Certain Lambert Series Identities and their Proof via Trigonometry: Part 2

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In the last post we saw that using a trigonometric identity Ramanujan was able to express the functions $S_{2n - 1}(x)$ or equivalently $\Phi_{0, 2n - 1}(x)$ in terms of simpler functions $P(x), Q(x), R(x)$. Continuing our journey further we start with the equation: \begin{align}&\left(\frac{1}{4}\cot\frac{\theta}{2} + \frac{x\sin\theta}{1 - x} + \frac{x^{2}\sin 2\theta}{1 - x^{2}} + \frac{x^{3}\sin 3\theta}{1 - x^{3}} + \cdots\right)^{2}\notag\\ &\,\,\,\,= \left(\frac{1}{4}\cot\frac{\theta}{2}\right)^{2} + \frac{x\cos\theta}{(1 - x)^{2}} + \frac{x^{2}\cos 2\theta}{(1 - x^{2})^{2}} + \frac{x^{3}\cos 3\theta}{(1 - x^{3})^{2}} + \cdots\notag\\ &\,\,\,\,+\frac{1}{2}\left\{\frac{x(1 - \cos\theta)}{1 - x} + \frac{2x^{2}(1 - \cos 2\theta)}{1 - x^{2}} + \frac{3x^{3}(1 - \cos 3\theta)}{1 - x^{3}} + \cdots\right\}\tag{1}\end{align} which was established in the last post.

Certain Lambert Series Identities and their Proof via Trigonometry: Part 1

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Introduction

This is yet another post based on a paper of Ramanujan titled "On certain arithmetical functions" which appeared in Transactions of the Cambridge Philosophical Society in 1916. In this paper Ramanujan provided a lot of identities concerning Lambert series and thereby deduced many relations between various divisor functions. Apart from the amazing results proved in this paper, what I liked most is the very elementary approach followed by Ramanujan compared to the methods of modern authors who are seduced by the modular form.

Rogers-Ramanujan Identities: A Proof by Ramanujan

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Introduction

The history of Rogers-Ramanujan identities is well described in various books and papers. In brief these identities were first discovered and proved by L. J. Rogers in 1894 and then later re-discovered (but not proved) by Ramanujan in 1913. Later in 1919 Ramanujan published a proof. It is this proof which will be described here.

Ramanujan was not used to publishing proofs of many of his discoveries and hence there is a feeling (even now) that his methods were mystical and often inspired by his dreams. However he did publish some proofs and when we study these it becomes at once very clear that Ramanujan possessed proofs of almost all the results he found, but it was just lack of time and resources due to which he did not record the proofs. Unfortunately this is a big loss for mathematics because from the nature of his formulas it seems that his methods were highly efficient and startling at the same time. The proof we present in this post also has the same qualities and I hope the reader will enjoy going through these.