On page 366 of his lost notebook [15], Ramanujan recorded a cubic continued fraction and several theorems analogous to Rogers-Ramanujan's continued fractions. In this paper, we derive several general formulas for explicit evaluations of Ramanujan's cubic continued fraction, several reciprocity theorems, two formulas connecting V (q) and V (q 3) and also establish some explicit evaluations using the values of remarkable product of theta-function.

Let pj,k(n) denote the number of (j, k)-regular overpartitions of n in which none of the parts congruent to j (mod k). In this paper, we obtained infinitely many families of congruences modulo powers of 2 for p3,6(n), p5, 10(n) and p9, 18(n). For example, for all n ≥ 0 and α, β ≥ 0, p9, 18(34α+1 ∙ 52β+1(24 (5n+i)+23)) ≡ 0 (mod 64) where i=0, 1, 2, 4.