
q − 1 + q 6n+1
n=0
6n+5
n=0 ∞ X
q + 1 + q 6n+5
Replacing q by −q in (13) yields (4).
n=0
6n+2
n=0 ∞ X
q − 1 + q 6n+2
n=0
! q 6n+4 . (13) 1 + q 6n+4
relating a theta function to a sum of lambert series 7
2
29
4
To prove (5), take q for q, and set a, b, c = −q, −q , −q in (1). We obtain ∞ X
∞ X
2
(−1)n q n
n=−∞
(q; q)∞ (q 7 ; q 7 )∞ (−q; q)∞ (−q 7 ; q 7 )∞
2
(−1)n q 7n =
n=−∞
=2
[q, q 2 , q 3 ; q 7 ]∞ (q 7 ; q 7 )2∞ . [−1, −q, −q 2 , −q 3 ; q 7 ]∞
(14)
Replacing q by q 7 , and setting a := −1, b := −q, c := −q 2 in (1), we obtain [q, q 2 , q 3 ; q 7 ]∞ (q 7 ; q 7 )2∞ [−1, −q, −q 2 , −q 3 ; q 7 ]∞ ∞ ∞ ∞ ∞ X X X X q 7n q 7n q 7n+1 q 7n+6 =1− + − + 1 + q 7n 1 + q 7n 1 + q 7n+1 1 + q 7n+6 n=0
−
∞ X n=0
n=1 ∞
n=0
n=0
∞
∞
n=0
n=0
X q 7n+5 X q 7n+3 X q 7n+4 q 7n+2 + + − . 7n+2 7n+5 7n+3 1+q 1+q 1+q 1 + q 7n+4 n=0
(15)
Amalgamating (14) and (15), and replacing q by −q, gives (5). For our proof of Theorem 3, we shall need the following two lemmas. Lemma 4 was given by Bailey [3]. We derive this result from Theorem 1. Lemma 4. ∞
∞
X X bq n b−1 q n b [a/b, ab; q]∞ (q; q)4∞ = + [a, b; q]2∞ (1 − bq n )2 (1 − b−1 q n )2 n=1
−
n=0
∞ X n=1
= Proof.
∞
X aq a−1 q n − n 2 (1 − aq ) (1 − a−1 q n )2 n
n=0
∞ X
∞ X
n
bq aq n − . (1 − bq n )2 n=−∞ (1 − aq n )2 n=−∞
(16)
Divide each side of (1) by 1 − ca, and then let c → a−1 . We obtain (16).
Noting that ∞ X
aq n a = + n 2 (1 − aq ) (1 − a)2 n=−∞ = =
∞ X n=1
aq n q n /a + (1 − aq n )2 (1 − q n /a)2
a + (1 − a)2
∞ ∞ X X
a + (1 − a)2
∞ X m(am + a−m )q m
!
m(am + a−m )q mn
n=1 m=1
m=1
1 − qm
,
we see that the identity (16) can be written as ∞
X n(bn + b−n − an − a−n )q n a b [a/b, ab; q]∞ (q; q)4∞ b − + = . (17) 2 2 n (1 − b) (1 − a) 1−q [a, b; q]2∞ n=1
30
george e. andrews, richard lewis and zhi-guo liu
If we divide both sides of (17) by b − a and let b → a, then we obtain ∞
n2 q n a [a2 ; q]∞ (q; q)6∞ a(1 + a) X n + (a − a−n ) = . 3 n (1 − a) 1−q [a; q]4∞
(18)
n=1
In the proof of (7), we also use the following. Lemma 5. [−q; q 2 ]4∞ = [q; q 2 ]4∞ + q [−1; q 2 ]4∞ .
(19)
(A more familiar form of (19) is the identity θ34 = θ24 + θ44 .) Proof.
See [5] and [8, Section 83.3].
Proof of Theorem 3. We first dispose of (6). Put a = −i, b = −1 in (17), and we obtain !4 ∞ X (q; q)4∞ n n2 (−1) q = (−q; q)4∞ n=0
=1+8
∞ X
(−1)n
n=1
and changing q to −q in (20) yields !4 ∞ ∞ X X n2 q =1+8 n=0
n=1
=1+8
∞ X n=1
nq n , 1 + qn
(20)
nq n 1 + (−q)n ∞
2n X nq n n nq − 8 (−1) , 1 − q 2n 1 − q 2n
(21)
n=1
which is (6). Now divide each side of (18) by 1 + a, and then set a = −1. We obtain !8 ∞ ∞ X X (−1)n n3 q n (q; q)8∞ n n2 = = (−1) q , 1 + 16 1 − qn (−q; q)8∞ −∞ n=1
and, changing q to −q, we have (8). √ Finally, we establish (7). First we take a = i = −1 in (18), obtaining 1+4
∞ X
(−1)n
n=1
(q; q)6∞ (−q; q)2∞ (2n − 1)2 q 2n−1 = 1 − q 2n−1 (−q 2 ; q 2 )∞ = (q 2 ; q 2 )6∞ [−q; q 2 ]2∞ [q; q 2 ]4∞ .
(22)
On the other hand, using the elementary identity ∞
a(1 + a) X 2 n = n a, (1 − a)3 n=1
we can rewrite (18) as ∞ X n2 (an − a−n q n ) n=1
1−
qn
=
a [a2 ; q]∞ (q; q)6∞ . [a; q]4∞
(23)
relating a theta function to a sum of lambert series
31
4
Now change q to q in (23), and replace a by q, to obtain ∞ X q [q 2 ; q 4 ]∞ (q 4 ; q 4 )6∞ n2 q n = 16 16 2n 1+q [q; q 4 ]4∞ n=1
= q (q 2 ; q 2 )6∞ [−q; q 2 ]2∞ [−q 2 ; q 2 ]4∞ .
(24)
Adding (22) and (24), we have ∞ X (2n − 1)2 q 2n−1 n2 q n + 16 1 − q 2n−1 1 + q 2n n=1 n=1 = (q 2 ; q 2 )6∞ [−q; q 2 ]2∞ [q; q 2 ]4∞ + q [−q 2 ; q 2 ]4∞
1+4
∞ X
(−1)n
= [−q; q 2 ]6∞ (q 2 ; q 2 )6∞ !6 ∞ X n2 = q , n=−∞
by (19) and (11). This is (7). References 1. G. E. Andrews, ‘The theory of partitions’, Encyclopedia of mathematics and its applications (AddisonWesley, 1976). 2. G. E. Andrews and R. P. Lewis, ‘An identity of F. H. Jackson and its implications for the theory of partitions’, Preprint, Sussex University (1997). 3. W. N. Bailey, ‘A further note on two of Ramanujan’s formulae’, Quart. J. Math. Oxford (2) 3 (1952) 158–160. 4. B. C. Berndt, Ramanujan’s notebooks III (Springer, New York, 1991). 5. P. R. Hammond, R. P. Lewis and Z-G. Liu, ‘Hirschhorn’s identities’, Bull. Austral. Math. Soc. 60 (1999) 73–80. 6. R. P. Lewis and Z. G. Liu, ‘New proofs of some old results on sums of squares’, Preprint, Sussex University (2000). 7. L. Lorenz, ‘Contribution a` la th´eories des nombres’, Oeuvres scientifiques, Vol. II (ed. H. Valentiner, Librarie Lehmann & Stage, Copenhagen, 1904) 403–431. 8. H. Rademacher, Topics in analytic number theory, Grundlehren Math. Wiss. 169 (Springer, New York, 1973). 9. L. J. Slater, Generalized hypergeometric functions (Cambridge University Press, 1966).
Mathematics Department Pennsylvania State University 228 McAllister Building University Park, PA 16802 USA
Mathematics Department Xinxiang Education College Xinxiang Henan 453000 P. R. China
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