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. 39-40, 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) " H-Q) '
(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)]
(2-18)
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(n-1)/2 and Vn :=
a
n(n-i)/26n(n+i)/2
Then [5, p. 48, Entry 31]
n - l
(2-19) /(t/l,Fl) = E^/(l^' %
The Rogers-Ramanujan functions are intimately associated with the Rogers-
Ramanujan continued fraction, defined by
(2-20)
*(«)-T!+
f
+
T
+
T
+
- '
| 9 | 1
'
which first appeared in a paper by Rogers [31] in 1894. Using the Rogers-Ramanujan
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)].
Previous Page Next Page