10 BERNDT, CHOI, CHOI, HAHN, YEAP, YEE, YESILYURT, YI

ENTRY 3.22.

(3.26) G(-g)G(-g 4 ) + qH(-q)H(-q4) =

X

(g2).

ENTRY 3.23.

(3.27)

G(-q2)G(-q3)

+

qH(-q2)H(~q3)

=

x(r)x(r)vf}{^X

ENTRY 3.24.

(3.28) G(-q«)H(-q) - qH{-q*)G{-q) = ^ ^

Bressoud [14] established the three previous entries.

ENTRY 3.25.

(3.29) G(-q)G(q9) - q2H(~q)H(q9) = X{~^«*}.

Equality (3.29) is a corrected version of that given by Watson [29] and was

first proved by Bressoud [14].

ENTRY 3.26.

(3.30)

G(qu)H(-q)

+

q2G(-q)H(qn)

X(q2)x(q22) 2.J3

x(-q2)x(-Q22) x(-q2)x(-q4M-q22M-q44)'

Watson [34] established (3.30). The minus sign in front of the second expression

on the right side of (3.30) is missing in Watson's list [29].

Our formulations of Entries 3.27 and 3.28 are slightly different from those of

Ramanujan, who had reversed the hypotheses in each entry. In other words, he

intended that the formulas for U and V be the conclusions in each case, with the

pairs of equations, (3.33), (3.34) and (3.35), (3.36) being the conditions under which

the formulas for U and V should hold. Watson proved Entry 3.27 under the same

interpretation as we have given.

ENTRY 3.27. Define

(3.31) U := U(q) :=

G(q)G(q44)

+

q9H(q)H(q44)

and

(3.32) V := V(q) := G(q4)G(q11) + q3H{q4)H{q11).

Then

(3.33)

U2

+

qV2

=

X

3(q)x3(q11)

(3.34) UV + q =

X

2(q)x2(qU)-

ENTRY 3.28. Define

U~G(q17)H(q2)-q3G(q2)H(q17)

and V :=

G(q)G(q34)

+

q7H(q)H(q34).

Then

/, o ^ U X(-g)

(3'35)

v = xJ^)