CHAPTER 2

Definitions and Preliminary Results

We first recall Ramanujan's definitions for a general theta function and some

of its important special cases. Set

oo

(2.1) f(a,b):= Yl ara(n+1)/26n("-1)/2, \ab\ 1.

n = — oo

Basic properties satisfied by /(a, b) include [5, p. 34, Entry 18]

(2.2) f(a,b)=f(b,a),

(2.3) / ( l , a ) = 2 / ( a , a 3 ) ,

(2.4) / ( - l , o ) = 0 ,

and, if n is an integer,

(2.5) /(a, b) = a

n n + 1

)/V

( n

~

1 ) / 2

/(a(ab)

n

,

b{ab)~n).

The basic property (2.2) will be used many times in the sequel without comment.

The function /(a, 6) satisfies the well-known Jacobi triple product identity [5, p. 35,

Entry 19]

(2.6) /(a, b) = (-a; a&)oo(-&; ab)oo(ab] ab)^.

The three most important special cases of (2.1) are

oo

(2.7) p(q) := f(q, q) = £

qn*

= (-q;

q2)Uq2; q2)^

n— — oo

(2.8) m := /(,, ,3) = £ 9"(n+1)/2 = ^ T

n=0

and

(?;72)oo

'

(2.9) / ( - g ) := /(-*, -g

2

) = £ ( - l ) V

( 3 n

"

1 ) / 2

^ (9; ?)oo =:

T

1/24

T7(T),

n = —oo

where g = exp(27rir), Im r 0, and 7 7 denotes the Dedekind eta-function. The

product representations in (2.7)-(2.9) are special cases of (2.6). Also, after Ra-

manujan, define

(2.10) X(Z) :=("?; Z2)oo.

Using (2.6) and (2.9), we can rewrite the Rogers-Ramanujan identities (1.2) in

the forms

(2-11) G(q) = ^ f l and *(,) = 4 = * ^ .