Ramanujan’s Continued Fractions of Order Six and Twelve with Their Interesting Properties

Abstract

In the present paper, we derive three continued fractions: R(q), which is of the sixth order, and S(q) and T(q), both of the twelfth order. Each of these arises as a particular case of the general continued fraction mentioned by Ramanujan in his notebook. For these continued fractions, we also establish several theta function identities. In addition, we derive matching coefficients for the continued fraction S(q), and we obtain color partition identities as a result of the theta function identities of S(q) and T(q). Finally, we present 2-, 4-dissections of the continued fraction S∗(q) = q-1S(q).