Let
denote the number of representations
of
as
,
where
are positive
integers,
,
,
,
,
are
arbitrary nonnegative integers,
and let
denote the number of representations
of
as
,
where this time
are integers.
In a recent paper, Sun not only discovered
many relations between
and
, but also posed a number of
conjectures
on the relations between
and
.
In this paper, we confirm
some of
Sun's conjectures by
using Ramanujan's
theta function identities
and
-parametrization
of theta functions.