DOI: 10.1017/s0004972726101695 ISSN: 0004-9727
ON AN IDENTITY ASSOCIATED WITH JACOBI’S THETA FUNCTIONS
HENG HUAT CHAN, SONG HENG CHAN Abstract
Recently, Yang [‘On modular equations for the Jacobi theta functions’, Preprint] discovered an identity that involves the sum of squares of the theta functions
ϑ
j
(
0
|
τ
)
ϑ
j
(
0
|
N
τ
)
$\displaystyle \vartheta _j(0|\tau )\vartheta _j(0|N\tau )$
theta Subscript j Baseline left parenthesis 0 vertical bar tau right parenthesis theta Subscript j Baseline left parenthesis 0 vertical bar upper N tau right parenthesis
for
j
=
2
,
3
,
4.
$j=2,3,4.$
j equals 2 comma 3 comma 4 period
In this article, we prove this identity and its generalisations. Our study also leads us to a four-variable extension of an identity discovered by Chan, Chua and Solé [‘Quadratic iterations to
π
$\pi $
pi
associated to elliptic functions to the cubic and septic base’,
Trans. Amer. Math. Soc.
355
(4) (2002), 1505–1520].