Some results on Ramanujan's continued fractions of order ten and applications

We derive two continued fractions I (q) and J (q) of order ten from a general continued fraction identity recorded by Ramanujan in his notebook. We prove theta-function identities for the continued fractions I (q) and J (q). Using Ramanujan's parameter k(q) = R(q) R-2(q(2)), where R(q) is the Rogers-Ramanujan continued fraction, together with a new parameter u(q), we prove general theorems for the explicit evaluations of I (+/- q) and J (+/- q) and give examples. As applications of some of the identities of I (q) and J (q), we derive some partition identities using colour partition of integers. We establish 2-dissections for the continued fraction I* (q) := q(-3/4) I (q) and J* (q) = q(-1/4) J(q) and their reciprocals.