On Ramanujan's Continued Fractions of Orders Five, Ten, and Twenty and Associated Lambert Series Identities

Abstract

In this work, we establish several new identities connecting Ramanujan's continued fractions of order twenty. By employing product representation for Jacobi's theta function ${\theta }_1$, we derive a family of new relations connecting the continued fractions of order twenty with continued fractions of order ten and Rogers-Ramanujan continued fraction. Further, utilizing certain mock theta functions and their logarithmic derivatives, we obtain beautiful relations between Lambert series and theta functions of level twenty. Using Ramanujan's ${}_1{\psi }_1$ summation formula, we establish Lambert series identities associated with the continued fractions of order twenty. These results extend earlier work on continued fractions of order 6, 12, and 16 and contribute to theory of $q$-series.