We use the theory of Jacobi-like forms to construct modular forms for a congruence subgroup of $\text{SL}\left( 2,\,\mathbb{R} \right)$ which can be expressed as linear combinations of products of certain theta functions.

In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R. Russell in 1887. We give a proof of Russell’s main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. Motivated by Russell’s theorem, we state and prove its cubic analogue which allows us to construct Russell-type modular equations in the theory of signature 3.

For the $q$-series $\sum\nolimits_{n=0}^{\infty }{{{a}^{n}}{{q}^{b{{n}^{2}}+cn}}/}\,{{(q)}_{n}}$ we construct a companion $q$-series such that the asymptotic expansions of their logarithms as $q\,\to \,{{1}^{-}}$ differ only in the dominant few terms. The asymptotic expansion of their quotient then has a simple closed form; this gives rise to a new $q$–hypergeometric identity. We give an asymptotic expansion of a general class of $q$-series containing some of Ramanujan's mock theta functions and Selberg's identities.

In his first and lost notebooks, Ramanujan recorded several values for the Rogers-Ramanujan continued fraction. Some of these results have been proved by K. G. Ramanathan, using mostly ideas with which Ramanujan was unfamiliar. In this paper, eight of Ramanujan's values are established; four are proved for the first time, while the remaining four had been previously proved by Ramanathan by entirely different methods. Our proofs employ some of Ramanujan's beautiful eta-function identities, which have not been heretofore used for evaluating continued fractions.

There are three modular forms a(q), b(q), c(q) involved in the parametrization of the hypergeometric function analogous to the classical θ2(q), θ3(q), θ4(q) and the hypergeometric function We give elliptic function generalizations of a(q), b(q), c(q) analogous to the classical theta-function θ(z, q). A number of identities are proved. The proofs are self-contained, relying on nothing more than the Jacobi triple product identity