Elementary Approach to Modular Equations: Ramanujan's Theory 6

The Fundamental Formulas

In this post we will continue our journey of modular equations and derive a host of these mostly by using Lambert series for various theta functions. The following formula (see equation $ (14)$ of this post) will be of great help here: $$\phi^{2}(-ab)\,\frac{f(a, b)}{f(-a, -b)} = 1 + 2\sum_{n = 1}^{\infty}\frac{a^{n} + b^{n}}{1 + a^{n}b^{n}}\tag{1}$$

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Elementary Approach to Modular Equations: Ramanujan's Theory 5

Quintuple Product Identity

We first establish an identity similar to Jacobi's Triple Product which involves five factors and is quite useful in establishing various other identities involving q-series and products. This was first introduced in the mathematical literature by G. N. Watson in order to prove some of Ramanujan's theorems. The quintuple product identity is given by \begin{align}&\prod_{n = 1}^{\infty}(1 - q^{n})(1 - q^{n}z)(1 - q^{n - 1}z^{-1})(1 - q^{2n - 1}z^{2})(1 - q^{2n - 1}z^{-2})\notag\\ &\,\,\,\,\,\,\,\,= \sum_{n = -\infty}^{\infty}q^{n(3n + 1)/2}(z^{3n} - z^{-3n - 1})\notag\end{align}

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Elementary Approach to Modular Equations: Ramanujan's Theory 4

Lambert Series

In this post we will focus our attention on series of the form: $$\sum_{n = 0}^{\infty}a_{n}\cdot\frac{q^{b_n}}{1 \pm q^{c_n}}$$ which are more popularly known as Lambert Series. We will not deal with the general theorems concerning such series but will restrict ourselves to the Lambert series for the theta functions and study some identities involving these series.

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