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 = * ^ .
Previous Page Next Page