6 BERNDT, CHOI, CHOI, HAHN, YEAP, YEE, YESILYURT, YI
We shall use (2.11) many times in the remainder of the paper. A useful consequence
of (2.11) in conjunction with the Jacobi triple product identity (2.6) is
(2.12) G{q)H{q) = l t & .
Basic properties of the functions (2.7)(2.10) include [5, pp. 3940, Entries 24,
25(iii)]
(2.13)
f(q) _ ip{q) _ x(v) / v(q)
f(q) ip{q) x{q) \
P(Q)'
(2.17) V(9) =
x(q)f(q4)
= rbHff' x(q)f(~q) =
p(~q2)
{
'
X(q

f(12)
" V ^(9) " f(q) " HQ) '
(2.15) f{q2) = P(Q)^2(q), X(q)x(~q) = x(q2),
(2.16) p(q)p(~q) = ^(/ 2 )
It is easy to deduce from (2.14) or (2.6) that
We shall use the famous quintuple product identity, which, in Ramanujan's
notation, takes the form (2.1) [5, p. 80, Entry 28(iv)]
(218)
f^a2,~a~y,
=
TT1^
{f(~a\ a"V) + af(a~\
a3*2)}
,
f(a,a
1q)
f(q)
where a is any complex number.
The function /(a, b) also satisfies a useful addition formula. For each positive
integer n, let
Un := an(n+1/26n(n1)/2 and Vn :=
a
n(ni)/26n(n+i)/2
Then [5, p. 48, Entry 31]
n  l
(219) /(t/l,Fl) = E^/(l^' %
The RogersRamanujan functions are intimately associated with the Rogers
Ramanujan continued fraction, defined by
(220)
*(«)T!+
f
+
T
+
T
+
 '
 9  1
'
which first appeared in a paper by Rogers [31] in 1894. Using the RogersRamanujan
identities (1.2), Rogers proved that
(2"21) R{q)q W)~q
(MUWU
This was independently discovered by Ramanujan and can be found in his notebooks
[28], [5, p. 79, Chap. 16, Entry 38(iii)].