2 Partitions 547We h av e
F(x)−G(x)
=∑
n≥ 0(− 1 )nx^2 nq^5 n(n+^1 )/^2 {q−^2 n−x^2 q^2 (n+^1 )−q−n+xqn+^1 }/(q)n(xqn+^1 )∞=
∑
n≥ 0(− 1 )nx^2 nq^5 n(n+^1 )/^2 {q−^2 n( 1 −qn)+xqn+^1 ( 1 −xqn+^1 )}/(q)n(xqn+^1 )∞=
∑
n≥ 1(− 1 )nx^2 nq^5 n(n+^1 )/^2 −^2 n/(q)n− 1 (xqn+^1 )∞+xq∑
n≥ 0(− 1 )nx^2 nq^5 n(n+^1 )/^2 +n/(q)n(xqn+^2 )∞=−x^2 q^3∑
n≥ 0(− 1 )nx^2 nq^5 n(n+^1 )/^2 +^3 n/(q)n(xqn+^2 )∞+xq∑
n≥ 0(− 1 )nx^2 nq^5 n(n+^1 )/^2 +n/(q)n(xqn+^2 )∞=xq∑
n≥ 0(− 1 )n(xq)^2 nq^5 n(n+^1 )/^2 −n{ 1 −(xq)q^2 n+^1 }/(q)n(xqn+^2 )∞=xqG(xq).Similarly,
G(x)=∑
n≥ 0(− 1 )nx^2 nq^5 n(n+^1 )/^2 {q−n−xqn+^1 }/(q)n(xqn+^1 )∞=
∑
n≥ 0(− 1 )nx^2 nq^5 n(n+^1 )/^2 {q−n( 1 −qn)+ 1 −xqn+^1 }/(q)n(xqn+^1 )∞=
∑
n≥ 1(− 1 )nx^2 nq^5 n(n+^1 )/^2 −n/(q)n− 1 (xqn+^1 )∞+
∑
n≥ 0(− 1 )nx^2 nq^5 n(n+^1 )/^2 /(q)n(xqn+^2 )∞=
∑
n≥ 0(− 1 )n(xq)^2 nq^5 n(n+^1 )/^2 −^2 n{ 1 −(xq)^2 q^2 (^2 n+^1 )}/(q)n(xqn+^2 )∞=F(xq).Combining this with the previous relation, we obtain
F(x)=F(xq)+xq F(xq^2 ).But we have seen that thisq-difference equation has a unique holomorphic solution
f(x)such thatf( 0 )=1. HenceF(x)=f(x).