Number Theory: An Introduction to Mathematics

548 XIII Connections with Number Theory

The Rogers–Ramanujan identities may now be easily derived:

Proposition 6If|q|< 1 ,then

∑

n≥ 0

qn

2

/( 1 −q)( 1 −q^2 )···( 1 −qn)=

∏

m≥ 0

( 1 −q^5 m+^1 )−^1 ( 1 −q^5 m+^4 )−^1 ,

∑

n≥ 0

qn(n+^1 )/( 1 −q)( 1 −q^2 )···( 1 −qn)=

∏

m≥ 0

( 1 −q^5 m+^2 )−^1 ( 1 −q^5 m+^3 )−^1.

Proof PutP=

∏

k≥ 1 (^1 −q

k). By Proposition 5 and its proof we have

∑

n≥ 0

qn

2

/( 1 −q)( 1 −q^2 )···( 1 −qn)=F( 1 )

=

[

1 +

∑

n≥ 1

(− 1 )n{qn(^5 n+^1 )/^2 +qn(^5 n−^1 )/^2 }

]/

P

and, sinceF(q)=G( 1 ),
∑

n≥ 0

qn(n+^1 )/( 1 −q)( 1 −q^2 )···( 1 −qn)=F(q)

=

[

1 +

∑

n≥ 1

(− 1 )n{qn(^5 n+^3 )/^2 +qn(^5 n−^3 )/^2 }

]/

P.

On the other hand, by replacingqbyq^5 /^2 andzby−q^1 /^2 ,resp.−q^3 /^2 , in Jacobi’s
triple product formula (Proposition XII.2), we obtain
∑

n∈Z

(− 1 )nqn(^5 n+^1 )/^2 =

∏

m≥ 1

( 1 −q^5 m)( 1 −q^5 m−^2 )( 1 −q^5 m−^3 )

=P

/∏

m≥ 0

( 1 −q^5 m+^1 )( 1 −q^5 m+^4 )

and
∑

n∈Z

(− 1 )nqn(^5 n+^3 )/^2 =

∏

m≥ 1

( 1 −q^5 m)( 1 −q^5 m−^1 )( 1 −q^5 m−^4 )

=P

/∏

m≥ 0

( 1 −q^5 m+^2 )( 1 −q^5 m+^3 ).

Combining these relations with the previous ones, we obtain the result.

The combinatorial interpretation of the Rogers–Ramanujan identities was pointed
out by MacMahon (1916). The first identity says that the number of partitions of a
positive integerninto parts congruent to±1 mod 5 is equal to the number of partitions
ofninto parts that differ by at least 2. The second identity says that the number of par-
titions of a positive integerninto parts congruent to±2 mod 5 is equal to the number
of partitions ofninto parts greater than 1 that differ by at least 2.