Number Theory: An Introduction to Mathematics

518 XII Elliptic Functions

whereq,z∈Candz=0. Both series on the right converge if|q|<1, both diverge
if|q|>1, and at most one converges if|q|=1. Thus we now assume|q|<1.
A remarkable representation for the series on the left was given by Jacobi (1829),
in§64 of hisFundamenta Nova, and is now generally known asJacobi’s triple product
formula:

Proposition 2If|q|< 1 and z= 0 ,then

∑∞

n=−∞

qn

2

zn=

∏∞

n= 1

( 1 +q^2 n−^1 z)( 1 +q^2 n−^1 z−^1 )( 1 −q^2 n). (30)

Proof Put

fN(z)=

∏N

n= 1

( 1 +q^2 n−^1 z)( 1 +q^2 n−^1 z−^1 ).

Then we can write

fN(z)=c 0 N+c 1 N(z+z−^1 )+···+cNN(zN+z−N). (31)

To determine the coefficientscnNwe use the functional relation

fN(q^2 z)=( 1 +q^2 N+^1 z)( 1 +q−^1 z−^1 )fN(z)/( 1 +qz)( 1 +q^2 N−^1 z−^1 )

=( 1 +q^2 N+^1 z)fN(z)/(qz+q^2 N).

Multiplying both sides byqz+q^2 N and equating coefficients ofzn+^1 we get,
forn= 0 , 1 ,...,N−1,

q^2 n+^1 cnN+q^2 N+^2 n+^2 cNn+ 1 =cNn+ 1 +q^2 N+^1 cNn,

i.e.,

q^2 n+^1 ( 1 −q^2 N−^2 n)cnN=( 1 −q^2 N+^2 n+^2 )cnN+ 1.

But, since

∑N

n= 1 (^2 n−^1 )=N

(^2) , it follows from the definition offN(z)thatcN
N=q

N^2.

Hence, for 0≤n≤N,

cnN=( 1 −q^2 N+^2 n+^2 )( 1 −q^2 N+^2 n+^4 )···( 1 −q^4 N)qn

2

/D,

whereD=( 1 −q^2 )( 1 −q^4 )···( 1 −q^2 N−^2 n).
If|q|<1andz=0, then the infinite products

∏∞

n= 1

( 1 +q^2 n−^1 z),

∏∞

n= 1

( 1 +q^2 n−^1 z−^1 ),

∏∞

n= 1

( 1 −q^2 n)

are all convergent. From the convergence of the last it follows that, for each fixedn,

lim

N→∞

cnN=qn

2

/∏∞

k= 1

( 1 −q^2 k).