Number Theory: An Introduction to Mathematics

546 XIII Connections with Number Theory

In the same way that we proved Proposition 3 we may show that, if|x|<1, then

1 /( 1 −x)( 1 −x^2 )···( 1 −xm)= 1 +

∑

n≥ 1

pm(n)xn,

wherepm(n)is the number of partitions ofninto parts not exceedingm.

From the vast number of formulas involving partitions and their generating func-
tions we select only one more pair, the celebratedRogers–Ramanujan identities.The
proof of these identities will be based on the following preliminary result:

Proposition 5If|q|< 1 and|x|<|q|−^1 ,then

1 +

∑

n≥ 1

xnqn

2

/(q)n=

∑

n≥ 0

(− 1 )nx^2 nq^5 n(n+^1 )/^2 −^2 n{ 1 −x^2 q^2 (^2 n+^1 )}/(q)n(xqn+^1 )∞,

where(a) 0 = 1 ,

(a)n=( 1 −a)( 1 −aq)···( 1 −aqn−^1 ) if n≥ 1 , and

(a)∞=( 1 −a)( 1 −aq)( 1 −aq^2 )···.

Proof Consider theq-difference equation

f(x)=f(xq)+xq f(xq^2 ).

A formal power series

∑

n≥ 0 anx

nsatisfies this equation if and only if

an( 1 −qn)=an− 1 q^2 n−^1 (n≥ 1 ).

Thus the only formal power series solution witha 0 =1is

f(x)= 1 +xq/( 1 −q)+x^2 q^4 /( 1 −q)( 1 −q^2 )

+x^3 q^9 /( 1 −q)( 1 −q^2 )( 1 −q^3 )+···.

Moreover, if|q|<1, this power series converges for allx∈C.

If|q|<1, the functions

F(x)=

∑

n≥ 0

(− 1 )nx^2 nq^5 n(n+^1 )/^2 −^2 n{ 1 −x^2 q^2 (^2 n+^1 )}/(q)n(xqn+^1 )∞,

G(x)=

∑

n≥ 0

(− 1 )nx^2 nq^5 n(n+^1 )/^2 −n{ 1 −xq^2 n+^1 }/(q)n(xqn+^1 )∞

are holomorphic for|x|<|q|−^1.