FORTY IDENTITIES
9
ENTRY 3.15.
(3.16) G(q3)G(q7) + q2H(q3)H(q7) = G(q21)H(q)  q*G{q)H(q21)
(3.17)  ^x(V /2 )x(g 3/2 )x(7 7/2 )x(z 21/2 ).
The only known proofs of (3.16) and (3.17) are by Biagioli [10], who used the
theory of modular forms.
ENTRY 3.16.
(3.18)
G(q2)G(q13)
+
q3H(q2)H(q13)
=
G(q26)H(q)

q"G(q)H(q26)
(3.19)
=J^^qX{~q)13)'X(Qx(q)
The only known proof of (3.18) is by Bressoud [14], while Biagioli, using the
theory of modular forms, has established the only known proof of (3.19). Biagioli's
[10] formulation of (3.19) contains two misprints; the formula is also misnumbered
as #17 instead of #18.
Proofs of the next four identities, (3.20)(3.23), have been given by Bressoud
[14].
ENTRY 3.17.
G(q)G(q19)
+
q4H(q)H(q™)
= J  ^ / V ( ?
1 9 / 2
) 
~X2{q1,2)X2{qW'2)
q2
(3.20) H
X2(q)x2(q19Y
ENTRY 3.18.
G(q31)H(q)

q6G(q)H(q31)
= ^x(?)x(?
3 1
) 
^*(/)x(?31)
q3
(3"21) + x(q2)x(q62Y
ENTRY 3.19.
{G(q)G(q39)
+
q*H(q)H(q39)}
f(q)f(^)
(3.22) = {G(ql3)H(q3)  q2G(q3)H(q13)} f(q3)f(q™)
(3.23) = 1 (p(q3Mq13) ~ ¥(zMZ39)) •
ENTRY 3.20.
2 2rP(q2)
(3.24) G(q)H(q) + G(q)H(q) = ^{_q2) 
/ (
_ ^
2 )
.
ENTRY 3.21.
(3.25) G(q)H(q)  G(q)H(q) = J ^  
Watson [34] constructed proofs of both (3.24) and (3.25).