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^)
Previous Page Next Page