FORTY IDENTITIES
and
(3.36) U4V4 - qU2V2 = )^{~q") (l + q2 - £ ^
x'y-i) V x
3
( - r
7
) ,
Bressoud proved (3.35) in his thesis [14]. Biagioli claimed in [10] that he was
going to prove (3.36), but a proof of (3.36) does not appear in his paper.
ENTRY 3.29.
{G(q2)G(q23)
+
q5H(q2)H(q23)} {G(qi6)H(q)
-
q9G(q)H(q4i)}
(3.37)
=
x(-q)x(-q™)
+ q +
2q2
(3.38)
E
(3.39)
E
(3.40)
x(-g)x(-g23)'
ENTRY 3.30.
G(q™)H(q4)
-
q3G(q4)H(q™) X(-Q2)
G(q™)H(-q) + q™G(-q)H(q™)
X
(-q 3 8 ) '
ENTRY 3.31.
G(q2)G(q33)
+
g7H(q2)H(q33)
=
X(~93)
G{q**)H(q) - q*H(q")G{q)
X
( - ?
U
) '
ENTRY 3.32.
G(q3)G(q22) + q5H(q3)H(q22)
X
(-
9
3 3 )
G(q")H(q«) - qG(q«)H(q")
X
(-g ) '
Using the theory of modular forms, Biagioli [10] constructed proofs of Entries
3.29-3.32. No other proofs are known.
ENTRY 3.33.
(

4 n
G(q)G(q54) + qnH(q)H(q5i)
= X
(-?3)x(-?27)
1
'
G(q2?)H(q2)
-
q^G(q2)H(q^)
x(-q)x(-q9) '
ENTRY 3.34.
{G(q)G(-qw)
-
q4H(q)H(-qw)} {G(-q)G(q19)
-
q*H(-q)H(q19)}
(3.42) = G(q2)G(q38) + q8H(q2)H(q38).
Proofs of (3.41) and (3.42) have been found by Bressoud [14], who corrected a
misprint in Watson's [29] formulation of (3.42).
ENTRY 3.35.
{G(q)G(q94) + qwH(q)H(q94)} {G(q47)H(q2) - qsG(q2)H(q47)}
=
X(-?)x(-?47)
+
2?2+
^
X(-q)x(-q47)
(3.43)
+ 9
y 4
X
( -
g ) x (
- 0
+
V
+ x(
_^
(
_
g47)
.
The only known proof [10] of Entry 3.35 employs the theory of modular forms.
Observe that in most of the forty identities, G(q) and H(q) occur in the com-
binations,
(3.44) G(qr)G(qs) + q^s)/5H(qr)H(qs), when r + s = 0 (mod 5),
(3.45)
G{qr)H(qs)
-
q(r-s)/5H(qr)G{qs),
when r - s = 0(mod5),
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