Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 304

q 1 / 2 (substituted into the sum in f ) yields Similarly, the value u = −

2 ∞

1
2
(1 − q n−

q n−

−

2 ) 2 .

(9.33)

1

n =1

Since

1
2

1
2

q n− q n− 4 q 2 n− 1

2 ) 2 −

=

q 2 n− 1 ) 2 ,

1 1

2 ) 2

(1 −

(1 − q n− (1 + q n−

the sum of (9.31), (9.32), (9.33) is

(1 − q 2 n− 1 ) 2 .

8 ∞

q n + q 2 n + q 3 n
(1 − q 2 n ) 2

q 2 n− 1

− + (9.34)

n =1

Differentiating the series for 1 / (1 − X ) yields the identity

= ∞

1

mX m− 1 .

(1 − X ) 2

m =1

Substituting X = q 2 n− 1 , multiplying by q 2 n− 1 , and summing over n yields

∞

= ∞

∞

q 2 n− 1
(1 − q 2 n− 1 ) 2

mq (2 n− 1) m

(9.35)

n =1 m =0 n =1

⎛ ⎞

= ∞

⎝ ⎠ q N .

d (9.36)

N =1

d | N, N/d odd

Similarly, we obtain

∞

= ∞

∞

q n

mq n (2 m− 1)

(1 − q 2 n ) 2

n =1 m =0 n =1

⎛ ⎞

(9.37)

= ∞

d +1
2

⎝ ⎠ q N

N =1 d | N, d odd

and

∞

= ∞

∞

q 3 n
(1 − q 2 n ) 2

mq n (2 m +1)

n =1 m =0 n =1

⎛ ⎞

(9.38)

= ∞

d − 1
2

⎝ ⎠ q N .

N =1

d | N, d odd

Also, the method yields

⎛ ⎞

∞

= ∞

q 2 n

⎝ ⎠ q N .

d (9.39)

(1 − q 2 n ) 2

n =1 N =1

d | N, N/d even