BERNDT, CHOI, CHOI, HAHN, YEAP, YEE, YESILYURT, YI
Entry 3.4 is the second identity offered by Ramamijan without proof in [26],
[27, p. 231]. The first published proof was given by Rogers [32]. Watson [34] also
gave a proof. R. Blecksmith, J. Brillhart, and I. Gerst [12] have shown that (3.5)
follows from a very general theta function identity established by the authors.
Proofs of the next seven entries were first given by Rogers [32]. N. D. Baruah,
J. Bora, and N. Saikia [4] and Baruah and Bora [3] have also found proofs of Entry
3.6.
G(q16)H(q)-q3G(q)H(q16)
=
X(q2).
f2(~Q3)
ENTRY 3.5.
(3.6)
ENTRY 3.6.
(3.7)
ENTRY 3.7.
(3.8)
ENTRY 3.8.
(3.9)
ENTRY 3.9.
(3.10)
ENTRY 3.10.
(3.11)
ENTRY 3.11.
(3.12)
ENTRY 3.12.
G(q)G(q») + q2H(q)H(q») =
f(~q)f(-q9Y
G{q2)G{qi)+qH{q')H{qi) =
3^ _ X(~q3)
x(-q)'
G(q6)H(q) - qG(q)H(q6) = % ( q)
G(q7)H(q2) - qG(q2)H(q7)
x(-q3)'
x(-q)
x(-q7Y
G(q)G(qu)+q3H(q)H(qu) = *£-£.
x(-q)x(-q4)
G(q8)H(q3) - qG(q3)H(qs)
x(-q3)x(-q12)'
(3.13)
ENTRY 3.13.
(3.14)
ENTRY 3.14.
(3.15)
G(q)G(q2i)+q5H(q)H(q^)
=
G(q9)H(q4) - qG(q4)H(q9) =
,2- _
X(~q3)x(~q12)
x{-q)x{-qA) '
x(-q)x(-q6)
x(~q3)x(-q18)'
G(q3e)H(q)-q7G(q)H(q^ = ^ ^ .
Entries 3.12-3.14 were first proved by Bressoud in his doctoral dissertation [14].
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